Some Musings on Theta, Eta, and Zeta: From E8 to Cold Plasma to an lnhomogeneous Universe [1 ed.] 9789819953356, 9789819953363

Table of contents :
Preface
Contents
1 A Theta Function Attached to a Positive Definite Matrix
2 Jacobi Type Inversion Formulas
3 A Theorem of Minkowski: Enter E8
4 Modular Properties of Theta and Eta
5 An Epstein Zeta Function Attached to A
6 An Inhomogeneous Epstein Zeta Function
7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff
8 The Modular j-Invariant and Powers of Its Cube Root: Enter E8 Again
9 Modular Forms of Non-positive Weight: Exact Formulas and Asymptotics of Their Fourier Coefficients
10 More on Logarithmic Corrections to Black Hole Entropy
11 A Dedekind Type Eta Function Attached to the Hecke Group Γ0(N)
12 Elementary Particles, the E8 Root Lattice, and a Patterson–Selberg Zeta Function
13 The Uncontroversial Mathematics Behind Garrett Lisi's Controversial ``Theory of Everything''
13.1 Introduction to Lie Groups and Lie Algebras
13.2 Grand Unified Theories and the Standard Model
13.3 A Theory of Everything
13.4 Conclusion
14 The Elliptic Functions s n(x, κ), c n(x, κ) , and d n(x, κ) of C. Jacobi
15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma, and the J-T Black Hole
16 The Weierstrass mathcalP-Function and Some KdV Solutions
17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections
18 A Finite Temperature Zeta Function
19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look at the BTZ Black Hole
20 A Cold Plasma-Sine-Gordon Connection
21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space: Computation of the One-Loop Effective Potential
Appendix References
References

Citation preview

Mathematical Physics Studies

Floyd L. Williams

Some Musings on Theta, Eta, and Zeta From E8 to Cold Plasma to an lnhomogeneous Universe

Mathematical Physics Studies Series Editors Giuseppe Dito, Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France Edward Frenkel, Department of Mathematics, University of California at Berkley, Berkeley, CA, USA Sergei Gukov, California Institute of Technology, Pasadena, CA, USA Yasuyuki Kawahigashi, Department of Mathematical Sciences, The University of Tokyo, Tokyo, Japan Maxim Kontsevich, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France Nicolaas P. Landsman, Chair of Mathematical Physics, Radboud Universiteit Nijmegen, Nijmegen, Gelderland, The Netherlands Bruno Nachtergaele, Department of Mathematics, University of California, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan

The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathematics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geometry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory.

Floyd L. Williams

Some Musings on Theta, Eta, and Zeta From E 8 to Cold Plasma to an Inhomogeneous Universe

Floyd L. Williams Department of Mathematics and Statistics University of Massachusetts Amherst Amherst, MA, USA

ISSN 0921-3767 ISSN 2352-3905 (electronic) Mathematical Physics Studies ISBN 978-981-99-5335-6 ISBN 978-981-99-5336-3 (eBook) https://doi.org/10.1007/978-981-99-5336-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

With an overflow of ceaseless love and esteem this work is dedicated to my mother, Mrs. Lee Edna Rollins, whose 96 years of Godly wisdom continues to guide me so very well.

Preface

The applications of theta, eta, and zeta functions, and of modular forms and elliptic functions have increasingly become a part of the landscape of various facets of mathematical physics. The modular invariance (under the action of SL(2, Z)) of partition functions in string theory is indispensable for consistency, finiteness, and the cancelation of spacetime anomalies results. The modular invariant (one-loop) heterotic string partition function for a Z2 orbifold compactification is expressed in terms of theta functions, the Dedekind eta function, and the E 8 root lattice; this lattice will be discussed in Chap. 12. E 8 , which Prof. Bertram Kostant calls the most magnificent object in all of mathematics, enters the picture here because modular invariance restricts the gauge group to only two possibilities: E 8 × E 8 or SO(32)— groups much larger than the gauge group U(1) × SU(2) × SU(3) of the standard model. Formulas that relate theta functions, more specifically their derivatives, and modular forms (like G2 (z), G4 (z), G6 (z), that are discussed in Chaps. 4 and 17) turn out to be useful for the computation of traces in string theory. The partition function in statistical mechanics (which is a sum over states) defined on a torus, by way of periodic boundary conditions imposed, is also modular invariant, by way of conformal invariance at a second-order phase transition. General conformal field theory (CFT) partition functions (again where SL(2, Z)—invariance is of critical importance) are given in terms of V ir × V ir characters of irreducible representations ¯ c) L(h, c), L(h, ¯ with highest weight h and central charge c, and with a multiplicity ¯ that counts the number of primary fields with conformal weight function m(h, h) ¯ (h, h)—weights that label highest weight states; Vir means Virasoro. Of particular interest/importance are the CFT’s called minimal models where the partition function is a finite sum of such characters, and where the central chargec = cp,q is parametrized by coprime integers p, q  2, given by cp,q = 1 − 6( p − q)2 pq. These include the Ising model, which is useful for understanding phase transition. Specifically (p, q) = (4, 3), (5, 4), (6, 5), for example, gives the critical Ising model, tricritical Ising model, 7 4 , 5, tetracritical Ising model (or 3-states Potts model) with central charges c = 21 , 10 respectively. Also, the number of irreducible representations is ( p − 1)(q − 1), and the character formulas involve the Dedekind eta function. By a concrete computation of the partition function of the critical Ising model, for example, in terms of theta vii

viii

Preface

functions and the Dedekind eta function, one sees immediately that it is indeed modular invariant. The holomorphic sector of the one-loop gravity partition function can also be expressed in terms of the Dedekind eta function, as we show in Chap. 10, where logarithmic corrections to the holomorphic sector BTZ black entropy are computed. The one-loop graviton and gravitino partition functions have a beautiful expression in terms of a Patterson-Selberg zeta function, as shown in Chap. 12, where these and other elementary particles are discussed and where, moreover, a Patterson– Selberg (P–S) zeta function formula for higher spin fermionic particles is presented. In the case of spin 3/2, this formula reduces to the aforementioned gravitino partition function for N = 1 supergravity. By constructing a suitable deformation of the P–S zeta function such that its logarithm at a special value coincides with the non-divergent term in the effective action of the BTZ black hole with a conical singularity, the author was able to express the one-loop ultraviolet quantum correction to Bekenstein–Hawking entropy in terms of a family of zeta functions. The discussion/ approach in Chap. 10, however, is by way of the asymptotics of the Fourier coefficients bk,n , as n → ∞, of the (modular invariant) partition function of a holomorphic conformal field theory with central charge c = 24k(k = 1, 2, 3, . . .). These coefficients, remarkably, are positive integers that count the number of states with Virasoro energy equal to n. Ed Witten considered the behavior of bn,k for large n with n/k fixed. The asymptotic formula (10.1) that we prove, however, works simply for any fixed k. Powers of the cube root of the Dedekind–Klein modular j-invariant also occur as partition functions of a holomorphic conformal field theory, more generally, with central charge c divisible by 8—the above case c = 24k being an important example that includes the famous Frenkel–Lepowsky–Meurman construction of a CFT with c = 24. Such powers are computed in Chap. 8—in Eqs. (8.19), (8.29) and (8.34). 1

j (z) 3 , in particular, is a vaccum character of affine E 8 . In Theorem 8.1, a classical formula for j(z) in terms of the E 8 theta function is recalled. The Jacobi elliptic functions and the Weierstrass phi, sigma, and zeta functions are introduced in Chaps. 14 and 17. These elliptic functions play a key role in the study of a magnetoacoustic wave propagated in a cold plasma, in the construction of a cold plasma metric on the continuous Heisenberg model, and in the construction of a change of variables by which this metric is transformed to a Jackiw–Teitelboim black hole metric of constant Ricci scalar curvature, for example,—matters discussed in Chap. 15. This change of variables coupled with another change of variables provides for a transformation of the cold plasma metric to a sine-Gordon metric associated with soliton and oscillating kink-antikink soliton solutions, for example, of the sine-Gordon equation, as indicated in Chap. 20. On the other hand, the Weierstrass phi, sigma, and zeta functions provide solutions to the KdV equation—a point of interest for plasma physics. These functions will be used in Chap. 19 in the construction of the Szekeres–Szafron metric, which is a very general solution of the Einstein gravitational field equations for an inhomogeneous universe. These Weierstrass functions are of relevance, of course, for homogeneous cosmological models as well. An infinite family of new solutions of the Einstein

Preface

ix

equations in terms of them for non-isotropic (homogeneous) Bianchi IX models were found by the author in joint work with Jennie D’Ambroise. The partition function (as a path integral) for a scalar field in thermal equilibrium at a finite temperature T can be rendered a clear mathematical meaning by way of a suitable zeta function. Such is the case, more generally, for a charged, massive field with a nonzero chemical potential, as will be discussed in Chap. 21. More specifically, we consider such a quantum field over a Kaluza–Klein spacetime modeled on a locally symmetric space \X where  is a discrete group of isometries of a non-compact symmetric space X. Here the suitable zeta function is given by definition (21.4) in terms of theta functions that depend on T, , and the chemical potential μ. Its meromorphic continuation is given by Eq. (21.23). Using the Harish-Chandra c-function, which controls the spherical harmonic analysis on X, we also attach to X a zeta function and a theta function in definition (21.29). In particular, for X of complex type, we give an explicit formula for all of the Minakshisundaram–Pleijel coefficients in the asymptotic expansion (as t → 0+ ) of this theta function, where it turns out in fact that this expansion result is a statement of equality. We also present an explicit calculation of the one-loop effective potential of X, for X of complex type, which includes the E. Cartan exceptional spaces like X = E 8C /E 8 , for example, of dimension 248 and of rank 8. Thus, the results extend to higher rank some known in the case of a rank 1 symmetric space. The effective potential, which is a quantum replacement of the classical potential, allows for the study of spontaneous symmetry breaking. It is expressed in terms of a zeta regularized determinant of some elliptic differential operator—like a Laplace– Beltrami operator on some manifold, for example, at the one-loop level. Spontaneous symmetry breaking is essential toward the generation of mass from the vacuum expectation value of the Higgs field, for example, this field which has the Higgs boson (the so-called “God particle”) as a building block. It was Peter Higgs back in 1964 who posited the existence of an invisible field in the universe that gave mass to elementary particles. He and Francois Englert shared the 2013 Nobel Prize in Physics for work on the origin of the mass of subatomic parties. It turns out that among an international team of some 6000 scientists involved in the search for and the 2012 discovery of the Higgs boson, two of them (who were part of the ATLAS team, in particular, of 3000) are employed here at the University of Massachusetts at Amherst: Benjamin Brau and Stephane Willocq, who were involved in writing the software program that tracks and analyzes the muon particle—a bi-product particle of boson decay, after the boson is created by protons crashing into each other. The theta function and the Epstein zeta function attached to a positive definite matrix, and a general Jacobi inversion formula for this theta functions, are basic topics of initial consideration, along with modular forms (of positive and non-positive weight), and Dirichlet and Hecke L-functions—material that comprise the first nine chapters. Throughout, an attempt is made to develop and to maintain an expository tone, where examples (not proofs necessarily) lead the way, even though some proofs or sketches thereof are provided. There is some balance between the presentation of well-known material and some quite new material—as already indicated in previous remarks here. A part of Chap. 11, on a Dedekind eta function for the Hecke group,

x

Preface

is the work of my former doctoral student Irina N. Vassileva. Chapter 13 is a special guest lecture contributed by Prof. Alfred G. Noël, of the Department of Mathematics at the University of Massachusetts, Boston. Regarding the lecture material presented, and the broader scope envisioned therein, some further remarks follow—some of which are summary or repetitive in nature since brief references to the content of the 21 Chapters, apart from Chap. 16, have already been made; comments on Chap. 16 also will be forthcoming. The lectures feature a focus on concrete, computational examples for the purpose of illustrating general theory and results. These include, for example, the explicit computations of theta functions and associated Jacobi inversion formulas of zeta and L-functions with their explicit meromorphic continuation and functional equations, the computation of powers of the cube root of the Dedekind–Klein j-invariant (mentioned on page v) with a discussion of the connection thereof to holomorphic Eisenstein seires, the Dedekind eta function, and to genus one partition functions of a holomorphic (or extremal) conformal field theory (ECFT), the explicit asymptotics of the Fourier coefficients of these partition functions with an application to the computation of logarithmic corrections to Bekenstein–Cardy–Hawking black hole entropy (see page iv), some concrete solutions of KdV, the modified KdV, and Duffing equations in terms of Jacobi elliptic functions and the Weierstrass phifunction (functions to be discussed in Chaps. 14 and 17 as has been noted above), and some concrete solutions of the Einstein gravitational field equations (in terms of the Weierstrass phi-function) for inhomogeneous cosmologies—where a particular focus is to be on the general Szekeres–Szafron solution (mentioned on page v). It is shown in Chap. 16 how a solution of the modified KdV equation provides for a solution of the KdV equation, by way of the Miura transformation. Lax pairs are also constructed in Chap. 16, and especially in Chap. 15 are various such pairs explicitly constructed to prove different gauge equivalences—such as that of the continuous Heisenberg model and a particular reaction-diffusion system, for example. Also, there is a useful reparametrization of the aforementioned Szekeres–Szafron solution (due to Charles Hellaby) that provides for a nice transition to Lemaitre–Tolman–Bondi and Friedmann–Lemaitre–Robertson–Walker limit solutions; this will be discussed in Chap. 19. The goal of the work here is two-fold: (i) One part of the goal is to continue the effort and direction of an earlier work “A Window Into Zeta and Modular Physics”, edited by the author and Prof. Klaus Kirsten, that explores interesting interconnections between domains of so-called “pure mathematics” (such as the theory of modular forms) and some of the exciting new developments in theoretical physicsand thus to further bridge the divide thereof. The E 8 root lattice, for example, (whose theta function is a modular form) plays a crucial, definitive role in heterotic string theory—as was indicated on page iv. In fact, its theta function is exactly the holomorphic Eisenstein series E 4 (z), as will be discussed in Chap. 12. In the aforementioned earlier work, it is noted that “the beautiful interlacing of theory and application, and cross-discipline interaction, leads as usual to be notable, fruitful, and bonus outcomes”. (ii) The second part of the goal is to present material not found in other monographs. For example, there is in Chap. 12 the author’s explicit formula that

Preface

xi

expresses the one-loop determinant for higher spin fermionic elementary particles in terms of the Patterson–Selberg zeta function of a three-dimensional hyperbolic cylinder. In Chaps. 15 and 20, for example, remarkable, explicit, new connections between the two-dimensionaI BTZ black hole (with positive mass M), sine-Gordon solutions, and cold plasma physics are established. In Chap. 15, the discussion of the cold plasma ↔ black hole connection mentioned on page v is also extended to the case of a naked singularity—i.e. to the case where M < 0. Some new, recent results are also presented in Chap. 21 that deal with the computation of the one-loop effective potential in the general context of a higher rank, non-compact symmetric space—i.e. the extension of known results in the rank one case to symmetric spaces of arbitrary rank is considered. In fact, to every such space, the author has attached, in a natural manner, a theta function and a zeta function, as was mentioned on page v. Particularly when the space is of complex type, quite explicit formulas are developed for all of the Minakshisundaram–Pleijel coefficients in the short-time asymptotic expansion of theta, and for the effective potential—the latter being expressed in terms of the computable value of the zeta function and of its derivative at the origin. The zeta function here is a Mellin transform of the theta function, which in turn is constructed by way of Lie algebraic data and (mainly) the Harish-Chandra c-function for the spherical harmonic analysis on the symmetric space. In particular, the inversion of the spherical Fourier transform is facilitated by integration against a Plancherel measure defined by the c-function. The latter inversion is a vast generalization of the classical inversion for the Mehler transform. The author is most grateful to Prof. Alfred Noël who kindly accepted the offer to participate in this project. Professor Noël is also a member of an ATLAS team of distinguished mathematicians (in contrast to the ATLAS team of physicists mentioned above) whose basic goal centers on the very ambitious and quite difficult problem of computing the irreducible unitary representations of a reductive real Lie group in terms of root data of its complexification. Gratitude is expressed moreover to the reviewers whose comments and advice have provided for further clarity and perspective of this work. Lastly, the author extends many thanks to Yaping Yuan for her most excellent work in preparing the manuscript. Yaping is a brilliant young mathematician, in her own right, whose professional assistance over the course of time has been of enormous benefit, for which the author expresses his boundless appreciation. Amherst, USA

Floyd L. Williams

Contents

1

A Theta Function Attached to a Positive Definite Matrix . . . . . . . . . .

1

2

Jacobi Type Inversion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

A Theorem of Minkowski: Enter E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

4

Modular Properties of Theta and Eta . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

5

An Epstein Zeta Function Attached to A . . . . . . . . . . . . . . . . . . . . . . . .

27

6

An Inhomogeneous Epstein Zeta Function . . . . . . . . . . . . . . . . . . . . . . .

35

7

Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

The Modular j-Invariant and Powers of Its Cube Root: Enter E8 Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Modular Forms of Non-positive Weight: Exact Formulas and Asymptotics of Their Fourier Coefficients . . . . . . . . . . . . . . . . . . .

63

10 More on Logarithmic Corrections to Black Hole Entropy . . . . . . . . .

79

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

8 9

12 Elementary Particles, the E8 Root Lattice, and a Patterson-Selberg Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 101 13 The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial “Theory of Everything” . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction to Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . 13.2 Grand Unified Theories and the Standard Model . . . . . . . . . . . . . . 13.3 A Theory of Everything . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 122 123 126

xiii

xiv

Contents

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma, and the J-T Black Hole . . . . . . . . . . . . . . . . . . . . 137 16 The Weierstrass P-Function and Some KdV Solutions . . . . . . . . . . . . 157 17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 18 A Finite Temperature Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look at the BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 20 A Cold Plasma-Sine-Gordon Connection . . . . . . . . . . . . . . . . . . . . . . . . 205 21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space: Computation of the One-Loop Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Chapter 1

A Theta Function Attached to a Positive Definite Matrix

In the initial sections that are presented the purpose is to attach to a symmetric, positive definite, m × m real matrix A a theta function θ A (z) and an Epstein zeta function E A (s). θ A (z) is defined on the upper half-plane where Im z (the imaginary part of z) is positive, and s is a complex variable with real part Re s > m2 . We consider modular properties of θ A (z) based on a Jacobi type inversion formula. We also consider the meromorphic continuation of E A (s) to the full complex plane and a functional equation that this zeta function satisfies. A certain Mellin transform relates θ A (z) and E A (s). Of particular interest is θ A (z) where A is the Cartan matrix of the complex, simple Lie algebra E 8 . An attempt is made to maintain an expository style coupled with the presentation of various concrete examples. The discussion is not proof oriented, as in many of the outstanding, standard texts, although some proofs and sketches of proofs in some instances are provided. Another purpose, particularly in later sections, is to illustrate some selected situations in physics where theta functions, or elliptic functions, etc, play an important, crucial role. A = [Ai j ] = At will denote a symmetric m × m matrix with entries Ai j in the field R of real numbers; At denotes the transpose of A. The corresponding quadratic form f A regarded as an R-valued function on Rm is given by def.

f A (x) = x Ax t =

m 

Ai j x i x j

(1.1)

i, j=1

for x = (x1 , · · · , xm ) ∈ Rm . Note that the sum over i, j in (1.1) is the sum over terms where i = j plus the sum over terms where i < j and i > j. However, in summing over the terms where i  > j we can  interchange i and j, and use that Ai j = A ji (as A is symmetric) to get i> j = i< j . In other words,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_1

1

2

1 A Theta Function Attached to a Positive Definite Matrix

f A (x) =

m 



Aii xi2 + 2

i=1

Ai j x i x j .

(1.2)

i< j

The eigenvalues of A are real since A is symmetric. If in fact they are all positive, then by definition A is a positive definite (p.d.) matrix. The following statements are all equivalent: (i) A is p.d. (ii) A = B t B for some matrix B with det B = 0. (iii) f A is p.d.: f A (x)  0 ∀x ∈ Rm with f A (x) = 0 ⇔ x = 0. (iv) ∃λ = λ(A) > 0 (λ depending only on A) such that f A (x)  λ(x12 + x22 + · · · + xm2 ) ∀x ∈ Rm . Definition 1.1 A is integral if all the Ai j ∈ Z = the ring of integers. A is even integral if A is integral and moreover the diagonal entries Aii ∈ 2Z. If A is even integral then of course for n = (n 1 , · · · , n m ) ∈ Zm it follows that n ) ∈ 2Z, by (1.2). f A ( The following are some examples/counter examples. ⎡ ⎤ 2 −1 0 √ √ 1. A = ⎣−1 2 −1⎦ = At is p.d. with eigenvalues 2, 2 + 2, 2 − 2 > 0. 0 −1 2 f A (x1 , x2 , x3 ) = 2(x12 + x22 + x32 ) − 2(x1 x2 + x2 x3 ) = x12 + (x1 − x2 )2 + (x2 − x3 )2 + x32 > 0 for x = (x1 , x2 , x3 ) = 0 = (0, 0, 0) ⇒ f A is p.d., in accord with the statement (iii) above.  equivalence 12 2. A = is symmetric but is not p.d. since its eigenvalues -1, 3 are not all 21 positive. Therefore by the equivalence statement (iii) f A is not p.d. In fact, 2 f A (x1⎡, x2 ) = x12 + ⎤ x2 + 4x1 x2 so that, for example, f A (1, −1) = −2 < 0. 2 3

3. A = ⎣− 23 1 3

2 3 1 3

1 3 2⎦ 3 − 23 23

is not symmetric so that its eigenvalues need not all be real.

In fact its eigenvalues are 1,

1 3

+

√ 2 2 i, 13 3

−

√ 2 2 i 3

for i 2 = −1.

The existence of the positive constant λ = λ(A) in the equivalence statement (iv) def. is crucial for convergence results ahead. One can in fact take λ(A) = the smallest eigenvalue of A, given that they are all positive. To check this, choose an orthogonal matrix P which diagonalizes A, since A is symmetric: ⎤ ⎡ 0 λ1 ⎥ def. ⎢ P A P −1 = ⎣ . . . (1.3) ⎦ = D 0

λm

where the λ j > 0 are the eigenvalues of A. Given x ∈ Rm , let y = P x t : x t = def.

def.

t t t −1 P −1 y = P t y ⇒ y t Dy = xm = y2 P so that f A (x) = x Ax = y P A P y =  m 2 t P x , P x t >= j=1 λ j y j  λ j=1 y j (by definition of λ)= λ < y, y >= λ <  λ < x t , x t > (as P preserves the inner product on Rm )= λ mj=1 x 2j , which is the desired inequality.

1 A Theta Function Attached to a Positive Definite Matrix

3

 As an example, for a general 2 × 2 symmetric matrix A =

a b/2 with a > 0 b/2 c

and discriminant D = b2 − 4ac (2ax1 + bx2 )2 − Dx22 4a

f A (x1 , x2 ) = ax12 + bx1 x2 + cx22 =

(1.4)

is clearly p.d. if D < 0. Conversely if f A (or A) is p.d., then as det A = product of the eigenvalues of A, D = −4 det A < 0. Also by (1.4), f A (1, 0) = a ⇒ a > 0 since (1, 0) = (0, 0) and f A is p.d. By definition λ(A) = the smallest eigenvalue of A. The eigenvalues of A in this example are 21 [a + c ± (a − c)2 + b2 ] so that λ(A) = 21 [a + c − (a − c)2 + b2 ]: f A (x1 , x2 ) ≥ λ(A)(x12 + x22 ) ∀(x1 , x2 ) ∈ R2 . The main point of this section is to introduce the theta function θ A (z) attached to a symmetric, p.d. matrix A. θ A (z) is defined on the upper 21 −plane π + , Im z > 0: 

def.

θ A (z) =

eπi z f A (n) .

n=(n 1 ,··· ,n m

(1.5)

)∈Zm

The equivalence statement (iv) above provides for the inequalities |eπi z f A (n) | = e−π Im z f A (n) ≤ e−π Im zλ =

m

e−π Im zλn k , 2



k=1

n 2k

|eπi z f A (n) |

(1.6)

n∈Zm

k=1

≤



m

m

e−π Im zλn k = 2

m



e−π Im zλn k

2

k=1 n k ∈Z

n∈Zm k=1

where λ = λ(A) and 

e−π Im zλn = 1 + 2 2

∞ 

e−π Im zλn < ∞ 2

(1.7)

n=1

n∈Z

for Im z, λ > 0, which gives the absolute convergence of θ A (z). For the uniform convergence of θ A (z) on any compact subset L ⊂ π + , we use that continuous function z → Im z on π + has a positive lower bound β on L: In (1.6) we get for z ∈ L |eπi z f A (n) | ≤ m  n∈Zm k=1

e

−πβλn 2k

=

m

k=1 m

e−πβλn k , 2



k=1 n k ∈Z

(1.8) e

−πβλn 2k

4

1 A Theta Function Attached to a Positive Definite Matrix

with



e−πβλn < ∞, 2

(1.9)

n∈Z

as in (1.7), for β, λ > 0. Thus we may conclude the uniformity of convergence of θ A (z) on L and the holomorphicity of θ A (z) on π + by standard results of Weierstrass. If A is also even integral, we have noted that (by (1.2)) f A (n) ∈ 2Z for n ∈ Zm . def. For  ∈ Z+ = {1, 2, 3, · · · } fixed, it makes sense therefore to consider the set def.

S = {n ∈ Zm | f A (n) = 2}

(1.10)

of integral solutions of the equation f A (x) = 2. Let def.

a = a(, m; A) = |S |

(1.11)

denote the cardinality of S . One then has the nice Fourier series expression θ A (z) = 1 +

∞ 

a e2πiz

(1.12)

=1

of θ A (z), with Fourier coefficients a . The leading term 1 here corresponds to the term in (1.5)for n = 0; i.e. f A (n) = 0 ⇔ n = 0 (as f A is p.d.). 21 For A = , which is p.d. and even integral (with eigenvalues 1, 3), f A (x1 , x2 ) 12 = 2(x12 + x22 + x1 x2 ) ⇒ θ A (z) =



e2πi z(n 1 +n 1 n 2 +n 2 ) 2

2

(n 1 ,n 2 )∈Z2

=1+

∞ 

(1.13) a e2πiz

=1

for

def.

a = |{(n 1 , n 2 ) ∈ Z2 |n 21 + n 1 n 2 + n 22 = }|.

(1.14)

We shall revisit this interesting example with further details in Chap. 7. For now we remark that Ramanujan, in his letter to G. H. Hardy from Fitzroy House (a London def. sanitarium) [10, 57], asserts that for q = e2πi z ∞   q 3n−2 q 3n−1 . − θ A (z) = 1 + 6 1 − q 3n−2 1 − q 3n−1 n=1

(1.15)

1 A Theta Function Attached to a Positive Definite Matrix

By (1.13) of course θ A (z) =

∞ 

5

q n 1 +n 1 n 2 +n 2 . 2

2

(1.16)

n 1 ,n 2 =−∞

The assertion (1.15) is a result actually discovered by Lorenz [74] years prior to the great master S. Ramanujan, and it leads to the Dirichlet character expression/computation (7.33) of the number of solutions al in (1.14). See Table 7.1 in Chap. 7. For future reference we state again that f A (x)  λ(A)(x12 + x22 + · · · + xm2 ) def.

(1.17)

for every x = (x1 , x2 , · · · , xm ) ∈ Rm , where 0 < λ(A) = smallest eigenvalue of A.

Chapter 2

Jacobi Type Inversion Formulas

For the world’s simplest theta function def.

θ(t) =



e−πn t = 1 + 2 2

∞ 

e−πn t , t > 0, 2

(2.1)

n=1

n∈Z

a Jacobi inversion formula relates its values at t and 1t , the inverse of t:   √ 1 = tθ(t) : t  √  −πn 2 t n2 e−π t = t e . θ

n∈Z

(2.2)

n∈Z

The simple statement (2.2) and more elaborate versions of it follows by way of a Poisson summation formula. Of interest here is the following version, where again we write n for (n 1 , · · · , n m ) ∈ Zm and for the inner product on Rm : For z ∈ π + , x ∈ Rm  n∈Zm

e

− πiz f A (n)+2πi

 ( zi )m  πi z f −1 (n+x) =√ e A . det A n∈Zm

(2.3)

For the square root in (2.3), the logarithm is specified by choosing for any w ∈ C − {0}(C = the field of complex numbers) arg w ∈ (−π, π]. A (as in Chap. 1) is symmetric and positive definitive and therefore so is A−1 . In Chap. 11 of [28], for example, towards the proof of (2.3) it is first established by induction on m that

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_2

7

8

2 Jacobi Type Inversion Formulas



 1 e−π f A (n)+2πi = √ e−π f A−1 (n+x) , det A n∈Zm n∈Zm

(2.4)

an equation independent of z. For m = 1, (2.4) follows in fact from the Jacobi inversion formula  √  −πn 2 t−2πnwt ni 2 eπt (w− t ) = t e (2.5) n∈Z

n∈Z

for t > 0, w ∈ C, which reduces to (2.2) for w = 0. f A (x1 ) = A11 x12 , f A−1 (x1 ) = x12 /A11 (where A11 > 0 as A is p.d.), and (2.5) gives (2.4) for the choices t = A11 , w = −i x/A11 , for m = 1. For t > 0, (2.4) applied to 1t A (which is also symmetric and p.d.) gives 

π

e− t

f A (n)+2πi

n∈Zm

t m/2  −πt f −1 (n+x) A =√ e det A n∈Zm

(2.6)

since det 1t A = t −m det A, f ( 1t A)−1 = t f A−1 . Equation (2.6) implies (2.3) for z = it ∈ π + , and then (2.3) follows for all z ∈ π + by analytic continuation. By (1.1), f A clearly extends to a function f A : Cm → C. Then in (2.3) one can replace x ∈ Rm by (w1 , · · · , wm ) ∈ Cm . In particular for the choice x = 0 in (2.3), the Jacobi inversion result  ( z )m 1 θ A (− ) = √ i θ A−1 (z) z det A

(2.7)

follows immediately by the definition in (1.5). If, moreover, A is even integral as in Definition 1.1 of Chap. 1 so that f A (n) ∈ 2Z(as we have noted), then also θ A (z + 1) = θ A (z).

(2.8)

The map L : Rm → Rm given by L(x) = x A is a linear isomorphism for det A = 0 that satisfies f A−1 (L(x)) = L(x)A−1 L(x)t = x A A−1 At x t = x Ax t = f A (x) def.

def.

(2.9)

since At = A. If A is integral, then L : Zm → Zm but this restriction need not be surjective: the linear system of equations x A = y for y ∈ Zm might not have a solution x ∈ Zm . However, by Cramer’s rule it does if det A = 1. That is, A integral and det A = 1 ⇒ L : Zm → Zm is a bijection so that def.

θ A−1 (z) =



eπi z f A−1 (n) =

n∈Zm

=



n∈Zm

e

 n∈Zm

πi z f A (n)

= θ A (z)

eπi z f A−1 (L(n)) (2.10)

2 Jacobi Type Inversion Formulas

9

by (2.9), which gives (by (2.7), (2.8)) 1 θ A (− ) = z

 m z θ A (z), θ A (z + 1) = θ A (z) i

for A even integral with det A = 1.

(2.11)

Chapter 3

A Theorem of Minkowski: Enter E8

Suppose A is symmetric, p.d., and even integral. Then if also det A = 1, so that  z m 1 ) θ A (z), θ A (z + 1) = θ A (z) θ A (− ) = ( z i

(3.1)

by (2.11), a theorem of H.Minkowski says that necessarily m must be divisible by 8—a proof of which can be obtained using (3.1). It would be interesting, however, to find a purely linear algebraic proof, whose existence at this point the author is unaware of. If m = 2, for example, which is not divisible by 8, A has the form  A=

 2n  , n, m,  ∈ Z  2m

(3.2)

and Minkowski’s theorem says that one cannot have det A = 1: the equation 4nm − 2 = 1 has no solution in Z. Indeed if  = 21 ∈ 2Z is even, the contradiction 41 = nm − 21 ∈ Z follows. On the other hand if  = 21 + 1 is odd, then nm = (1 + 2 )/4 = (4(21 + 1 ) + 2)/4 = 21 + 1 + 21 also forces a contradiction: 21 ∈ Z. Conversely if m is divisible by 8, one can find m × m real symmetric, p.d., even integral matrices A with det A = 1. A very important example, for m = 8 in fact, is the Cartan matrix of the 248 dimensional complex, simple, exceptional Lie algebra E 8 —a matrix which we also denote by E 8 : ⎡

2 ⎢−1 ⎢ ⎢0 ⎢ 0 def. ⎢ A = E8 = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

−1 2 −1 0 0 0 0 0

0 −1 2 −1 0 0 0 0

0 0 −1 2 −1 0 0 0

0 0 0 −1 2 −1 −1 0

0 0 0 0 −1 2 0 0

0 0 0 0 −1 0 2 −1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ = At . 0⎥ ⎥ 0⎥ ⎥ −1⎦ 2

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_3

(3.3)

11

12

3 A Theorem of Minkowski: Enter E 8

A Maple program (LinearAlgebra), for example, immediately computes that the characteristic polynomial is p(λ) = λ8 − 16λ7 + 105λ6 − 364λ5 + 714λ4 − 784λ3 + 440λ2 − 96λ + 1, (3.4) the eigenvalues are 2+ 2− 2+ 2− 2+ 2− 2+ 2−

  √ √ 1 7 + 5 + 30 + 6 5 = 3.9890438, 2  √ √ 1 7 + 5 + 30 + 6 5 = 0.0109562, 2  √ √ 1 7 + 5 − 30 + 6 5 = 2.8134733, 2  √ √ 1 7 + 5 − 30 + 6 5 = 1.1865267, 2  √ √ 1 7 − 5 + 30 − 6 5 = 3.4862897, 2  √ √ 1 7 − 5 + 30 − 6 5 = 0.5137103, 2  √ √ 1 7 − 5 − 30 − 6 5 = 2.4158234, 2  √ √ 1 7 − 5 − 30 − 6 5 = 1.5841766, 2

(3.5)

and that the determinant of E 8 is 1. The determinants for all the complex, simple Lie algebras, in fact, are known. For example, we have the tabulation (Table 3.1) Table 3.1 Cartan matrix determinants

Lie algebra

Cartan determinant

s(n + 1, C) = An , n  1 so(2n + 1, C) = Bn , n  2 sp(n, C) = Cn , n  3 so(2n, C) = Dn , n  4 E6 E7 E8 F4 G2

n+1 2 2 4 3 2 1 1 1

3 A Theorem of Minkowski: Enter E 8

13

The Cartan matrices of the 45 and 14 dimensional Lie algebras D5 and G 2 , for example, are ⎡ ⎤ 2 −1 0 0 0 ⎢−1 2 −1 0 0 ⎥   ⎢ ⎥ ⎢ 0 −1 2 −1 −1⎥ , 2 −1 (3.6) ⎢ ⎥ ⎣ 0 0 −1 2 0 ⎦ −3 2 0 0 −1 0 2 respectively, whose determinants are 4, 1 as in the tabulation. In (1.17) we see that the positive number λ(A) for which f A (x) ≥ λ(A) < x, x > on Rm (i.e. the smallest eigenvalue of A) for A = E 8 is given by (3.5): λ(E 8 ) = 0.0109562. By (3.1) 1 (3.7) θ E8 (− ) = z 4 θ E8 (z), θ E8 (z + 1) = θ E8 (z). z More generally, since m ∈ 8Z by Minkowski’s theorem, ( zi )m = z m/2 ⇒ 1 θ A (− ) = z m/2 θ A (z), θ A (z + 1) = θ A (z) z by (3.1).

(3.8)

Chapter 4

Modular Properties of Theta and Eta

The theory of modular forms is like a perennial garden—evergreen from its historic roots. It is a beautiful subject that flourishes with magnificent results. The limited intent here is to collect a few facts and examples pertaining thereto, which are useful for our interest, where the focus is largely (but not exclusively) on forms with respect to the subgroup  = S L(2, Z) of G = S L(2, R). Modularity of the Dedekind eta function η(z) is also considered as, in particular, its multiplier ε :  → {z ∈ C||z| = 1} is related to that of the Jacobi theta function θ(w|z). Also, with η(z) serving as a prototype, we present later in Chap. 11 a Dedekind type eta function η N (z) attached (more generally) to the Hecke subgroup 0 (N ) of . The explicit construction of η N (z) (where η N (z) = η(z) for N = 1), and of its multiplier, and the derivation of its modularity with respect to 0 (N ) (at the cusp ∞) is work of my thesis student Vassileva [101]. Excellent, comprehensive discussions (with proofs) of modular and automorphic forms are presented in the Refs. [4, 14, 50, 60, 98] for example. Some further exposition (and proofs) drawn from some of these references are found in my lecture in [114]. def. The group G = S L(2, R) of 2 × 2 real matrices with determinant equal to 1 acts + on π in the standard way and that action restricts to an action of the subgroup of def. matrices  = S L(2, Z) with entries in Z: g=

  ab def. az + b . ∈ G, z ∈ π + ⇒ g · z = cd cz + d

(4.1)

In particular       −1 0 def. 1 1 def. 0 −1 −1 = , T = , S = ∈ , z ∈ π + ⇒ 0 −1 01 1 0 def.

(4.2)

− 1 · z = z, T · z = z + 1, S · z = −1/z.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_4

15

16

4 Modular Properties of Theta and Eta

The matrices T, S generate -a proof of which is given in Appendix B of [114], and of course in other places as well: every γ ∈  is a finite product γ1 · · · γ with γ j = T n j S m j for some n j , m j ∈ Z. A result of this fact, given (4.2), is that the following two conditions (M1) : f (z + 1) = f (z) and (M2) : f (−1/z) = z k f (z), for a function f (z) on π + and k ∈ Z, k  0, are equivalent to the single condition   ab (M1) : f (γ · z) = (cz + d) f (z) for γ = ∈ . cd k

(4.3)

Let f (z) be a holomorphic function on π + . If f (z) satisfies the periodicity condition def. (M1) , then f (z) has a Fourier expansion (or q-expansion for q = e2πi z ) on π + f (z) =



an e2πinz =

n∈Z



an q n

(4.4)

n∈Z

—a proof of which is also given in [114]. f (z) is holomorphic at infinity if the coefficients an for n ≤ −1 are zero: f (z) =

∞ 

an e2πinz .

(4.5)

n=0

f (z) is a modular form, with respect to , of weight k if it satisfies condition (M1) (or equivalently conditions (M1) and (M2) ), and is holomorphic at infinity—the latter condition being about the holomorphicity of f (z) at the single cusp ∞ of . Since −1 · z = z in (4.2), f (z) = (−1)k f (z) by (M1)⇒ f (z) = 0 for k odd. One assumes therefore that k ∈ 2Z is even. Let M(, k) denote the C− vector space of modular forms, with respect to , of weight k. By (3.7), θ E8 (z) satisfies (M1) and (M2) for k = 4. More generally, so does θ A (z) by (3.8) for A which satisfies Minkowski’s theorem. That is, summarizing some of the preceding discussion, we have Theorem 4.1 Let A = [Ai j ] be a symmetric, positive definite, m × m real matrix which is even integral; i.e. Ai j ∈ Z and the diagonal entries Aii ∈ 2Z. Suppose also that det A = 1, which by Minkowski’s theorem in Chap. 3 means that necessarily m ∈ 8Z. Let θ A (z) be the theta function on π + (= the upper 1/2-plane) attached to A in (1.5), which in particular is holomorphic at infinity by (1.12). Then by (3.8), θ A (z) ∈ M(, m2 ). In particular θ E8 ∈ M(, 4) by (3.7) for E 8 given in (3.3). We plan to add further remarks regarding E 8 in Chaps. 8 and 12. If the assumption det A = 1 (which is a bit strict) is relaxed in Theorem 4.1, then for  replaced by the Hecke subgroup      ab def.  (4.6) ∈   N divides c 0 (N ) = cd

4 Modular Properties of Theta and Eta

17

of , for a suitable N ∈ Z, N > 1, θ A (z) satisfies (in place of condition (M1)), for m ∈ 2Z, (4.7) θ A (γ · z) = χ(d)(cz + d)m/2 θ A (z)   ab for γ = ∈ 0 (N ) and for a suitable character χ(d) of 0 (N ). One can take N cd to the smallest positive integer such that N A−1 is also even integral. χ(d) explicited in terms of Kronecker’s symbol. See Appendix I to this chapter. Also see Theorem 10.9 of [60]. Another nice result, before further examples are considered, is the following. Theorem 4.2 (E. Hecke) Let A = [Ai j ] be as in Theorem 4.1. Then for the Fourier expansion ∞  al e2πiz (4.8) θ A (z) = 1 + =1

in (1.12) on π + , where (in (1.11)) def.

n ∈ Zm | f A ( n ) = 2}|, a = a(, m; A) = |{

(4.9)

one has the formulas a = 240σ3 (), a = 480σ7 ()

(4.10)

for m = 8, 16 respectively, where 

def.

σk () =

dk

(4.11)

d>0,d|

for k ∈ Z, k  0. As an application, θ E8 (z) = 1 + 240

∞ 

σ3 ()e2πiz

(4.12)

=1

on π + . On the other hand, we have the holomorphic Eisenstein series 

def.

G k (z) =

(m,n)∈Z2 −{(0,0)}

1 (m + nz)k

(4.13)

on π + for k = 4, 6, 8, 10, 12, · · · , which are modular forms of weight k, and which have the Fourier expansion [4] ∞

G k (z) = 2ζ(k) +

2(2πi)k  σk−1 ()e2πiz (k − 1)! =1

(4.14)

18

4 Modular Properties of Theta and Eta

on π + , where def.

ζ(s) =

∞  1 , Re s > 1 ns n=1

(4.15)

is the Riemann zeta function. The G k (z) serves as building blocks for other modular def. forms, and we use E k (z) = G k (z)/2ζ(k) as a particular normalization of G k (z): ∞

E k (z) = 1 +

 (2πi)k σk−1 ()e2πiz . ζ(k)(k − 1)! =1

(4.16)

The point is that for k = 4, in particular, using the special value ζ(4) = π 4 /90, we see that ∞

E 4 (z) = 1 +

(2π)4 90  σ3 ()e2πiz 3! π 4 =1

= 1 + 240

∞ 

σ3 ()e

(4.17)

2πiz ∴

= θ E8 (z) !

=1

by (4.12). The normalization in (4.16), which is used in my lecture in [114], differs from that in the lecture of Mason and Tuite in [78] given on page 197. As was indicated in the initial remarks of this section, the Dedekind eta function η(z) (apart from its independent importance) has a connection to the Jacobi theta function θ(w|z) by way of the multiplier systems of these two functions. This connection emerges as the modular property of these functions is considered. We start with ∞ 

def. η(z) = eπi z/12 1 − e2πinz (4.18) n=1 def.

for z ∈π + . η(z) is a holomorphic function on π + since for an (z) = −e2πinz the −2πn Im z with Im z > 0), and the series ∞ n=1 an (z) converges absolutely (|an (z)| = e convergence is uniform on any compact subset K of π + : The continuous function Im π + has a positive lower bound B so that |an (z)| ≤ e−2πn B on K , with ∞z on−2πn B e = a convergent geometric series for B > 0. n=1 R. Dedekind’s profound modular result for η(z) is that η(γ · z) = eπiχ(γ) (cz + d) 2 η(z) 1

for

(4.19)

4 Modular Properties of Theta and Eta

19



 ab ∈ , c = 0, c d 1 c c def. a + d − − s(d, |c|), χ(γ) = 12c 4 |c| |c| |c|−1  j jd

def. , s(d, |c|) = |c| |c| j=1

γ=

(4.20)

def.

s(0, 1) = 0, where for x ∈ R def.



((x)) =

x − [x] − 0

1 2

 for x ∈ /Z , for x ∈ Z

(4.21)

[x] being the largest integer which does not exceed x. Actually, ∞ 1  sin 2πnx . ((x)) = − π n=1 n

(4.22)

def.

Note here that for U = C− negative x−axis including 0, w → log w = log |w| + i arg w for −π < arg w < π is a holomorphic function on U such that elog w = w. For c = 0 in (4.20) and z ∈ C with Im z = 0 (for example z ∈ π + ) we have cz + d ∈ U . In particular z → log(cz + d) is a holomorphic function on π + , and the square root in (4.19) is well-defined. In case c = 0, det γ = 1 ⇒ a = d = ±1, and def.

χ(γ) = ±

b , η 12



 ±1 b · z = e±πib/12 η(z). 0 ±1

(4.23)

In particular, for T, S in (4.2) 1 η(z + 1) = η(T · z) = eπi/12 η(z), χ(S) = − , 4

1 = η(S · z) = e−πi/4 z 1/2 η(z) ⇒ η − z

1 = z 12 η 24 (z). η 24 (z + 1) = η 24 (z), η 24 − z

(4.24)

That is, η 24 (z) satisfies conditions (M1) , (M2) (with k = 12) and thus equivalently condition (M1): η 24 (γ · z) = (cz + d)12 η 24 (z)   ab for γ = ∈  ⇒ η 24 (z) ∈ M(, 12). c d

(4.25)

20

4 Modular Properties of Theta and Eta

Actually, η 24 (z) is a cusp form—meaning that in (4.5) the zeroth Fourier coefficient a0 vanishes: ∞  τ (n)e2πinz (4.26) η 24 (z) = n=1

on π + , where τ (n) is the Ramanujan tau function. Remarkably the τ (n) are all integers (τ (1) = 1, τ (2) = −24, τ (3) = 252, τ (9) = −113643, τ (10) = −115920, τ (18) = 2727432, τ (27) = −73279080, for example), and P. Deligne proved the Ramanujan conjecture |τ (n)| ≤ σ0 (n)n 11/2 for n  1; σ0 (n) is the number of positive divisors of n. The G k (z) are not cusp forms since in (4.14) the special value ζ(k) = 0. In fact, the Riemann zeta function ζ(s) is nonvanishing for Re s > 1. However the discriminant form def.

(z) = [60G 4 (z)]3 − 27 [140G 6 (z)]2

(4.27)

is a cusp form of weight 12, and since it is known that the C− vector space S(, k) of cusp forms of weight k (a subspace of M(, k)) is one-dimensional for k = 12 it follows that (z) must be a multiple of η 24 (z). In fact (z) = (2π)12 η 24 (z).

(4.28)

The statement made that the weight of (z) was 12 relied on the fact that the product of two modular forms with weights k1 and k2 is a modular form of weight k1 + k2 . Also the statement (4.29) M(, k) = S(, k) ⊕ CG k (z) holds, where this vector space sum is indeed direct since G k (z) is not a cusp form. Before getting to θ(w|z) which is yet to be defined, we make three observations about the function χ(γ) on  in (4.19). (i) s(d, |c|) in (4.20) is a Dedekind sum. Such sums and their properties (their reciprocity law, for example) are elaborated on in [4], for example. By (4.19), (4.25) for c = 0, (cz + d)12 η 24 (z) = η 24 (γ · z) = e24πiχ(γ) (cz + d)12 η 24 (z) ⇒ e24πiχ(γ) = 1, def.

(4.30)

which also holds if c = 0 since then χ(γ) = ±b/12 by (4.23). Thus (ii) eπiχ(γ) is a 24th root of unity for every γ ∈ . (iii) 12χ(γ) ∈ Z for every γ ∈  by (4.30), since ew = 1 ⇔ w ∈ 2πiZ. As another example of the modular properties of theta we now consider the Jacobi theta function  1 2 def. (−1)n eπi z (n+ 2 ) eπiw(2n+1) (4.31) θ(w|z) = −i n∈Z

4 Modular Properties of Theta and Eta

21

for w ∈ C, z ∈ π + . With z fixed, θ(w|z) is an entire function of w. The alternate expression θ(w|z) = 2

∞  1 2 (−1)n eπi z(n+ 2 ) sin(2n + 1)πw

(4.32)

n=0

is valid. For b, n ∈ Z, n 2 + n ∈ 2Z ⇒ b(n 2 + n) ∈ 2Z ⇒ θ(w|z + b) = eπib/4 θ(w|z). 

For γ=

ab c d

(4.33)

 ∈  with c > 0



w  az + b πicw 2 1 = e3πiχ(γ) e cz+d (cz + d) 2 θ(w|z), θ  cz + d cz + d

(4.34)



(4.35)

where χ(γ) is given by (4.20); see [82]. Here e3πχ(γ) is an 8th root of unity for every γ ∈ , by (4.30). If c = 0 in (4.34) with a = d = 1, then Eq. (4.35) still holds since def. in this case it is precisely Eq. (4.33), as then χ(γ) = b/12 by (4.23). def. Of course we now see that the multipliers εη , εθ :  → C given by εη (γ) = def.

eπiχ(γ) in (4.19), (4.20) and εθ (γ) = e3πiχ(γ) in (4.35) are indeed related. Actually, ∂θ (w|z). Then η(z) and θ(w|z) are more directly related as follows. Let θ (w|z) = ∂w θ (0|z) = 2πη 3 (z)

(4.36)

on π + . One can employ (4.36) as an important ingredient, in fact, towards the proof of (4.35). When Eqs. (4.19) and (4.35) are compared with the condition (M1) in (4.3), it is natural to think of η(z) and θ(w|z) as modular forms of weight 1/2—or of 1/η(z) as a form of negative weight −1/2. In Chap. 9 we will consider in general forms of negative weight—subject to a precise definition. We mentioned the strictness of the assumption in Theorem 4.1 that det A = 1, and that with the relaxing of this assumption the modular property of θ A (z), which would be with respect to the subgroup 0 (N ) in (4.6) (instead of ), would then be expressed by Eq. (4.7)—N being the smallest positive integer for which N A−1 is also even integral. Before closing out this section we look at two examples with det A = 1. Take 1    0 20 −1 −1 2 , A = 4A = (4.37) A = 02 0 21 so clearly N = 4 is this case. Also

22

4 Modular Properties of Theta and Eta

x 2 + x22 on R2 f A (x1 , x2 ) = 2 x12 + x22 , f A−1 (x1 , x2 ) = 1 2  2 2 ⇒θ A (z) = e2πi z (m +n ) , (m,n)∈Z2



θ A−1 (z) =

eπi z

(m2 +n2 ) 2

(m,n)∈Z2

(4.38)

z = θ A ( ) on π + . 4

For T ∈  in (4.2) we clearly have that def.

θ A (T · z) = θ A (z + 1) = θ A (z) = θ A (z − 1)

(4.39)

since m 2 + n 2 ∈ Z in (4.38). In place of S ∈  in (4.2), consider the element   0 − 21 def. ∈ . (4.40) S˜ = 2 0 In the inversion formula (2.7), replace z there by 4z:

1 4z def. ˜ θ A ( S · z) = θ A − = θ A−1 (4z) = −2i zθ A (z) 4z 2i

(4.41)

by (4.38). Replace z ∈ π + by − 4z1 − 1 = S˜ · z − 1 ∈ π + : ˜ − 1) = S˜ · ( Sz

  −1 z −1 def. 1 0

= = 1 = · z. 41 ˜ − 1) 4z + 1 4 − 4z − 1 4( Sz

(4.42)

Then by (4.41), (4.39) 

   −1 10 ˜ ˜ ˜ − 1 θ A Sz θA · z = −2i( Sz − 1)θ A ( Sz − 1) = −2i 41 4z

(4.43) 1 = −2i − − 1 (−2i z)θ A (z) = (4z + 1)θ A (z). 4z (4.39) and (4.43) show that   d−1 ab θA · z = (−1) 2 (cz + d)θ A (z) c d

(4.44)

in particular for 

ab c d



def.

=T =



   11 def. 1 0 , R = ∈ 0 (4). 01 41

(4.45)

4 Modular Properties of Theta and Eta

23



 ab = −1 in (4.2), the right hand side being (−1)(−1)θ A (z) c d in this case, which is the left hand side of (4.44). Now the elements −1, T and R of 0 (4) are  known  to generate 0 (4). Thus we can conclude that Eq. (4.44) holds for ab all γ = ∈ 0 (4), which is an example of Eq. (4.7) where the weight m2 there c d

Also (4.44) holds for

def.

d−1

is 1, and where χ(γ) = χ(d) = (−1) 2 on 0 (4). In general, for m even χ(d) in (4.7) is given by

(−1)m/2 det A (4.46) χ(d) = d

for the Kronecker symbol (-). In the present example, this says that χ(d) = −4 , d d−1 which we have said is (−1) 2 . Let’s check this for d = an odd prime p, for example– noting that d is always odd, since c = 4k ∈ 4Z by definition of 0 (4) and since ) is computed as a Legendre symbol 1 = det γ = ad − 4kb ⇒ ad is odd. Then ( −4 d (see Appendix I):

since

p−1 2 2 = (−1) 2 p p

(4.47)

2 2  p 2 −1 p 2 −1 2 8 = (−1) = (−1) 4 = 1; p

(4.48)

−4 p

=

−1 p

indeed p = 2n + 1, n ∈ Z ⇒ p 2 − 1 /4 = n(n + 1) ∈ 2Z. Thus Eq. (4.47) checks the consistency of the two formulas for χ(d) when d is an odd prime, for example. In general χ(d) in (4.46) would be computed by writing d as a product of primes.   As an a second example we take m = 1, A = [2], A−1 = 21 ; f A (x) = 2x 2 , f A−1 (x) = x 2 /2 on R:  2 e2πin z = θ A−1 (4z). (4.49) θ A (z) = n∈Z

Again N = 4, and this time we replace z in (2.7) by 2z:   2z

1 z z i −1 = √ θ A (2z) = θA , θA − 2z i 2 2 θ A (z ± 1) = θ A (z).

(4.50)

In the end, the following version of (4.7) holds: 1

θ A (γ · z) = χ(c, d)(cz + d) 2 θ A (z),

(4.51)

24

4 Modular Properties of Theta and Eta

for a suitable character χ(γ) = χ(c, d) on 0 (4). Thus one can think of θ A (z) as modular form (with respect to 0 (4)) of weight 21 . Forms of half-integral weight however need to be defined with some precision. χ(c, d) can be expressed in terms of the Kronecker symbol ( dc ). More simply, it holds that χ2 (c, d) = sin πd/2. We have

noted that d = 2n + 1 is an odd integer. Thus χ2 (c, d) = sin πn + π2 = cos πn = (−1)n ⇒ χ4 (c, d) = 1. That is χ(γ) is a 4th root of unity for every γ ∈ 0 (4) ⇒ χ(γ) = 1, −1, i, or − i. In addition to the Jacobi theta function θ(w|z) in (4.31), there are the Jacobi theta functions θ1 (w|z), θ2 (w|z), θ3 (w|z) given by def.

θ1 (w|z) =



eπi

n∈Z ∞ 

=2

(2n+1)2 4

eπi

z (2n+1)πiw

(2n+1)2 4

e z

cos(2n + 1)πw,

n=0

def.

θ2 (w|z) =



(−1)n eπin z e2πinw 2

n∈Z ∞  2 =1+2 (−1)n eπin z cos 2nπw, def.

θ3 (w|z) =



(4.52)

n=1

e

πin 2 z 2πinw

e

n∈Z

=1+2

∞ 

eπin z cos 2πnw. 2

n=1

The notation for these four theta functions varies quite widely. The notation here is that adopted in [28] , except that a “comma” is used there in place of the bar “|” used here. Given the Jacobi formula θ (0|z) = πθ1 (0|z)θ2 (0|z)θ3 (0|z) in Theorem 5 of [28], Eq. (4.36) says that η 3 (z) =

1 θ1 (0|z)θ2 (0|z)θ3 (0|z) 2

(4.53)

on π + , which also directly relates theta and eta. As a matter of fact, even more directly, there is the formula 1 z η(z) = eπi z/12 θ3 ( + | 3z). 2 2

(4.54)

4 Modular Properties of Theta and Eta

25

Appendix I 

 ab The character χ(d) of 0 (N ) in (4.7), for γ = ∈ 0 (N ), was expressed in c d terms of the Kronecker symbol (−) in (4.46)—a symbol yet to be defined, which we now do. This would complete the statement of the modular property θ A (γ · z) = χ(d)(cz + d)m/2 θ A (z)

(4.55)

of θ A (z) in (4.7), where the assumption that det A = 1 in Theorem 4.1 is no longer imposed. Towards the definition of (−), we first recall the Legendre symbol. Let m, k ∈ Z be relatively prime elements with m > 0 : (k, m) = 1. k is a quadratic residue modulo m if there exists x ∈ Z such that x 2 ≡ k(mod m). For example, m = 7 and k = 2 are relatively prime and x = 3 satisfies x 2 ≡ 2(mod 7), which says that 2 is a quadratic residue modulo 7. Also k = 4 is a quadratic residue modulo m = 7 since x = 5 solves x 2 ≡ 4(mod 7). Note that k = 3 is not a quadratic residue modulo 7 since the equation x 2 − 3 = 7n, n ∈ Z has no solution x ∈ Z. If p  3 is an odd prime, and a ∈Z is such that a and p are relatively prime (i.e. p  a) then the Legendre symbol ap is defined as follows: ⎡ ⎤ 1 if a is a quadratic residue modulo p a def. ⎣ = −1 if a is not a quadratic residue modulo p ⎦ . p 0 if p | a

(4.56)

One of the most famous theorems in number theory is of course the following law of quadratic reciprocity, proved by Gauss (at age 18) in 1796. Theorem 4.3 (K. Gauss) If p and q are distinct odd primes (in particular p = q ⇒ p and q are relatively prime), then    p−1 q−1 p q 2 2 = (−1) . q p

(4.57)

p  3 has been assumed to be an odd prime. For p = 2, the only non-odd prime, one sets   a  a 2 −1 def. (−1) 8 if a is odd = . (4.58) 2 0 if a is even

Now given n and a in Z the Kronecker symbol an is defined as follows. If n = 0, set   a  def. 1 if a = ±1 = . (4.59) 0 otherwise 0

26

4 Modular Properties of Theta and Eta

Assume therefore n = 0, and write n as a product of primes p j , up to units u = ±1 : n = up1e1 p2e2 · · · pe . Then

 a   a ej = u j=1 p j

(4.60)

⎤ 1 if u = 1 = ⎣ 1 if u = −1, a  0 ⎦ , −1 if u = −1, a < 0

(4.61)

a  n where

a  u

a pj

 =

def.

⎡

def.

 the usual Legendre symbol in (4.56) , if p j is odd

(4.62)

and where ( paj ) is given by (4.58) in case p j = 2. We saw that 4 was a quadratic residue modulo 7 but 3 was not, which means that ( 47 ) = 1, ( 37 ) = −1 by (4.56). Also ( 41 ) = ( 31 ) = 1 by (4.61). Therefore by (4.60), (4.62) the Kronecker symbols ( 47 ), ( 37 ) are the Legendre symbols 1, −1, respectively.

Chapter 5

An Epstein Zeta Function Attached to A

In addition to the theta function θ A (z) attached to a symmetric, positive definite, m × m real matrix A (in Chap. 1) there is an Epstein zeta function def.

E A (s) =

 n∈Zm −{0}

1 . f A ( n )s

(5.1)

attached to A, which is defined initially for s ∈ C with Re s > m/2 [37]. We consider the meromorphic continuation of E A (s) to the full complex plane—which in accomplished by way of the Jacobi inversion result (2.3) for θ A (z). Epstein’s functional equation, which relates the values E A (s) and E A−1 (m/2 − s) in derived; see Eq. (5.20). In particular, for m = 2, we note that the meromorphic continuation and functional equation for the nonholomorphic Eisenstein series E ∗ (s, z) = y s def.

 (m,n)∈Z2 −{(0,0)}

1 |m + nz|2s

(5.2)

follows, where this series is defined initially for Re s > 1 and for z = x + i y ∈ π + . Of course these results for E ∗ (s, z) are generally obtained independently and directly, as in [14, 98, 103], for example. One can compare the definition of the holomorphic Eisenstein series G k (z) in (4.13) with that of the nonholomorphic Eisenstein series in (5.2). The absolute convergence of the series in (5.1) for Re s > m/2 and its uniform convergence on any domain of the form Re s > m/2 +  for  > 0 (by which it would follow that E A (s) is holomorphic on Re s > m2 ) can be reduced to that of known properties of the simpler series  n∈Zm −{0}

1  2 s . 2 n 1 + n 2 + · · · + n 2m

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_5

(5.3)

27

28

5 An Epstein Zeta Function Attached to A

Indeed by the inequality (1.17) again 1 1   2 Re s Re s | f A ( n )s | n 1 + · · · + n 2m λ(A)

(5.4)

in particular for all n = 0. Of course for m = 1, the series in (5.3) is 2ζ(2s) with Re s > 21 ; see (4.15). If m = 2, for example, (an important case as we shall see) 1 1 = ,  2  Re s |n 1 i + n 2 |2 Re s n 1 + n 22

(5.5)

and the absolute convergence of (5.3) (with Re s > 1) is proved in Appendix G of [114], for example. We concentrate on the analytic continuation of E A (s) to the full complex plane. By definition (1.5) and by (2.3), for the choices z = i/t, it, t > 0, x = 0, θ A (i/t) =

 n

t m/2 θ A−1 (it) e−π/t f A (n ) = √ det A

(5.6)

where the sum is oven Zm . Thus (via the change of variables t → 1/u ) (5.6) gives 



1

∞



  i θA − 1 t −s−1 dt = t

[θ A (it) − 1] t dt = 1  ∞  m/2 t θ A−1 (it) − 1 t −s−1 dt = √ det A 1  ∞  m/2 t t m/2 − 1 t −s−1 dt = √ (θ A−1 (it) − 1) + √ det A det A 1  ∞  m/2 t 1 1 √ (θ A−1 (it) − 1) t −s−1 dt + √ −  m s det A det A s − 2 1 s−1

0

for Re s >

m , 2

since



∞

t w dt =

1

−1 w+1

for Re w < −1. In other words, if we set  ∞ def. Z A (s) = [θ A (it) − 1] t s−1 dt 0

(which is the Mellin transform of θ A (it) − 1) for Re s > m/2, and define

(5.7)

(5.8)

(5.9)

5 An Epstein Zeta Function Attached to A def.



∞

F1 (s, A) =

def.

F2 (s, A) =

1 ∞

29

[θ A−1 (it) − 1] t 2 −s−1 dt , m

(5.10) [θ A (it) − 1] t s−1 dt ,

1

then



1

Z A (s) =

 [θ A (it) − 1] t s−1 +

0

∞

[θ A (it) − 1] t s−1 =

1

F1 (s, A) 1 + F2 (s, A) + √ √  det A det A s −

− m 2

(5.11)

1 s

by (5.7). On the other hand (again as f A (0) = 0) θ A (it) = 1 +



e−πt f A (n ) ,

(5.12)

n=0

and we can commute the summation here with the integration in (5.9) to write Z A (s) =

 n=0

=

(s)  πs

n=0

∞

e−πt f A (n ) t s−1 dt

0

1 def. (s) = E A (s) f A ( n )s πs

(5.13)

by (5.1), and by way of the general Mellin transform formula 

∞ 0

e−wt t s−1 dt =

(s) ws

(5.14)

for Re s > 0, Re w > 0, where (s) is the gamma function of course. Also the function F2 (s, A) in (5.10) is an entire function of s—and thus so is the function F1 (s, A). Putting the pieces together we see, in summary, that for Re s > m2 πs E A (s) = (s)



F1 (s, A) 1 + F2 (s, A) + √ √  det A det A s −



πs (s)s = (s + 1) 2 (5.15) by (5.13), (5.11), where the right hand side here now provides for the meromorphic 1 continuation of E A (s) to C, since (s) is also an entire function of s. We can read off immediately from (5.15) that E A (s) has exactly one pole—a simple pole located at s = m/2 with residue equal to  − m

π m/2 πs |s=m/2 = √ . √ (s) det A det A(m/2)

(5.16)

30

5 An Epstein Zeta Function Attached to A

That E A (s) satisfies a functional equation (F.E.) is also immediate once one notes that, by (5.10), m



∞

m m def. [θ A (it) − 1] t 2 −( 2 −s )−1=s−1 dt = F2 (s, A) , 2 (5.17) 1 ∞ m

m def. −1 def. F2 = [θ A−1 (it) − 1]t 2 −s−1 dt = F1 (s, A) . − s, A 2 1

F1

− s, A−1

def.

=

Then by (5.15) ( since E A−1

√

√ det A = ( det A−1 )−1 ),

 √ m √ π 2 −s det A  F2 (s, A) det A + F1 (s, A) − − s = m 2 s  2 −s

m

−π 2 −s  m   =  2 − s + 1 = 2 − s  m2 − s  m √ π 2 −s m  F2 (s, A) det A + F1 (s, A) −  2 −s √ m π 2 −s det A  , − m  2 −s s m

m

or

m 2

1 −s

   m π s E A−1 m2 − s π2 F1 (s, A)  F2 (s, A) + √ = m √  2 − s (s) (s) det A det A  m 1 π2   +√ . −   m  2 − s (s)s det A s − m2

That is, by (5.15),

  m π s E A−1 m2 − s π 2 −s  E A (s) , = m √  2 −s (s) det A



(5.18)

(5.19)

(5.20)

which is the F.E. for E A (s). Note the special values of E A (s): E A (0) = −1, E A (−l) = 0 for l = 1, 2, 3, . . . .

(5.21)

by (5.15), since (1) = 1, (s)−1 = 0 for s =  −1, −2, −3, . . .. a b/2 The special case m = 2 is of interest. A = is positive definite ⇔ a > b/2 c def.

0 and the discriminant D = b2 − 4ac = −4 det A < 0 (as was noted in Chap. 1), which we assume.

5 An Epstein Zeta Function Attached to A def.

E A (s) =

31



1  s 2 am + bmn + cn 2 (m,n)∈Z2 −{(0,0)}

(5.22)

(for Re s > m2 = 1) extends meromorphically to C with a single simple pole at s = 1, √ with residue π/ det A by (5.16). 

cx 2 − bx1 x2 + ax22 c −b/2 A ⇒ f A−1 (x1 , x2 ) = 1 −b/2 a det A  1 ⇒ E A−1 (s) = (det A)s  s cm 2 − bmn + an 2 (m,n)=(0,0)  1 = (det A)s  s 2 cn − bnm + am 2 (m,n)=(0,0)  1 m→−m = (det A)s  s 2 cn + bnm + am 2 (m,n)=(0,0) −1

1 = det A

(5.23)

∴

= (det A)s E A (s) . The F.E. (5.20) for m = 2 therefore reduces to π s E A (1 − s) (det A)

s− 21

(s)

=

π 1−s E A (s) . (1 − s)

(5.24)

Now given z = x + i y ∈ π + , choose def.

A = A(z) =



1 x , x x 2 + y2

(5.25)

which is p.d. with discriminant D = −4 det A = −4y 2 < 0. In the preceding notation, a = 1, b = 2x, and c = x 2 + y 2 = z z¯ so that for (m, n) ∈ Z2 (or R2 ), |m + nz|2 = (m + nz)(m + n z¯ ) = m 2 + mn(z + z¯ ) + n 2 z z¯ = m 2 + mn2x + n 2 c = am 2 + bmn + cn 2 . That is, by (5.22), (5.2) y s E A(z) (s) = y s

 (m,n)∈Z2 −{(0,0)}

1 = E ∗ (s, z) |m + nz|2s

(5.26)

for Re s > 1, which provides for the meromorphic continuation of E ∗ (s, z) (the nonholomorphic Eisenstein series) to the complex plane. Also by (5.24) it follows that E ∗ (s, z) has the following F.E.: π 1−s E ∗ (s, z) π s E ∗ (1 − s, z) = , (s) (1 − s)

(5.27)

32

5 An Epstein Zeta Function Attached to A

which, as was pointed out, one can derive of course by other direct methods. Choosing A = [1] with m = 1, so that f A (x) = x 2 on R, we see that for Re s > ∞  1 1 =2 = 2ζ(2s) 2s (n 2 )s n n=1



def.

E A (s) =

n∈Z−{0}

1 2

(5.28)

for ζ(s) = E A (s/2)/2 = the Riemann zeta function (Re s > 1). It follows in fact (as if one didn’t already know) that ζ(s) has a meromorphic continuation to C with F.E. 1−s

π 2 ζ(s) π s/2 ζ(1 − s) =  1−s  , (s/2)  2

(5.29)

by (5.20). Going back to A(z) defined in (5.25), we see that on R2 , and for t > 0   f A(z) (x1 , x2 ) = x12 + 2x x1 x2 + x 2 + y 2 x22 ⇒  2 2 2 2 e−πt [m +2xmn+(x +y )n ] θ A(z) (it) = (5.30)

(m,n)∈Z2



=

e−πt|m+nz|

2

(m,n)∈Z2

by (1.5), and by the 2nd and 3rd lines following (5.25). Similarly on R2 ,

−1

A(z)

x 2 +y 2 y2 − yx2

=

− yx2 1 y2

 ,

  1  2 x + y 2 x12 − 2x x1 x2 + x22 ⇒ 2 y  − πt [(x 2 +y 2 )m 2 −2xmn+n 2 ] e y2

f A(z)−1 (x1 , y1 ) = θ A−1 (it) =

(m,n)∈Z2



=

e

− yπt2 [(x 2 +y 2 )n 2 −2xmn+m 2 ]

(5.31)

(m,n)∈Z2



m→−m

=

e

− yπt2 [m 2 +2xmn+(x 2 +y 2 )n 2 ]

(m,n)∈Z2

=



(m,n)∈Z2

e

− yπt2 |m+nz|2

= θ A(z) (i

t ) y2

by (5.30) for t > 0. Here m = 2 and det A(z) = y 2 so that in this case the inversion formula (5.6) (with t there replaced by yt and the last equation of (5.31) give for t >0

5 An Epstein Zeta Function Attached to A

θ A(z)

33

    it i 1 = θ A(z) . y t ty

(5.32)

To simplify the notation, for z = x + i y ∈ π + and for t > 0, we set   t def. θz (t) = θ A(z) i = y



πt

e− y |m+nz|

2

(5.33)

(m,n)∈Z 2

by (5.30). Then Eq. (5.32) reads θz (t) =

1 1 θz ( ) , t t

(5.34)

which compares with Eq. (2.2). For Re s > 1, z ∈ π + y s πs Z A(z) (s) E ∗ (s, z) = y s E A(z) (s) = (z)   y s πs ∞  = θz (yt) − 1 t s−1 dt (z) 0  ∞   πs = θz (t) − 1 t s−1 dt (z) 0

(5.35)

by (5.26), (5.13), (5.9), (5.33), and the change of variables t → yt. Thus Eq. (5.35) provides for an integral representation of the nonholomorphic Eisenstein series E ∗ (s, z)—from whence (also) the meromorphic continuation in s can be worked out. One is lead, in fact, to the Fourier expansion of E ∗ (s, z):   √  s − 21 1−s y + E (s, z) = 2ζ(2s)y + 2ζ(2s − 1) π (s) ∞  4π s 1  1  y2 σ1−2s (n)K s− 21 (2πny)n s− 2 e2πnxi + e−2πnxi , (s) n=1 ∗

s

(5.36)

where K v (r ) is the K-Bessel function (see (6.11) in the next section), and σv (n) is the divisor function in (4.11). An alternate, direct proof of (5.36) is set up in Chap. 15 of [103], for example.

Chapter 6

An Inhomogeneous Epstein Zeta Function

For A = the m × m identity matrix Im , f A (x) =

m 

xi2 , θ A (z) =



eπi z (n 1 +···+n m ) 2

2

n∈Zm

i=1

   it 2 2 ⇒ θA = e−t (n 1 +···+n m ) , t > 0. π m

(6.1)

n∈Z

Fix x = (x1 , . . . , xm ) ∈ Rm with each x j > 0 and fix a > 0. Then we can consider the theta function  2 2 def. (6.2) e−t (x1 n 1 +···+xm n m +a ) θ(t; x, a) = n∈Zm

for t > 0, which is an inhomogeneous generalization of θ A the Mellin transform of θ(t; x, a): By (5.14) 

∞



θ(t; x, a)t s−1 dt =

0

n∈Zm

∞

 it  , and we can consider π

e−t (x1 n 1 +···+xm n m +a ) t s−1 dt 2

2

(6.3)

0

= (s)E(s; x, a) where def.

E(s; x, a) =

 n∈Zm



1 x1 n 21

+ · · · + xm n 2m + a

s

(6.4)

for Re s > m2 is the Epstein zeta function [37] attached to θ(t; x, a). In applying (5.14) we use that a and the x j are all positive. The statement

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_6

35

36

6 An Inhomogeneous Epstein Zeta Function

1 (s)

E(s; x, a) =



∞

θ(t; x, a)t s−1 dt

(6.5)

0

for Re s > m2 corresponds to that of (5.13), given (5.9). The Jacobi inversion formula (5.6) was the key towards deriving the meromorphic continuation of the Epstein zeta function E A (s) in (5.15). For the zeta function E(s; x, a) we use the simple inversion formula (2.2). Namely, for b > 0 replace t > 0 by tb/π and thus write (2.2) as 

 e

−n 2 bt

=

n∈Z

π  −π2 n 2 /bt e . tb n∈Z

(6.6)

For the choice b = x j we can use (6.6) to write θ(t; x, a) = e

−ta

m 

e−n j x j t 2

j=1 n j ∈Z

=e

−ta

m  π −π2 n 2j /x j t e t xj j=1 n ∈Z j





 − πt2 nx21 +···++ nxm2m π t e 1 = √ e x1 · · · xm m n∈Z ⎡  2 ⎤ m m n1 n 2m − 2 −ta −π 2  2 +···+ π t e t x1 xm ⎣1 + ⎦, = √ e x1 · · · xm m m 2

− m2 −ta

(6.7)

n∈Z −{0}

so that by (6.5) we write  ∞ m π2 m e−ta t s− 2 −1 dt E(s; x, a) = √ (s) x1 · · · xm 0 m   ∞ −ta− π2 m n2j 2 π m j=1 x j s− −1 t + e t 2 dt. √ (s) x1 · · · xm 0 m

(6.8)

n∈Z −{0}

By (5.14) again, the 1st term here is    s − m2 π m/2 √ m (s) x1 · · · xm a s− 2 for Re s >

m . 2

(6.9)

Regarding the 2nd term the formula 

∞

e 0

−γt−β/t v−1

t

 v/2  β dt = 2 K v (2 βγ) γ

(6.10)

6 An Inhomogeneous Epstein Zeta Function

37

for γ, β > 0 applies for the K-Bessel (or Macdonald–Bessel) function 1 K v (r ) = 2 def.



∞

e− 2 (t+ t ) t v−1 dt, r

1

(6.11)

0

r > 0, v ∈ C. For example, see formula 9 on page 340 of [49] . Thus for n = 0 the 2nd integral in (6.8) has the value ⎡

m 2 

π 2⎣ a

⎤ s−2m2

n 2j ⎦

j=1

K s− m2

xj

⎛  ⎞  m  n 2j  ⎝2π 2 a ⎠. x j=1 j

(6.12)

The pieces (6.8), (6.9), (6.12) when put together lead to the conclusion   m  s − m2 π2 + E(s; x, a) = s− m √ a 2 x1 · · · xm (s)  1 s− m   n2 2π s 1 n 2m 2 ( 2 ) 1 + ··· + 1 m √ xm a 2 (s− 2 ) x1 · · · xm (s) n∈Zm −{0} x1 ⎛ ⎞  √ n 21 n2 + ··· + m⎠. K s− m2 ⎝2π a x1 xm

(6.13)

for Re s > m2 . The pole structure of E(s; x, a) in (6.13) is governed by that of the first term as the second term involving the K-Bessel function is entire in s—the latter function being exponentially damped for large values of its argument:  K v (r ) ∼

π −r e as r → ∞. 2r

(6.14)

The potential poles for the first term (all of which are simple) are located at the points s for which s − m2 = −,  = 0, 1, 2, 3, . . .. The corresponding residue is   m    s − m2 π2 m , lim s − ( − ) s− m √ s→ m2 − 2 a 2 x1 · · · xm (s)

(6.15)

which is computed using that the residue of (z) at − is (−1) /!. Thus for z = s − m/2, the limit in (6.15) is m

lim [z + ] (z)

z→−

1 π2  √ z a x1 · · · xm  z +

1 (−1) π 2 a  . √ ! x1 · · · xm ( m2 − ) (6.16) m

= m 2

38

6 An Inhomogeneous Epstein Zeta Function

Since 1/ (z) vanishes at z = 0, −1, −2, −3, . . ., the residues in (6.16) vanish for m −   0, say for m even. That is, if m is even then s = m2 −  is not a pole for 2   m2 : The poles are at s = m2 , m2 − 1, m2 − 2, m2 − 3, . . . , 1, for m ∈ 2Z even, and thus are finite in number. Formula (6.16) gives the residue of E(s; x, a) at a typical pole located at s = m2 − ,  ∈ Z,   0, again with 0    m2 − 1 in case m is even. In the meromorphic continuation of E(s; x, a) given by Eq. (6.13) , one can also use formula  α v √  ∞  v− 1 π  e−αt t 2 − 1 2 dt (6.17) K v (α) = 2 1  v+ 2 1     √ for Re v + 21 > 0, α > 0; see page 958 of [49] where we use that  21 = π. Using the functional equation K v (α) = K −v (α), we take v = −s + m/2 so that for (that is Re (v + 21 ) > 0) we can write for n = 0 Re s < m+1 2 ⎛ K s− m2 ⎛ ⎝

⎞  2 √ n 21 n ⎝2π a + ··· + m⎠ = x1 xm

√ 2π a



n 21 x1

n2

+···+ xmm

2

  −s +  ⇒

⎞−s+ m2 ⎠

√ π



 m+1 2

n 21 n2 + ··· + m x1 xm

∞

e

√ −2π a

1



n2 1 x1

n2

+···+ xmm t

2 −s+ m−1 2 t −1 dt

⎛

 21 (s− m2 ) K s− m2

(6.18)

⎞

 √ n 21 n2 ⎝2π a + ··· + m⎠ = x1 xm

 √  m√ 2 −s+ m−1 (π a)−s+ 2 π ∞ −2π√a nx11 +···+ nxm2m t  2 2   t e − 1 dt .  −s + m+1 1 2

We therefore write the second term in (6.13) as m+1

2π 2   s− m2 √ a x1 · · · xm (s) m+1 −s 2





∞ 1

e

√ −2π a



n 21 x1

n2

+···+ xmm t

2  m−1 −s t −1 2 dt

n∈Zm −{0}

(6.19) . If m = 1, for example, a special case that will arise in Chap. 18 for Re s < m+1 2 in the discussion of a finite temperature zeta function, this second term in (6.19) simplifies quite a bit of course. We can in fact sum a geometric series to write the integrand as ∞ √  −2π xa t n 2 −s 1 ) (t − 1) 2 (e = n=1

2e

−2π

1−e

√a

−2π

x1

t

√a x1

t

−s  −s 2 2 t2 − 1 = t −1 . (6.20) √ 2π xa t 1 − 1 e

6 An Inhomogeneous Epstein Zeta Function

39

Also (s)(1 − s) = π/ sin πs for s = 0, ±1, ±2, ±3, . . .. Thus by (6.13), (6.19), the meromorphic continuation of E(s; x, a) to the domain Re s < 1 for m = 1, x, a > 0 is given by E(s; x, a) =



1 1

a s− 2

   −s  4 sin πs ∞ t 2 − 1 dt π  s − 21 + . √a 1√ x (s) a s− 2 x 1 e2π x t − 1

(6.21)

Chapter 7

Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

The theta function

def.

θ A (z) =



e2πi z (m

2

+mn+n 2 )

(m,n)∈Z2

=1+

∞ 

(7.1) a e

2πiz

=1

 in (1.13), with A =

 21 , has Fourier coefficients a given by (1.14): 12

  a =  (m, n) ∈ Z2 | m 2 + mn + n 2 =   .

(7.2)

It is well-known, as will be reviewed in this chapter, that these coefficients can be expressed/computed in terms of a suitable Dirichlet character χ3 . A connection of the corresponding L-function L (s, χ3 ) to the Epstein zeta function E A (s) will be indicated-with a similar discussion carried out for the (simpler) theta function θ⎡

⎤ (z) def. =

2 0⎦ ⎣ 02



e2πi z (m

2

+n 2 )

(m,n)∈Z2

=1+

∞ 

(7.3) a e

2πiz

,

=1

in (4.38), with

  a = {(m, n) ∈ Z2 | m 2 + n 2 = } ;

(7.4)

see (1.10), (1.11), (1.12).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_7

41

42

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

In this second example, one of course is considering the number of integral solutions (x1 , x2 ) of the Diophantine equation x12 + x22 = , for  ∈ Z,   1. The study of which prime numbers  could be expressed as such (as a sum of two squares) was initiated by Fermat. In his most celebrated August, 1659 letter to Pierre de Carcavi, Fermat asserts that every prime number  of the form 4k + 1 can be written as a sum of two squares. However, since he basically didn’t offer proofs of his statements, one might assign to them the status of possibly being correct. Later, Euler would prove that an odd prime  could be written as a sum of two squares (indeed) if and only if  ≡ 1(mod4). Which integers   1 can be written as a sum of n squares? Fermat also studied this question for n = 3-with a solution given by Legendre in 1798. Lagrange looked at the case n = 4, and Jacobi looked at the cases n = 4, 6 and 8. Precise results involving these and other cases (for example, n = 10 or 12) will be mentioned later in this chapter. We begin with a few definitions, and examples. Definition 7.1 Fix b ∈ Z. A Dirichlet character χ(mod b) is a function χ : Z → C that satisfies the following three conditions for any m, n ∈ Z : (i) χ(n + b) = χ(n), (ii) χ(mn) = χ(m)χ(n), and (iii) χ(n) = 0 ⇔ n and b are relatively prime : (n, b) = 1. Note that since (1, b) = 1, χ(1) = 0 by (iii) so that by (ii) χ(1)χ(1) = χ(1 · 1) = χ(1) ⇒ χ(1) = 1. As an example, define χ, which will be denoted χ4 , by def.

χ(n) =



0 if n ∈ 2Z is even n−1 2 if n ∈ 2Z + 1 is odd (−1)

 .

(7.5)

Then it is directly checked that (ii) holds and that (i) holds for b = 4. If (n, 4) = 1, then n has to be odd, for otherwise 2 > 1 would be a common factor of n and 4; thus χ(n) = ±1 = 0. Conversely, if χ(n) = 0 then n is odd and if d > 0 is a common divisor of n and 4, the cases d = 2 or 4 would force n to be even. That is, d = 1, and we see that condition (iii) also follows directly so that χ4 is a Dirichlet character (mod 4). For example, χ4 (1) = 1, χ4 (2) = 0, χ4 (3) = −1, χ4 (4) = 0, χ4 (5) = 1, χ4 (6) = 0, χ4 (7) = −1, χ4 (8) = 0, χ4 (9) = 1, χ4 (10) = 0, χ4 (11) = −1, χ4 (12) = 0, . . . . . .

(7.6)

ie.

Following Eq. (4.51) we noted that for d = 2n + 1 odd, sin πd/2 = (−1)n = d−1 (−1) 2 . Since sin πd/2 = 0 for d even, we see that also

χ4 (n) = χ(n) = sin πn/2 for n ∈ Z. The next example is of a Dirichlet character χ3 (mod 3):

(7.7)

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

⎤ −1 if n ≡ −1 (mod 3) χ3 (n) = ⎣ 0 if n ≡ 0(mod 3) ⎦ . 1 if n ≡ 1(mod 3)

43

⎡

def.

(7.8)

Note that by the division algorithm n = 3q + r for a unique pair (q, r ) ∈ Z2 such that 0  r < 3. Thus r = 0, 1, or 2, where r = 0 or 1 ⇒ n ≡ 0(mod 3) or n ≡ 1(mod 3), respectively, and r = 2 ⇒ n + 1 = 3(q + 1) ⇒ n ≡ −1(mod 3). χ3 (n) = 0 ⇔ n + 1 = 3k or n − 1 = 3k for some k ∈ Z—which says that 3k + n(−1) = 1 or 3(−k) + n(1) = 1—which says that n and 3 are relatively prime—i.e. 3x + ny = 1 has a solution (x, y) ∈ Z2 . Thus χ3 satisfies condition (iii). Condition (i) is directly checked. To check condition (ii), one could test various cases. Suppose, for def. def. example, that m ≡ 1(mod 3) and n ≡ −1(mod 3) : χ3 (m) = 1 and χ3 (n) = −1. Write m − 1 = 3k, n + 1 = 3 for some (k, ) ∈ Z2 . Then mn + 1 = (3k +  − def. k)3 ⇒ mn ≡ −1(mod 3) ⇒ χ3 (mn) = −1 = χ3 (m)χ3 (n). Other cases could be tested similarly. In the end χ3 is a Dirichlet character (mod 3). χ3 (1) = 1, χ3 (2) = −1, χ3 (3) = 0, χ3 (4) = 1, χ3 (5) = −1, χ3 (6) = 0, χ3 (7) = 1, χ3 (8) = −1,

(7.9)

χ3 (9) = 0, χ3 (10) = 1, χ3 (11) = −1, χ3 (12) = 0, . . . Assume that b = 0 and set def.

Mb = max |χ( j)|

(7.10)

|χ(n)|  Mb , ∀n ∈ Z.

(7.11)

0 j 0, then χ(n) = χ(b + · · · + b + r ) = χ(r ) by applying condition (i) (χ(b + ) = χ() on Z) q times. If q < 0, −n = (−q)b − r for which, as just argued, χ(−n) = χ((−q)b − r ) = χ(−r ) since −q > 0. That is, χ(−1)χ(n) = χ((−1)n) = χ((−1)r ) = χ(−1)χ(r ) (by condition (ii)) and with −1 and b relatively prime we see that χ(−1) = 0 by condition (iii), which implies that again χ(n) = χ(r ) and that |χ(n)|  Mb . The Dirichlet L-function L(s, χ) attached to the Dirichlet character χ (mod b), say for b = 0, is given by ∞  χ(n) def. L(s, χ) = (7.12) ns n=1

44

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

for Re s > 1, where the inequality in (7.11) insures the absolute convergence of this series on this domain, and also the uniformity of its convergence of the domain Re s > 1 + δ for any δ > 0. That is, L(s, χ) is holomorphic on the domain Re s > 1. By (7.6) and (7.9) 1 1 1 1 1 + s − s + s − s + ··· , 3s 5 7 9 11 1 1 1 1 1 1 1 L(s, χ3 ) = 1 − s + s − s + s − s + s − s + · · · , 2 4 5 7 8 10 11 L(s, χ4 ) = 1 −

(7.13)

for example, for Re s > 1. For the Hurwitz zeta function given by def.

ζ(s, a) =

∞  n=0

1 (n + a)s

(7.14)

for a > 0 fixed and Re s > 1, note in particular that for Re s > 1 ∞

  ∞  1 1 2 def. 1  1 1 ζ(s, ) − ζ(s, ) = s − 3s 3 3 3 n=0 (n + 13 )s (n + 23 )s n=0  ∞   1 1 1 1 1 =1− s + s − s = − s s (3n + 1) (3n + 2) 2 4 5 n=0 +

(7.15)

1 1 1 1 − s + s − s + · · · = L (s, χ3 ) . 7s 8 10 11

by (7.13). The following theorem has a wide range of applications as it allows for the expression of important Dirichlet series as an Euler product. Let Z+ = {1, 2, 3, . . . .} = the set of positive integers, and let P = the set of positive primes. Theorem 7.1 Let f : Z+ → C and α : P → C be functions subject to the multiplicative condition f (n) f ( p) =

f (np)

  if p  n f (np) + α( p) f np if p|n

(7.16)

for (n, p) ∈ Z+ × P. Assume that f is not identically zero, and consider the Dirichlet series ∞ def.  f (n) φ f (s) = (7.17) ns n=1 defined by  f which on some subset D ⊂ C. Then  is assumed to converge absolutely  f (1) = 0, p∈P 1 + α( p) p −2s − f ( p) p −s and φ f (s) are non-zero on D, and φ f (s) has the Euler product expansion

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

φ f (s) = 

f (1) −2s − f ( p) p −s ] p∈P [1 + α( p) p

45

(7.18)

on D. Also (in particular) if f : Z+ → C is a function that is not identically zero and is completely multiplicative, in that f (mn) = f (m) f (n) for all m, n ∈ Z+ , then for any s ∈ C for which φ f (s) converges absolutely one has that φ f (s) = 

1 p∈P (1

f ( p) ) ps

−

.

(7.19)

def.

Here we take α( p) = 0 for every p ∈ P. A complete, detailed, step by step, self-contained proof of Theorem 7.1 is given in [114], for example. We consider a few examples of its application. First, for each n ∈ Z+ , there is a Hecke operator T (n) − a linear operator on the space of modular forms M(, k) of weight k. For a function f 1 (z) in M(, k) def.

(T (n) f 1 ) (z) = n

k−1







f1

d|n, d>0 a∈Z/dZ

nz + da d2



d −k ,

(7.20)

where the integers a range over a complete set of representatives for the cosets Z/dZ. As in Chap. 4, we take k to be an even integer—say k = 4, 6, 8, 10, . . .. If f 1 (z) =

∞ 

an e2πinz

(7.21)

n=0

is the Fourier expansion of f 1 (z) on π + , as in (4.5), then the series def.

L (s, f 1 ) =

∞  an n=1

ns

(7.22)

is the Hecke L-function attached to f 1 (z). One has an estimate of the form |an | < C ( f 1 , k) n k−1 for n  1, and even better |an | < C( f 1 , k)n k/2 for n  1 if f 1 is a cusp form (again meaning that a0 = 0 ) where C( f 1 , k) is some positive constant; a sketch of the proof of this is provided in [114], for example. It follows that L(s, f 1 ) converges absolutely for Re s > k. L(s, f 1 ) is holomorphic on this domain, and it admits a meromorphic continuation to C, and it satisfies an appropriate functional equation— all by Hecke theory. If f 1 is a cusp form, we have holomorphicity on the domain Re s > 1 + k/2 (by the above estimate), etc. Now getting back to Theorem 7.1, the point is the following. The case of interest is when f 1 (z) is a normalized simultaneous eigenform. This means that a1 = 1 (the normalization condition) and that f 1 (z) is an eigenfunction of all the Hecke operators T (n) : T (n) f 1 = λ(n) f 1 for some λ(n) ∈

46

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

C, and f 1 = 0. For such an eigenform, Hecke theory says that λ(n) = an for n  1 and, moreover, the Fourier coefficients an of f 1 satisfy the multiplicative condition. 

an 1 an 2 =

d k−1 an 1 n 2 /d

(7.23)

d>0 d|n 1 ,d|n 2

for n 1 , n 2  1. In particular for n ∈ Z+ and n 2 = p ∈ P a prime. Equation (7.23) simplifies to the condition  an a p =

if p  n anp anp + p k−1 anp/ p2 if p|n

 (7.24)

which in fact is the motivation for considering condition (7.16). Namely, we now def. def. obviously choose f (n) = an and α( p) = p k−1 for (n, p) ∈ Z+ × P. Then Theorem 7.1 provides for the following Euler product expansion of a Hecke L-function. Theorem 7.2 As above, suppose f 1 (z) ∈ M(, k) is a normalized simultaneous eigenform of weight k, with Fourier expansion on π + given by (7.21). Let L(s, f 1 ) be the corresponding L-function given by (7.22) for Re s > k or (even better) for Re s > 1 + k/2 if f 1 happens to be a cusp form. Then on these respective domains L (s, f 1 ) =

 p∈P

1 1+

p k−1−2s

− a p p −s

.

(7.25)

For example, η 24 (z) ∈ M(, 12) is a cusp form with Fourier coefficients given by the Ramanujan tau function τ (n) on Z+ (see (4.26)), and it is normalized since τ (1) = 1. The Hecke L-function for this form is therefore ∞    τ (n) L s, η 24 = ns n=1

(7.26)

(by (7.22)) for Re s > 1 + 12 = 7. η 24 (z) is also a simultaneous eigenform—which 2 we state without proof. By (7.25), the following beautiful formula results, although it was actually obtained first in 1917 by L. Mordell, before E. Hecke: ∞  τ (n) n=1

ns

=

 p∈P

1 1 + p 11−2s − τ ( p) p −s

(7.27)

for Re s > 7. For the Riemann zeta function ∞  1 ζ(s) = s n n−1 def.

(7.28)

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

47

def.

in (5.28), with Re s > 1, the choice f (n) = 1 on Z+ in Theorem 7.1 leads to the classical Euler product expansion 

1 1 − p −s p>0

ζ(s) =

(7.29)

(by (7.19)) for Re s > 1. This expansion was obtained by Euler in 1737—thus in fact over 120 years before Riemann. Euler also conjectured that the functional equation (5.29) for ζ(s) should hold. If χ : Z → C is a Dirichlet character (mod b), then we know that χ|Z+ is not identically zero (χ(1) = 1) and χ|Z+ is completely multiplicative by condition (ii). By (7.19) therefore, the Dirichlet L-function L(s, χ) in (7.12) (say for b = 0) has an Euler product expansion ∞  χ(n) n=1

ns

=



1 1 − χ( p) p −s p>0

(7.30)

for Re s > 1. The expression/computation of the Fourier coefficients in (7.2), (7.4) in terms of Dirichlet character, regarding initial remarks of this chapter, is herewith addressed, along with some related matters. For , n ∈ Z+ , let ⎧ ⎫ ⎨ n ⎬   def. an () =  (m 1 , m 2 , . . . , m n ) ∈ Zn | m 2j =   , ⎭ ⎩ j=1

(7.31)

which is the number of representations of  as a sum of n squares. The Fourier coefficient a in (7.4) of the theta function in (7.3) is therefore a2 () and is given by the formula   d−1 χ4 (d) = 4 (−1) 2 ; (7.32) a2 () = 4 d|, d>0

d|, d odd d>0

see (7.5). If  = 5 for example, then independently of (7.32) one computes by hand that the integral solutions (m 1 , m 2 ) of the equation m 21 + m 22 = 5 are (1, 2), (1, −2), (−1, 2), (−1, −2), (2, 1), (2, −1), (−2, 1), (−2, −1), which is 8 in number. In fact if p is any odd prime such ≡ 1(mod 4), say p − 1 ≡ 4k ∈ 4Z then formula  that p p−1   (7.32) gives a2 ( p) = 4 1 + (−1) 2 = 4 1 + (−1)2k = 8. The number of integral solutions (m, n) of the equation m 2 + mn + n 2 =  is the Fourier coefficients a in (7.2) of the theta function in (7.1) and is given by the formula  χ3 (d). (7.33) a = 6 d|, d>0

48

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

For example, a9 = 6 [χ3 (1) + χ3 (3) + χ3 (9)] = 6 by (7.9). One can compute, similarly, by hand that a1 = 6, a2 = 0, a3 = 6, a4 = 6, a5 = 0, a6 = 0, a7 = 12, a8 = 0, a10 = 0, a11 = 0, a12 = 6, a13 = 12, a14 = 0, a15 = 0, a16 = 6, a17 = 0, a18 = 0, a19 = 12, a20 = 0, for example. By (7.1) then θ⎡



⎤ (z) def. =

2 1⎦ ⎣ 12

e2πi z(m

2

+mn+n 2 )

=

(m,n)∈Z2

1 + 6q + 6q 3 + 6q 4 + 12q 7 + 6q 9 + 6q 12 + 12q 13

(7.34)

+ 6q 16 + 12q 19 + . . . , which we will see in Chap. 12 is the theta function of the hexagonal lattice; here q = e2πi z , as usual. Similarly, a2 (1) = 4, a2 (2) = 4, a2 (3) = 0, a2 (4) = 4, a2 (5) = 8(as we have seen), a2 (6) = 0, a2 (7) = 0, a2 (8) = 4, a2 (9) = 4, a2 (10) = 8,

(7.35)

a2 (11) = 0, a2 (12) = 0 ⇒ θ⎡



(7.3) ⎤ (z) def.=

2 0⎦ ⎣ 02

e2πi z(m

2

+n 2 )

(m,n)∈Z2

(7.36)

= 1 + 4q + 4q + 4q + 8q + 4q + 4q + 8q 2

4

5

8

9

10

+ ....

By (1.10), (1.11), the an () in (7.31) are the Fourier coefficients of the theta function θ A (z) for the n × n matrix ⎡ def. ⎢ A = ⎣

2

..

0 f A (x) = 2

0 .

⎤ ⎥ ⎦:

2 n  j=1

x 2j , θ A (z) =

(7.37) 

e2πi z

"n j=1

m 2j

(m 1 ,··· ,m n )∈Zn

for x ∈ Rn , z ∈ π + . Also, one can deduce the Fourier coefficient formula (7.33) directly from the Lorenz formula (1.15)—obtained later by Ramanujan, as we have ⎤ (z) are shown in (7.34). remarked. A few of these Fourier coefficients a of θ⎡ 2 1⎦ ⎣ 12 A larger set of them, say non-zero ones, are presented in the Table 7.1, for which I thank my student Miss Yuqi Zhang who set up the computer program Python to do the computation.

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff Table 7.1 Some non-zero Fourier coefficients of θ 2 1

1 2

49

 (z)



a



a



a

1 3 4 7 9 12 13 16 19 21 25 27 28 31 36 37 39 43 48 49

6 6 6 12 6 6 12 6 12 12 6 6 12 12 6 12 12 12 6 18

52 57 61 63 64 67 73 75 79 79 81 84 91 93 97 100 103 108 109 111

12 12 12 12 6 12 12 16 12 12 6 12 24 12 12 6 12 6 12 12

112 117 121 124 127 129 133 139 144 147 148 151 156 157 163 169 171 172 175 181

12 12 6 12 12 12 24 12 6 18 12 12 12 12 12 18 12 12 12 12

The C. Jacobi expression (7.32) is followed up by his formulas for a4 (), a6 (), and a8 () in (7.31). For example, the number of representations of   1 as a sum of four squares involves a sum over its positive divisors that are not divisible by 4. More precisely,  d (7.38) a4 () = 8 d|, d>0 4d

with a4 () = 8

 d|, d>0

d or 24



d,

(7.39)

d|n, d>0 d odd

according to whether  is odd or even, respectively. Since d = 1 is a positive divisor of l which is not divisible by 4, it follows that a4 () > 0. That is, every integer   1 is a sum of four squares—which was proved by C. Bachet (1621) and by J. Lagrange (1770), by elementary methods. Jacobi’s method for evaluating ak () for k = 4, 6, 8, going back to 1828, involved working out identities for [θ3 (0 | z)]k where

50

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff def.

θ3 (0 | z) =



eπin

2

z def.

= θ[2] (z/2)

(7.40)

n∈Z

by (4.49) and (4.52), and then by matching various coefficients. The Jacobi formulas for a6 () and a8 () are a6 () = 16



χ4 (/d)d 2 − 4

d| d>0

a8 () = (−1) 16





χ4 (d)d 2 ,

d| d>0

(7.41)

(−1)d d 3 ,

d| d>0

where χ4 in the Dirichlet character (mod 4) in definition (7.5). Another example is the formula of Liouville and Glaisher [44]: a12 () = (−1)−1 8



(−1)d+/d d 5 + 16ω()

(7.42)

ω()eπiz .

(7.43)

d|, d>0

where ω() is an th Fourier coefficient: η 12 (z) =

∞  =1

Note here that, apriori, η 12 (z) has a Fourier expansion η 12 (z) =



ω eπiz .

(7.44)

∈Z def.



  12 · 2z = 01

For if f (z) = η (2z), then f (z + 1) = η (2z + 2) = η  πi2/12 12 e η(2z) (by (4.23)) = e2πi η 12 (2z) = f (z), and since f (z) is holomorphic on π +  ω e2πiz (7.45) η 12 (2z) = f (z) = 12

12

12

∈Z

by (4.4). Replacing z by z/2 in (7.45) we get (7.44), and in (7.43) we see that ω() = ω for   1. The L-functions L (s, χ3 ) , L (s, χ4 ) in (7.13) are related to appropriate Epstein zeta functions. For L (s, χ4 ), the Epstein zeta function is the zeta function def.

ζR (s) =



1 , Re s > 1 r s r ∈R−{0}

(7.46)

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

51 def.

of the ring R of Gaussian integers r = m + ni, (m, n) ∈ Z2 , with norm r  = m 2 + n 2 . R is a Euclidean ring and thus every r ∈ R has a unique factorization (up to units) into primes. This leads to the fact that ζR (s) has an Euler product expansion analogous to that of the Riemann zeta function ζ(s) in (7.29). By a further analysis of the primes in R (work of Fermat and Euler) the Euler product expansion of ζR (s) reduces to the statement (for Re s > 1) ζR (s) = 4ζ(s)L(s, χ4 ), where the 4 here is the number of units in R—the units being {±1, ±i}. On the other hand   def. 1 0 def. ⇒ f A (x1 , x2 ) = x12 + x22 on R2 A = 01  1 def. ⇒ E A (s) =  s 2 m + n2 (m,n)∈Z2 −{(0,0)}

(7.47)

(7.48)

by (1.1) and (5.1), for Re s > 1, and since r  = m 2 + n 2 for r = m + ni ∈ R (as noted above) we see that E A (s) = ζR (s). That is, for Re s > 1 E A (s) = 4ζ(s)L(s, χ4 )

(7.49)

by (7.47), for E A (s) in (7.48). Similarly for def.

A =



1 1 2

def.

E A (s) =

1 2

1



def.

, f A (x1 , x2 ) = x12 + x1 x2 + x22 on R2 , 

(m,n)∈Z2 −{(0,0)}

(m 2

1 , Re s > 1 , + mn + n 2 )s

(7.50)

one has that E A (s) = 6ζ(s)L(s, χ3 )

(7.51)

for Re s > 1, where the 6 here of units in the ring R = Z[ω], ω being  is the √number  −1+ 3i 2πi/3 = , and the units being the six roots of unity the cube root of unity e 2   2 ±1, ± ω, ± ω . Equations (7.49) and (7.51) provide, in particular, for both the meromorphic continuation and functional equations of the L-functions L (s, χ4 ) and L (s, χ3 ). For example, by (5.24) and (5.29)

52

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

πs ⎡ π 1−s ⎤ (1 − s) = ⎤ (s), E E⎡ (s) ⎣ 1 0 ⎦ (1 − s) ⎣ 1 0 ⎦ 01 01

(7.52)

1−s

π 2 π s/2 ζ(1 − s) = ζ(s), (s/2) ( 1−s ) 2 which with (7.49) gives the functional equation L (1 − s, χ4 ) π 2 −s L (s, χ4 )  1−s  =  s  .  2 (1 − s) (s) 2

(7.53)

  s+1 2s−1  s   , (s) = √  2 2 π s   s π   1− = , 2 2 sin πs 2

(7.54)

1

The formulas

the first one being the duplication formula for the gamma function, can be used for an alternate expression of Eq. (7.53). Namely, the first one gives √ s 1 π2  =  1−s   (1 − s)  2  1 − 2s

(7.55)

√ 1 L (1 − s, χ4 ) π 2 −s L (s, χ4 ) π2s     1−s  =  s   (s) 2  2  1 − 2s  1−s 2 √ s 1 −s 2 π L (s, χ4 ) π2 sin πs 2   = , π  1−s 2

(7.56)

so that (7.53) reads

by the second formula in (7.54). That is,  πs  L (1 − s, χ4 ) = 2s π −s sin L (s, χ4 ) (s) 2

(7.57)

is another version of the functional equation (7.53). By (5.24), the first equation in (7.52) can be replaced by the equation πs  3 s− 21 4

(s)

E⎡

1 ⎣ 1 2

⎤ 1 (1 2⎦

1

− s) =

π 1−s E⎡ (1 − s) ⎣ 1 1 2

⎤ 1 (s). 2⎦

1

Then by (7.51) and the second equation in (7.52), we obtain the version

(7.58)

7 Dirichlet and Hecke L-Functions, Sums of Squares, and Some Other Stuff

L (1 − s, χ3 ) π 2 −s L (s, χ3 )   = 1  3 s− 2  1−s   2s (1 − s) (s) 4 2

53

1

(7.59)

of (7.53)—the argument being exactly the same as that for L (s, χ4 ), which means also that we can use (7.55) and the second equation in (7.54) (again) to write the right hand side of (7.59) as √ 1 π 2 −s L (s, χ3 ) π2s sin    1 − 2s π

πs 2

=

2s π −s L (s, χ3 ) πs  sin .  2  1 − 2s

(7.60)

Thus, similarly, the functional equation (7.59) for L(s, χ3 ) can be written as  πs  L (1 − s, χ3 ) L (s, χ3 ) . = 2s π −s sin  3 s− 21 2 (s) 4

(7.61)

Equation (7.15) can be generalized to an arbitrary Dirichlet L-function L(s, χ) to obtain its meromorphic continuation and functional equation using the Hurwitz zeta function in (7.14). Theorem 7.1 also provides for the Euler product expansion, over “good” and “bad” primes, of the Hasse–Weil L-function of an elliptic curve over the rational field.

Chapter 8

The Modular j-Invariant and Powers of Its Cube Root: Enter E8 Again

The discriminant form (z) on π + defined in (4.27) in terms of the holomorphic Eisenstein series G 4 (z) and G 6 (z) in (4.13) is used now to define the modular Jinvariant def. (8.1) J (z) = (60G 4 (z))3 /(z) on π + . (z) = (2π)12 η 24 (z) by (4.28), which is never zero. We have noted that (z) has weight 12 and that the weight of G 4 (z) is 4 so that G 4 (z)3 also has weight 12. Thus the weight of J (z) is 0, meaning that J (γ · z) = J (z) ∀ (γ, z) ∈  × π + ,

(8.2)

which says that J (z) is automorphic—not just modular. J (z) was constructed by R. Dedekind in 1877, and by F. Klein a year later. Of equal importance is the modular j-invariant def. (8.3) j (z) = (12)3 J (z) = 1728J (z) on π + . We noted also that the Fourier coefficients of (z)/(2π)12 = η 24 (z) (the Ramanujan tau function τ (n) in (4.26)) were all integers. Remarkably as well is the fact that the Fourier coefficients an of j (z) (thanks to the normalization factor 1778 in (8.3)) also are all integers: j (z) = 1e−2πi z +

∞ 

an e2πinz

(8.4)

n=0

with a0 = 744, a1 = 196884, a2 = 21493760, a3 = 864299970, a4 = 20245856256, a5 = 333202640600, a6 = 4252023300096, for example. Powers of the cube root of the j-invariant appear in the description of genus one partition functions of a holomorphic conformal field theory with central charge © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_8

55

56

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

divisible by 8 [5, 6, 39, 58, 75, 123]. Such a theory was proposed by Witten [123] to be dual to 3-dimensional gravity with a negative cosmological constant. The cube root j 1/3 also describes a vaccum character at level one of the affine Lie algebra Eˆ 8 , and it is the partition function in the case when the central charge equals 8. The focus here is on a beautiful connection of j and j 1/3 , to theta functions; see (8.7) and Theorem 8.1. Cosmological matters will be considered later—in Chap. 19. Some m/3  for m = 1, 2 are of the initial terms in the power series expansion of e2πi z j (z) computed; see Eqs. (8.29) and (8.34). By (4.52)  πi 2 def. e 4 (2n+1) z θ1 (0 | z) = n∈Z

 2 θ2 (0 | z) = (−1)n eπin z def.

(8.5)

n∈Z

def.

θ3 (0 | z) =



eπin

2

z

n∈Z

on π + . In Chap. 6 of [28], for example, it is shown that 3  1 θ1 (0 | z)8 + θ2 (0 | z)8 + θ3 (0 | z)8 J (z) = 54 θ1 (0 | z)8 θ2 (0 | z)8 θ3 (0 | z)8

(8.6)

on π + . Since 1728/54 = 32, definition (8.3) gives 3  32 θ1 (0 | z)8 + θ2 (0 | z)8 + θ3 (0 | z)8 j (z) = θ1 (0 | z)8 θ2 (0 | z)8 θ3 (0 | z)8

(8.7)

on π + . On the other hand, by Eq. (4.53) the denominator in (8.7) is 28 η 24 (z) so that 3  θ1 (0 | z)8 + θ2 (0 | z)8 + θ3 (0 | z)8 j (z) = 23 η 24 (z)

(8.8)

on π + . By (4.28) again and by the definitions (8.3), (8.1), we can also write 3  θ1 (0 | z)8 + θ2 (0 | z)8 + θ3 (0 | z)8 = 23 η 24 (z) j (z) = 23

(z) (60G 4 (z))3 (12)3 = 36 · 103 (G 4 (z))3 /π 12 . 12 (2π) (z)

(8.9)

def.

Following (4.15), we used the normalization E k (z) = G k (z)/2ζ(k) of G k (z), and we also used that ζ(4) = π 4 /90 to establish in (4.17) the result E 4 (z) = θ E8 (z)

(8.10)

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

57

on π + . Therefore in (8.9) 36 · 103 (G 4 (z))3 36 · 103 = 12 π π 12  3 = (2E 4 (z))3 = 2θ E8 (z) .



2π 4 E 4 (z) 90

3 (8.11)

That is, by (8.9) and (8.8) we see that

3  3 8 Theorem 8.1 j (z) = E 4 (z)/η 8 (z) = θ(z) on π + . Thus E 8 /η (z) θ E (z) θ1 (0 | z)8 + θ2 (0 | z)8 + θ3 (0 | z)8 E 4 (z) = 88 = 8 η (z) η (z) 2η 8 (z)

(8.12)

def.

is a cube root of j (z), where E 4 (z) = G 4 (z)/2ζ(4) is the normalized holomorphic Eisenstein series with Fourier expansion given by (4.17), η(z) is the Dedekind eta function (see (4.18)), θ E8 (z) is the theta function attached to the positive definite  3 Cartan matrix E 8 in (3.3) (see (1.5)), and the theta functions θ j (0 | z) j=1 are defined in (8.5) (also see (4.52)). def.

For q = e2πi z , z ∈ π + , (as before) j (z)1/3 has an expansion of the form

1 3

j (z) = q

− 13

1+

∞ 

 βn q

n

(8.13)

n=1

which we check, along with the computation of some of the coefficients βn . More generally, for m = 1, 2, 3, . . ., we will note that for suitable coefficients βn (m) j (z)m/3 = q −m/3

∞ 

βn (m)q n E 4 (z)m

(8.14)

n=0

on π + . We start with two definitions. A partition of a positive integer n is a finite  sequence λ = (λ1 , . . . , λr ) of positive integers λ j  λ1  λ2  λ3  . . .  λr , rj=1 λ j = n. If n = 5, for example, (λ1 , λ2 ) = (3, 2) and (λ1 , λ2 , λ3 ) = (3, 1, 1) are partitions. The remaining ones are (4, 1), (5), (2, 2, 1), (2, 1, 1, 1) and (1, 1, 1, 1, 1). Such sequences define, of course, the partition function p(n)—the number of ways of writing n as a sum of positive integers, without regard to order. Thus p(5) = 7. def. One sets p(0) = 1.  If k is also a positive integer, we consider a multipartition (1) λ = λ , . . . , λ(k) with k components where λ( j) is a partition of some integer  n j  0 for which kj=1 n j = n. For example, take n = 5, k = 2. n 1 + n 2 = 5 for (2) n 1 = 3, n 2 = 2, and λ(1) = (2, 1),  λ = (1, 1) are partitions of n 1 and n 2 that define (1) (2) of 5 with 2 components. Also for n 1 = 4, n 2 = 1, a multipartition λ = λ , λ

58

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

(1) (2) n 1 + n 2 = 5 and the  λ = (2, 1, 1), λ = (1) of n 1 , n 2 define another  partitions (1) (2) multipartition λ = λ , λ of 5 with 2 components. Define pk (n) to be the number def.

of multipartitions of n with k components; pk (0) = 1. Note that p1 (n) = p(n). pk (n) is also referred to as the number of partitions of n into k “colors”. The point of interest here is that the Euler formula ∞ 

1 n n=1 (1 − w )

p(n)w n = ∞

n=0

(8.15)

for w ∈ C with |w| < 1 (which gives the generating function for p(n)) generalizes: ∞ 

1 − w n )k (1 n=1

pk (n)w n = ∞

n=0

(8.16)

for w ∈ C with |w| < 1, which for w = q = e2πi z , z ∈ π + , gives q −k/24

∞ 

pk (n)q n =

n=0

1 η(z)k

(8.17)

by (4.18). A few terms of the power series in (8.17) are given as follows: ∞ 

pk (n)q n =1 + kq +

n=0

+

k k (k + 3)q 2 + (k + 1)(k + 8)q 3 2 6

  k k (k + 1)(k + 3)(k + 14)q 4 + (k + 3)(k + 6) k 2 + 21k + 8 q 5 + . . . . 24 120

(8.18) This provides the values of pk (n) at least for 0 ≤ n ≤ 5; pk (0) = 1, pk (1) = k. For m = 1, 2, 3, . . . choose k = 8m. Then by Theorem 8.1 and (8.17) def.

∞  E 4 (z)m − m3 =q j (z) = 8m p8m (n)q n E 4 (z)m , η (z) n=0 m 3

(8.19)

def.

which is (8.14) for βn (m) = p8m (n). By (4.17) again, E 4 (z) =

∞ 

bn q n

(8.20)

n=0

where b0 = 1, bn = 240σ3 (n), n  1

(8.21)

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

59

def.  with σ3 (n) = d>0, d|n d 3 , by (4.11). In (8.19) we see that [q j (z)]m/3 is a product of m + 1 power series, which is therefore a power series. Getting back to the cube root j (z)1/3 , we compute some initial terms of this power series—which serves also as an example for the computation of the terms of the power series for j (z)2/3 , or of j (z)m/3 —which generally would require the help of a computer. For def. def. (8.22) A(n) = p8 (n)q n , Bn = bn q n ,

 1 3

j (z) = q

− 13

∞ 

An

 ∞ 

n=0

by (8.19), (8.20) where def.

Cn =

 Bn

= q− 3 1

n=0

n 

∞ 

Cn

(8.23)

n=0

A j Bn− j

(8.24)

j=0

 is a Cauchy sum which defines an absolutely convergent series ∞ n=0 C n —the product ∞ ∞ def. of the absolutely convergent series n=0 An and n=0 Bn . pk (0) = 1, and by (8.18) p8 (1) = 8,

p8 (2) =

8 (8 + 3) = 44, 2

p8 (3) =

8 (8 + 1)(8 + 8) = 192, 6

8 (8 + 1)(8 + 3)(8 + 14) = 726, 24 8 p8 (5) = (8 + 3)(8 + 6)(64 + 21 × 8 + 8) = 2464. 120

p8 (4) =

(8.25)

Also by (8.21), bn = 240σ3 (n) for n  1, b0 = 1, where σ3 (1) = 1, σ3 (2) = 1 + 23 = 9, σ3 (3) = 1 + 33 = 28, σ3 (4) = 1 + 23 + 43 = 73, σ3 (5) = 1 + 53 = 126 ⇒ b1 = 240, b2 = 2160, b3 = 6720, b4 = 17520, b5 = 30240 ⇒

(8.26)

E 4 (z) = 1 + 240q + 2160q 2 + 6720q 3 + 17520q 4 + 30240q 5 + . . . . Then by (8.22) and (8.24) def.

Cn = q n

n 

p8 ( j)bn− j ⇒ C0 = 1,

j=0

C1 = q [ p8 (0)b1 + p8 (1)b0 ] = q[240 + 8] = 248q, C2 = q 2 [ p8 (0)b2 + p8 (1)b1 + p8 (2)b0 ] = q 2 [2160 + 8 × 240 + 44] = 4124q 2 . (8.27) Similarly, the equations in (8.25) and (8.26) give

60

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

C3 = 34752q 3 , C4 = 213126q 4 , C5 = 1057504q 5 .

(8.28)

In the end, we see by (8.23) that  1 1  j (z) 3 = q − 3 1 + 248q + 4124q 2 + 34752q 3 + 213126q 4 + 1057504q 5 + . . . . (8.29) Next, by (8.29) (or by (8.23)) write q 2/3 j (z)2/3

∞  ∞  ∞  1   1    1 1 n 3 3 3 3 q j (z) = = q j (z) Cn Cn = C n=0

for n def. C =

n 

n=0

(8.30)

n=0

C j Cn− j ,

(8.31)

j=0

as in (8.24). Then by (8.27), (8.28) 0 = C0 C0 = 1, C 1 = C0 C1 + C1 C0 = 2C1 = 496q, C 2 = C0 C2 + C1 C1 + C2 C0 = 2C2 + C12 = 2(4124q 2 ) + (248q)2 = 69752q 2 . C (8.32) Similarly, 4 = 34670620q 4 , C 5 = 394460000q 5 3 = 2115008q 3 , C C

(8.33)

and consequently  j (z)2/3 = q −2/3 1 + 496q + 69752q 2 + 2115008q 3  +34670620q 4 + 394460000q 5 + . . .

(8.34)

by (8.30). A holomorphic conformal field theory (CFT) is also called an extremal conformal field theory (ECFT). j (z)2/3 in (8.34) is the partition function for such a theory with central charge c equal to 16 = 8 × 2. Even though such partition functions are constructed for c ∈ 8Z, the existence of the corresponding ECFT is generally quite def. an open issue. For c = 8 × 3 = 24 however, where the partition function is Z 1 (z) = j (z) − 744 (744 being the Fourier coefficient a0 in (8.4)), the corresponding ECFT is constructed in the ground-breaking paper [38]. Using Z 1 (z) with the help of the Hecke operators T (n) in (7.20) (where the weight k there is taken to be zero as Z 1 (z) is modular invariant), one can set up the ECFT partition functions Z  (z),  = 1, 2, 3, . . ., for c = 24. In Chap. 10 we consider asymptotics of the Fourier coefficients of Z  (z), which we use to compute quantum corrections to black hole entropy.

8 The Modular j-Invariant and Powers of Its Cube Root: Enter E 8 Again

61

The normalization factor 1728 in (8.3) plus 13 is the famous “taxi-cab” number 1729, which goes back to Hardy’s 1919 account of riding in a taxi numbered 1729 to visit Ramanujan at a hospital in Putney (England). On hearing of this number from Hardy (for whom it had no meaning), Ramanujan recognized its significance immediately—as the smallest number expressible as a sum of two positive cubes in two different ways: (8.35) 1729 = 13 + 123 = 93 + 103 . 2 = 13 + 13 is a smaller number but it is expressible as a sum of two positive cubes in only one way. Many numbers larger than 1729 can be expressed as a sum of positive cubes in different ways. For example 87539319 = (167)3 + (436)3 = (228)3 + (423)3 = (255)3 + (414)3 .

(8.36)

The number “8” has been somewhat special in this chapter. It occurs in the spectacular formula (8.7) for the j-invariant. The theta function of the exceptional Lie algebra E 8 , which is part of the Cartan-Killing classification in Table 3.1 of Chap. 3, occurs in Theorem 8.1—in an expression for the cube root of j (z), and it coincides with the Eisenstein series E 4 (z), as shown in (4.17). The affine Lie algebra Eˆ 8 and its product Eˆ 8 × Eˆ 8 have vacuum characters j (z)1/3 , j (z)2/3 at level 1 that are ECFT partition functions with central charge c = 8, 8 × 2, respectively. Speaking of “8”, there is an interesting little formula for this special whole number. Recall that for x ∈ R, [x] is the largest integer that does not exceed x-notation used in the definition of the Dedekind sums in (4.20). Then for def.

√

x =

3 log

√ π 142

√ 10+11 2 2

+

√ , 10+7 2 2

√

8 = [x]. This chapter therefore is approriately called “Chapter 8”.

(8.37)

Chapter 9

Modular Forms of Non-positive Weight: Exact Formulas and Asymptotics of Their Fourier Coefficients

In addition to the holomorphic Eisenstein series G k (z) in (4.13) of positive weight k ∈ {4, 6, 8, 10, . . .} and the j-invariant in (8.3) of weight 0, we have considered the Dedekind eta function η(z) in (4.18), which by (4.19) could be regarded as a form of weight 21 —with a multiplier eπiχ(γ) . More to the point, after a precise definition  1/η(z) would be a premier example of a modular form of negative weight is set up, = − 21 . Rademacher and Zuckerman developed in [91] an exact formula for the Fourier coefficient an of such forms. We present, in particular, a new formula for the coefficient a0 that greatly simplifies the Rademacher-Zuckerman formula; see Theorem 9.3. For forms of weight zero there is also an exact formula for their Fourier coefficients—due to Brisebarre and Philibert [13]. These exact formulas are reviewed, along with some asymptotics of Fourier coefficients. Since ∞  1 = e−πi z/12 p(n)e2πinz η(z) n=0

(9.1)

by (8.17), the Rademacher-Zuckerman (R-Z) formula provides for, in particular, an exact formula for the partition function p(n) due to Rademacher [90]. We close this chapter with the construction of logarithmic/sub-leading corrections to black hole entropy that extend results in [32]; see (9.79). Consider now a holomorphic function F(z) on π + such that (i) F(z + 1) = def. e2πiα F(z) for some fixed α ∈ R with 0  α < 1. Then f (z) = e−2πiαz F(z) is holomorphic on π + , and satisfies f (z + 1) = f (z). Therefore by (4.4) F(z) has a Fourier expansion  an e2πinz (9.2) F(z) = e2πiαz f (z) = e2πiαz n∈Z

on π + . Assume (ii): There is some integer μ ≥ 1 such that a−n = 0 for n > μ; that is

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_9

63

64

9 Modular Forms of Non-positive Weight … ∞ 

F(z) = e2πiαz

an e2πinz

(9.3)

n=−μ

on π + . Thirdly, assume (iii): There is some function ε˜ :  = S L(2, Z) → {w ∈ C | |w| = 1}, a multiplier for F(z),and some r > 0 such that (iii) F(γ · z) = ε(γ)(cz ˜ + ab −r + ∈  with c > 0, where d) F(z) on π for γ = c d 0 < arg(cz + d) < π.

(9.4)

Definition 9.1 A holomorphic function F(z) on π + that satisfies the conditions (i), (ii) and (iii) above, for some α ∈ R with 0  α < 1, for some integer μ  1, and for some r > 0, is a modular form (with respect to ) of negative −r , with  weight  ab multiplier ε(γ)—which ˜ is also written as  ε(a, b, c, d) for γ = ∈ . c d def.

1 on π + . Then by (4.24), F(z + 1) = As an initial example, take F(z) = η(z) 1 def. 1 23 e−πi/12 /η(z) = e2(1− 24 )πi F(z) ⇒ condition (i) holds for α = 1 − 24 = 24 ∈ [0, 1). Write Eq. (9.1) as ∞  1 = e−πi z/12 p(n + 1)e2πi(n+1)z η(z) n=−1 ∞ 

= e2(1− 24 )πi z 1

p(n + 1)e2πinz

(9.5)

n=−1

= e2πiαz

∞ 

p(n + 1)e2πinz ,

n=−1 def.

which shows that condition (ii) holds  forμ = 1, and that p(n + 1) is the nth Fourier ab coefficient an of 1/η(z). For γ = ∈  with c > 0, Eq. (4.19) gives c d F(γ · z) = ε(γ)(cz ˜ + d)− 2 F(z)

(9.6)

= e−πiχ(γ) = e−πi ( 12c − 4 −s(d,c)) ε(γ) ˜

(9.7)

1

for

def.

def.

a+d

1

by (4.20) since c > 0. Condition (iii) therefore holds also and by Definition 9.1 we see (as expected) 1/η(z) is a modular form with negative weight −r = − 21 . It’s multiplier  ε(γ) is given by Eq. (9.7). We should note that forms of negative weight −r in Definition 9.1 are called forms of positive dimension r in [91].

9 Modular Forms of Non-positive Weight …

65

For the next example, take def.

F(z) = Fk (z) =

1 , k ∈ {1, 2, 3, . . .} η(z)k

(9.8)

where the case k = 1 has just been covered. The largest integer [x] that does not exceed x ∈ R satisfies the inequalities x − 1 < [x]  x. In particular this says that −x + 1 > −[x], or x − [x] < 1. Choose x = −k/24 : −k/24 − [−k/24] < 1 and −k/24 − [−k/24]  0 since x − [x]  0. That is def.

α = −k/24 − [−k/24] ⇒ 0  α < 1.

(9.9)

Moreover since −[−k/24] ∈ Z ⇒ e−2πi[−k/24] = 1, Eq. (4.24) can be applied again to give k F(z + 1) = e−πik/12 F(z) = e2πi (− 24 −[−k/24]) F(z) (9.10) def. 2πiα = e F(z), which is condition (i). By (8.17) F(z) = e−πikz/12

∞ 

pk (n)e2πinz .

(9.11)

n=0

[x] is an integer  x so if x < 0, then [x] < 0 ⇒ [x]  −1 ⇒ −[x]  1. For x = −k/24 again def. (9.12) μ = −[−k/24] ≥ 1 and we can write (9.11) as F(z) = e−πikz/12

∞ 

pk (n + μ)e2πi(n+μ)z

n=−μ

= e2πi(−k/24+μ)z

∞ 

pk (n + μ)e2πinz

(9.13)

n=−μ

= e2πiαz

∞ 

pk (n + μ)e2πinz

n=−μ def.

(since α = −k/24 + μ by (9.9)), which is condition (ii), where the Fourier coefficient an of F(z) = Fk (z)  is pk (n + μ) for n ≥ −μ. ab Again for γ = ∈  with c > 0, we appeal to the Eqs. (4.19) and (4.20) to c d establish condition (iii):

66

9 Modular Forms of Non-positive Weight …

F(γ · z) = e−πikχ(γ) (cz + d)−k/2 F(z),

(9.14)

def.

which by Definition 9.1 shows that F(z) = Fk (z) = 1/η(z)k is a modular form of negative weight −r = −k/2 (for k = 1,2,3, ….) with multiplier  ε(γ) = e−πikχ(γ) = e−πik ( 12c − 4 −s(d,c)) def.

a+d

1

(9.15)

by (4.20) since c > 0. The Fourier expansion of F(z) is given in (9.13), where the parameters α, μ are defined in (9.9), (9.12): μ = −[−k/24], α = −k/24 + μ. As discussed in Chap. 4, pk (m) is the number of partitions of an integer m ≥ 0 into k def. colors; pk (0) = 1. The next example involves the normalized Eisenstein series def.

E 8 (z) = 1 +

∞ (2πi)8  σ7 (n)e2πinz ζ(8)7! n=1

(9.16)

in (4.16). By the special value of zeta formula ζ(2n) =

(−1)n+1 (2π)2n B2n 2(2n)!

(9.17)

discovered by L. Euler in 1736, where B j is the jth Bernoulli number defined by ∞

 Bj w = wj w e −1 j! j=0

(9.18)

for |w| < 2π, and where B8 = −1/30, ζ(8) =

(2π)8 , 60 × 8!

(9.19)

and the factor of the series in (9.16) is therefore 480: E 8 (z) = 1 + 480

∞ 

σ7 (n)e2πinz .

(9.20)

n=1

We will take def.

F(z) =

E 8 (z) . η(z)24

(9.21)

Since E 8 (z) and η(z)24 are modular forms of weight 8 and 12, respectively, (see (4.25)), they satisfy condition (M1) in (4.3):

9 Modular Forms of Non-positive Weight …

67

E 8 (γ · z) = (cz + d)8 E 8 (z), η(γ · z)24 = (cz + d)12 η(z)24 ⇒ F(γ · z) = (cz + d)−4 F(z) 

ab for γ = c d



def.



∈ , and in particular for γ = T =

(9.22)

 11 in (4.2) 01

F(z + 1) = F(z),

(9.23)

which is condition (i) above with α = 0, whereas (9.22) is condition (iii) with r = 4, and with the trivial multiplier ε(γ) ˜ = 1 on . To finish the argument that F(z) is modular of negative weight −r = −4 (given (9.22)) we need to check condition (ii) also. By the choice k = 24 in (9.11), we can write ∞

 1 = e−2πi z An 24 η(z) n=0 for

def.

An = p24 (n)q n , q = e2πi z . Similarly by (9.20), E 8 (z) =

∞ 

Bn

(9.24)

(9.25)

(9.26)

n=0

for

def.

B0 = 1,

def.

Bn = 480σ7 (n)q n , n  1 ⇒  ∞ ∞ ∞  1  1 An Bn = Cn F(z) = q n=0 q n=0 n=0 where def.

Cn =

n 

A j Bn− j =

j=0

n 

An− j B j

(9.27)

(9.28)

j=0

in a corresponding Cauchy sum—as in (8.24). A0 = 1, B0 = 1 ⇒ C0 = A0 B0 = 1. n  1 ⇒ ⎡ ⎤ n n   An− j B j = ⎣ p24 (n) + 480 p24 (n − j)σ7 ( j)⎦ q n . C n = An + j=1

j=1

(9.29)

68

9 Modular Forms of Non-positive Weight …

In other words, for def.

def.

a−1 = 1, an = p24 (n + 1) + 480

n+1 

p24 (n + 1 − j)σ7 ( j)

j=1

(9.30)

Cn+1 Cn+1 = n+1 , n  0, an q n = , n  −1 q q and we see that by (9.27) and (9.30) ∞ ∞  1  F(z) = Cn+1 = an q n , q n=−1 n=−1

(9.31)

which is condition (ii) with μ = 1, and which completes the argument that F(z) = E 8 (z)/η(z)24 is by Definition 9.1 a modular form of negative weight −r = −4, with α = 0, μ = 1, and with trivial multiplier  ε(γ) = 1 on . A statement of the exact R-Z formula for the Fourier coefficients an of F(z) in (9.3) requires some preliminary definitions/notation. The modified Bessel function of the first kind of order v ∈ C is defined by  v  ∞ (t/2)2k t Iv (t) = 2 k=0 k!(v + k + 1) def.

(9.32)

for t > 0. Here one could replace t by w ∈ C with |arg w| < π. We will be looking at a sum over pairs of relatively prime integers k, h with 0  h < k : (h, k) = 1. In particular, the congruence hh  ≡ −1(mod k) can be solved for h  ∈ Z. Namely, choose x, y ∈ Z  xh + yk = 1. Then h(−x) + 1 = yk so that h  = −x is a solution, and moreover hh  + 1 /k = y ∈ Z so that in fact def.



γk,h =

   h  − hh  + 1 /k ∈ k −h

(9.33)

since det γk,h = 1. Assigned to k ∈ Z, k > 0, 0  α < 1, (u, v) ∈ C2 is the generalized Kloosterman sum   −1 −2πi ((u−α)h  +(v+α)h ) (9.34) ε γk,h e k Ak,u (v; α) = Ak (v, u; α) = 0h 0 since both n, α  0 and not both are 0. Also j − α > 0 since 1 1 j ≥ 1 > α ⇒ ( j − α) 2 , (n + α) 2 > 0. Thus definition (9.32) is needed only for t > 0.   def. 0 −1 The asymptotic result, for S = in (4.2), 10 1

an

1

4π(μ−α) 2 (n+α) 2 a−μ r 1 e ∼ √ ε(S)(μ − α) 2 + 4 r 3 2 (n + α) 2 + 4

n→∞

(9.37)

References [64, 91] is known for a−μ = 0. It depends on the trivial inequality  Ak, j (n; α)  k and the asymptotic result lim

t→∞

√

2πt Iv (t)e−t = 1

(9.38)

for the modified Bessel function Iv (t) in (9.32). As an example, consider F(z) = 1/η(z)k again for k ∈ Z, k  1 − a form of negative weight −r = −k/2, with Fourier development (9.13) for μ = −[−k/24], α = −k/24 + μ, an = pk (n + μ), n  − − −μ, and with multiplier  ε(γ) = e−πikχ(γ) in (9.14). By (4.24), χ(S) = −πik/4 πik/4 e = 1. μ − α = k/24, r2 + 14 = k+1 , r2 + −1/4 so that by (9.35), ε(S) = e 4 def.

= k+3 , and a−μ = pk (0) = 1. Using this data, letting n → ∞ and thus replacing 4 n + α, n + μ by n, one gets by (9.37) 3 4

1 pk (n) ∼ √ 2 n→∞



k 24

 k+1 4

e4π( 24 ) k

n

k+3 4

1 2

1

n2

1 =√ 2



k 24

 k+1 4

eπ

√2

n

3 kn

k+3 4

,

(9.39)

which for k = 1 reduces to the 1918 Hardy-Ramanujan asymptotic formula for the partition function:

70

9 Modular Forms of Non-positive Weight …

√ 2n eπ 3 p(n) ∼ √ . 4 3n n→∞

(9.40)

The asymptotic result (9.39) also follows by a very general 1954 theorem of Meinardus [79]. The result (9.40) was also obtained independently by J.V. Uspensky in 1920 [99]. An exact Rademacher formula for pk (n) follows by Theorem 9.1. Suppose, for example, that 1  k  24. Then μ = −[−k/24] = 1 and α = −k/24 + 1. Since pk (n + μ) = an for n  −μ = −1, pk (n) = an−μ for n − μ  −μ, or for n  0. pk (1) = k (see (8.18)) and for n  2, n − μ = n − 1  1. That is, not both n − μ and α are zero, and therefore by Theorem 9.1 an exact formula for pk (n) = an−1 def. follows by way of the following data. a−1 = pk (0) = 1, r + 1 = k2 + 1, 1 − α = k/24 ⇒ n − 1 + α = n − k/24:    k4 + 21    21 ∞  k/24 k A,1 (n − 1; α) 4π k n− pk (n) = 2π I k2 +1  n − k/24  24 24 =1    k+2 1 ∞  4 k A,1 (n − 1; α) 4π (k(24n − k)) 2 = 2π I k2 +1  24n − k  24 =1 (9.41) for 1  k  24, n  2, where def.

A,1 (n − 1; α) =



e

πik 4

eπikχ(γ,h ) e−

2πi 



k ( kh 24 +(n− 24 )h)

(9.42)

0h 2; see (4.11). In the sum over a, d in (9.53) d is an inverse of a mod c. One has that Ak, j (n; 0) = (−1)r/2 S(−n, j; k) (9.55) where (again) r/2 ∈ Z for α = 0 by the integrality condition (9.52). Since r > 0—ie. r + 2 > 2, and also since r is even r ∞  σ−1−r ( j)(−1) 2 2(r + 2)! S(0, j; k) σ−1−r ( j) = = k r +2 ζ(r + 2) (2π)r +2 Br +2 k=1

(9.56)

by (9.54) and (9.17). Now σv (n) = n v σ−v (n) since d > 0 runs through the divisors of n as n/d does. By (9.55) and (9.56) then (again since r is even) j r +1

∞  Ak, j (0; 0) j r +1 σ−(1+r ) ( j)2(r + 2)! = k r +2 (2π)r +2 Br +2 k=1

2σ1+r ( j)(r + 2)(r + 1)! = . (2π)r +2 Br +2

(9.57)

By Theorem 9.2 we have therefore derived Theorem 9.3 For α = 0 in Theorem 9.1, μ

a0 =

2(r + 2)  a− j σr +1 ( j), Br +2 j=1

(9.58)

which is a much simpler version of the R-Z formula (9.45). In particular if μ = 1, then

9 Modular Forms of Non-positive Weight … Table 9.1 The non-zero Bernoulli numbers upto 1 B1 = − 21 B8 = − 30 5 B2 = 16 B10 = 66 1 691 B4 = − 30 B12 = − 2730 1 7 B6 = 42 B14 = 6

a0 =

73 B30 B16 B18 B20 B22

= − 3617 510 = 43867 798 = − 174611 330 = 854513 138

2(r + 2) a−1 . Br +2

B24 B26 B28 B30

= − 236364091 2730 = 8553103 6 = − 23749461029 870 = 8615841276005 14322

(9.59)

A motivation for our proof of Theorem 9.3, for an arbitrary modular form F(z) of negative weight −r , was the specific example F(z) = 1/η(z)k for k ∈ 24Z considered in Chap. 5 of [76], where the notation χ, pχ (χ/24) there corresponds to k, a0 here, with pχ (χ/24) computed in Eq. (5.12) of [76]. Thus I express credit to J. Manschot and G. Moore for this example that will be reviewed shortly. The Bernoulli numbers B j 0  j  30 were computed by Euler—all of the odd ones > 1 being equal to 0; B0 = 1. def. Here are two examples of Theorem 9.3—the first being F(z) = 1/η(z)k , k = 24,  ∈ Z, in [76] as previously mentioned. By (9.9), (9.12) α = − − [−] = 0 and μ = −[−] = . We know that r = k/2. By (9.58) 2(k/2 + 2) a0 = Bk/2+2

k/24 

a− j σk/2+1 ( j),

(9.60)

j=1

which is a simplified version of Eq. (5.12) in [76], where (for example) the special value ζ(k/2 + 2) of zeta appears which we have already evaluated (in (9.56)) in terms of the Bernoulli number Bk/2+2 . Actually we can say a bit more. Namely since α = 0, a− j in (9.60) is given by a− j = pk (k/24 − j), 1  j  k/24,

(9.61)

by (9.13). In particular for k = 24, Bk/2+2 = B14 = 7/6 by Table 9.1, μ =  = 1, 2 24 +2 def. a = p (0) = 1, and a = ( 2 ) · 1 = 24. −1

k

0

7/6

def.

For the second example of Theorem 9.3, we re-visit the example of F(z) = E 8 (z)/η(z)24 in (9.21), where the weight is −r = −4 and where α = 0, μ = 1. By 1 by Table 9.1 so by (9.59), a0 = 2 × 6 × 42 = 504. (9.30), a−1 = 1. Br +2 = B6 = 42 On the other hand, also by (9.30) a0 = p24 (1) + 480 p24 (0)σ7 (1) = 24 + 480 (by (8.18)) = 504. In the initial remarks of this chapter, it was asserted that an exact formula for the Fourier coefficients of a modular form F(z) of zero weight was also available—due to Brisebarre and Philibert [13]. Here is their result.

74

9 Modular Forms of Non-positive Weight …

Theorem 9.4 Suppose F(z) has the Fourier expansion on π + F(z) =

∞ 

an e2πinz .

(9.62)

n=−μ

for some μ ∈ Z, μ  1 (as in (9.3)). Then for n  1.   μ ∞   4π  S(n, − j; k) 2π  an = √ I1 a− j j jn , k k n j=1 k=1

(9.63)

where S(n, − j; k) is a classical Kloosterman sum defined in (9.53). Now S(−a, −b; c) = S(a, b; c) ⇒ S(n, − j; k) = S(−n, j; k) = Ak, j (n; 0) by (9.55) for the zero weight r . Therefore (9.63) corresponds exactly to the R-Z formula (9.36) on taking r = 0 and α = 0 there, given that (9.62) corresponds to α = 0 in (9.3). Theorem 9.4 was proved by Rademacher [89] in the case where F(z) = j (z)— the j-invariant in (8.3). If, moreover, cm (n) denotes the coefficient of e2πinz in the Fourier expansion of j (z)m (m = 1, 2, 3, . . .), then it is also established in [13, 84] that 1

√

m 4 e4π mn . cm (n) ∼ √ 2n 3/4 n→∞

(9.64)

In particular for m = 1, (9.64) reduces to the classical asymptotic result for an in (8.4) √ 4π n n→∞ e def. (9.65) an = c1 (n) ∼ √ 2n 3/4 of Petersson [88]—which was also obtained independently by Rademacher [89]. This chapter is closed out with a simple idea borrowed from [121], with a quick application. Further applications, to black hole entropy, for example, would be pursued in Chap. 10. Let C = 0 be a complex number and let {tn }∞ n=1 be a sequence ∞ of positive numbers such that (A1) limn→∞ tn = ∞. Also let {Bn }∞ n=1 , {An }n=1 be a sequence of non-zero complex numbers. Then the asymptotic statements (A2) √ n→∞ n→∞ An 2πtn Iv (tn ), for any choice v  0 are equivaBn ∼ C An etn , (A3) Bn ∼ C√ lent. Iv (t) > 0 for v  0 and etn / 2πtn Iv (tn ) → 1 as n → ∞ by (9.38) and (A1). Thus if (A2) is true—ie. Bn /C An etn → 1 as n → ∞, then so is (A3): Bn = √ C An 2πtn Iv (t)



Bn C An etn

Conversely given (A3), (A2) follows since



etn √ 2πtn Iv (tn )

 → 1.

(9.66)

9 Modular Forms of Non-positive Weight …

an = C An etn



an √ C An 2πtn Iv (tn )

75

 √

2πtn Iv (tn ) etn

 → 1,

(9.67)

again by (9.38) and by (A1). The usefulness of (A3), in contrast to (A2), is that the asymptotic expansion ∞ (v, k) et  n→∞ Iv (t) ∼ √ (−1)k (2t)k 2πt k=0

(9.68)

of the Bessel function Iv (t) can be exploited. Here the Hankel symbols (v, k) are defined by       2 − 12 2 − 32 2 − 52 · · · 4v 2 − (2k − 1)2 4v 4v 4v (v, k) def. def. , k  1. (v, 0) = 1, k = 2 8k k!

(9.69)

Now fix an arbitrary integer m  1 and let def.

χm (n; v) =

m  k=1

(−1)k

(v, k) (v, m + 1) + (−1)m+1 . k (2tn ) 2m+1 tnm+1

(9.70)

Since tn → ∞ as n → ∞ by (A1), we regard χm (n; v) as very small for large n and we therefore use the approximation 1 + χm (n; v) eχm (n;v) . In fact, given definition (9.70), we express the expansion in (9.68) as 

∴

2πtn Iv (tn ) etn (1 + χm (n; v)) = etn +χm (n;v)

(9.71)

for large n, and assuming (A2) (which we know is equivalent to (A3)), the point of it all is the approximation (by (A3), (9.71)) Bn C An etn +χm (n;v)

(9.72)

for large n, with m  (v, k) (−1)m+1 (v, m + 1) (−1)k k k + 2 tn 2m+1 tnm+1 k=1       m  (−1)k 4v 2 − 12 4v 2 − 32 4v 2 − 52 · · · 4v 2 − (2k − 1)2 def. = tnk 8k k! k=1

χm (n; v) =

+ O(

1 tnm+1

) (9.73)

76

9 Modular Forms of Non-positive Weight …

by (9.69). In the end, in (9.72), we have simply tacked on the factor eχm (n;v) to the asymptotics in (A2). However, in some practical physical situations it is the logarithm of Bn in fact which is of interest, where the χm (n; v) serve as correction terms, for example, to the leading contribution tn to the entropy expressed in terms of log Bn . def. As a concrete example, consider again F(z) = 1/η(z)k with negative weight def. −r = −k/2, and with Fourier coefficient an = pk (n + μ) for n  −μ = [−k/24]. def. Choose Bn = pk (n) so that Bn

1 ∼ √ 2

n→∞



k 24

 k+1 4

eπ

√2

n

3 kn

(9.74)

k+3 4

by (9.39), which is the statement (A2) for 1 C = √ 2 def.



k 24

 k+1 4

,

1

def.

An =

n

k+3 4

 def.

, tn = π

2 kn, 3

(9.75)

and which is the statement (16.57) on page 360 of [125], for example, where our notation k, n corresponds to b, N there—b being the number of transverse directions of a vibrating string. No discussion of negative weight modular forms, however, appears in the chapter there (on string thermodynamics and black holes), where the entropy formula for large n ∴

Sk = k B log pk (n) = k B log Bn

(9.76)

is derived; here k B is Boltzman’s constant. By (9.72) therefore, log Bn tn + log C + log An + χm (n; v)

(9.77)

 def. with tn = π 23 kn = a leading contribution to the entropy, and with sub-leading terms χm (n; v). For example, choose m = 4 and v = k/2 + 1 (= the weight +1 of F(z)). Since 1 def. log C = − log 2 + 2



 k+1 k (k + 3) def. log , log An = − log n, 4 24 4

4v 2 = k 2 + 4k + 4 we see that by definition (9.73)

(9.78)

9 Modular Forms of Non-positive Weight …

77

  k+1 k (k + 3) Sk 1 log − log n

tn − log 2 + kB 2 4 24 4   2    2 k + 4k + 3 k + 4k + 3 k 2 + 4k − 5 − + 8tn 2!82 t 2  2  2 n   2 k + 4k + 3 k + 4k − 5 k + 4k − 21 − 3!83 tn3  2     2   k + 4k + 3 k + 4k − 5 k 2 + 4k − 21 k 2 + 4k − 45 1 + +O 5 4!84 tn4 tn (9.79)  2kn for large n, where tn = π 3 . The computation in (9.79) also generalizes that ∴

in √ [32], for example, where the special case k = 24 is considered, with tn = th 2 4π n.Moreover,  we think that our 6 term (675 × 667)/ 128 × 16π n = (675 × 2 667)/ 2048π n in (9.79) (for k = 24) corrects the 5th term in [32] that reads (−675 × 9)/2048π 2 n.

Chapter 10

More on Logarithmic Corrections to Black Hole Entropy

In Chap. 8 some brief comments were made regarding partition functions Z k (z)(k = 1, 2, 3, . . .), of a holomorphic or extremal conformal field theory (ECFT) with central charge c = 24k. It was pointed out that such a theory was constructed for k = 1 in the ground-breaking paper of I. Frenkel, J. Lepowsky, and A. Meurman (FLM) [38], def. where Z 1 (z) = j (z) − a0 for a0 = 744—the Fourier coefficient in (8.4) and that Z k (z), k  2, could be defined in terms of Z 1 (z) with the help of Hecke operators, which will be done in this chapter. This is not to say however that given Z k (z), there exists a corresponding ECFT. It was shown in [39], for example, that for k = 2 (c = 48) there exists no self-dual ECFT with “monster symmetry”. Such symmetry does exist, as an important feature in fact, in the FLM construction—meaning that the states of the theory transform as a representation of the finite, simple group G called the “monster” (or the friendly giant) as it is so very large, containing about 1054 elements: G is the Fischer—Griess group with 246 · 320 · 59 · 76 · 112 · 132 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 elements. One can define, more generally, ECFT partition functions Z 8m (z) (m = 1, 2, 3, . . .) for central charges c = 8m divisible by 8. As was pointed out in Chap. 8, for m = 1, 2, for example, Z 8 (z) and Z 16 (z) are given by j (z)1/3 and j (z)2/3 in (8.29) and (8.34), respectively. Also pointed out there was that in this section the asymptotics of the Fourier coefficients of Z k (z) would be considered as this relates to the computation of correction terms to black hole entropy (in 3d gravity). These coefficients, which will be denoted by bk,n (since they also depend on k), amazingly are all whole numbers—like the Ramanujan—Fourier coefficients τ (n) in (4.26), and the Fourier coefficients an in (8.4). The bk,n in fact are positive and they count the number of states with Virasoro energy L 0 = n. The BekensteinCardy-Hawking black hole entropy (to which we present logarithmic corrections by √ way of log bk,n , for large n) is given by S = 4π kn = 2π cLo , since c = 24k. Once 6 Z k (z) and the bk,n are defined, the main effort of this chapter is to establish Theorem 10.1 which gives the asymptotic formula

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_10

79

80

10 More on Logarithmic Corrections to Black Hole Entropy √

bk,n

ke4π kn ∼ √ 2(kn)3/4

n→∞

(10.1)

for any fixed k. We start with the holomorphic sector     1 − e2πi z eπi z/12 def. 1 − e2πi z  = ∞  Z hol (z) = 2πinz η(z) n=1 1 − e def.

∞ 

∞

 1 1 = = 2πinz 1−e 1 − qn n=2 n=2

(10.2)

of the one-loop gravity partition function def. 1−loop Z gravity (z) = Z hol (z) Z¯ hol (z)

(10.3)

def.

on π + , [75, 114], page 336] where q = e2πi z as usual and where the bar “-” in (10.3) denotes complex conjugation. Define Z 0 (z) = q −k Z hol (z) def.

(10.4)

on π + . Then by (9.1) ∞ ∞   eπi z/12 = p(n)q n ⇒ Z 0 (z) = q −k (1 − q) p(n)q n η(z) n=0 n=0

= =

∞ 

p(n)q n−k −

n=0 ∞ 

∞ 

p(n)q n−k+1

(10.5)

n=0

p(r + k)q r −

r =−k

∞ 

p(r + k − 1)q r ,

r =−k

def.

where we use that p(−1) = 0. That is, Z 0 (z) =

∞ 

ar (k)q r

(10.6)

r =−1

for

def.

ar (k) = p(r + k) − p(r + k − 1), r  −k.

(10.7)

Next, the definition in (7.20) for the action of a Hecke operator T (n), n  1, on a modular form f 1 (z) of positive weight also works for a form of weight zero:

10 More on Logarithmic Corrections to Black Hole Entropy



(T (n)Z 1 ) (z) = n −1 def.



81

 Z1

nz + da d2





d|n,d>0 a∈Z/dZ

,

(10.8)

which, in particular, we can write as d−1  

def.

n (T (n)Z 1 ) (z) =

Z1

d|n,d>0 a=0

nz + da d2

.

(10.9)

Using the definition of ar (k) in (10.7) we can now define (again for k = 1, 2, 3, . . .) Z k (z): def.

Z k (z) = a0 (k) +

k 

a−r (k)r (T (r )Z 1 ) (z)

r =1

= p(k) − p(k − 1) +

k 

( p(k − r ) − p(k − r − 1)) r (T (r )Z 1 ) (z).

r =1

(10.10) Clearly (T (1)Z 1 ) (z) = Z 1 (z) in (10.9), and thus Z k (z) in (10.10) does reduce to def. def. def. Z 1 (z) for k = 1 there, again since p(−1) = 0, and p(0) = 1, p(1) = 1. That is, the definition in (10.10) extends that of the level 1 FLM partition function to level k. The computation of the Fourier expansion of Z k (z) requires that of (T (n)Z 1 ) (z), by (10.10). By (8.4) again, write def.

Z 1 (z) = j (z) − a0 = e

−2πi z

+

∞ 

am e2πimz

m=1

=e

−2πi z

+

∞ 

cm e

2πimz

(10.11) def.

def.

; c0 = 0, cm = am , m  1.

m=0

Although def.

f 1 (z) =

∞ 

cm e2πimz

(10.12)

m=0

is not a weight 0 modular form, the same argument for computing the Fourier expansion of (T (n) f )(z) for such a form f (z) works to show that (T (n) f 1 ) (z) =

∞  m=0

for

cm(n) e2πimz

(10.13)

82

10 More on Logarithmic Corrections to Black Hole Entropy

cm(n) =

 1 c mn . d d2 d|n,d|m

(10.14)

d>0

See the proof of Theorem 6.6 in [4], for example, where the main fact used is that d−1 

e2πima/d =

a=0

 d if d|m , 0 if d  m

(10.15)

and where the weight k there is taken to be 0. Going back to (10.9), (10.11) and (10.12), we see that (T (n)Z 1 ) (z) =

d−1 1  −2πi nz2  2πi(−1) a d + (T (n) f ) (z) d e e 1 n d|n a=0

(10.16)

d>0

1 = e−2πinz + (T (n) f 1 ) (z), n by (10.15) again. That is, by definition of the cm , m  0, in (10.11), and by (10.13), (10.14) ∞  1 cm(n) e2πimz (10.17) (T (n)Z 1 ) (z) = e−2πinz + n m=1 for cm(n) =

def.

 1 a mn , n  1, d2 d d|n,d|m

(10.18)

d>0

where the a are the Fourier coefficients in (10.11)—of the j—invariant. The notation cm(n) used here should not cause confusion with respect to the notation cm (n) used in Chap. 9 for the Fourier expansion of j (z)m . The Fourier expansion of Z k (z) now follows. By (10.10) and (10.17) Z k (z) = a−k (k)q

−k

+ · · · + a−2 (k)q

−2

+ a−1 (k)q

−1

+ a0 (k) +

∞ 

bk,m q m

m=1

(10.19) for def.

bk,m =

k 

a−r (k)r cm(r ) , m  1 .

r =1

Here a−k (k) = 1, by the definition in (10.7).

(10.20)

10 More on Logarithmic Corrections to Black Hole Entropy

83

Proposition 10.1 For r  1 fixed in (10.20), √

m→∞ cm(r ) ∼

e4π r m . √ 2(r m)3/4

(10.21)

To see this, we use the result √

e4π n an = √ 2n 3/4

 1−

3 .055 , n  1, √ + εn , |εn |  n 32π n

(10.22)

which follows from Theorem 1.1 in [13], where one takes m = 1 in the notation there—where the assumption n  1000 can be dropped by way of a comment of the authors there. (10.22), in particular implies (9.65) of course. Let √ √ def. c∞ () = e4π  /( 23/4 ) for   1. Then for d > 0 such that d|r , d|m √ √ rm  3d e4π d d 3/2 2(r m)3/4 √ 1 − =√ + ε √ m,r,d c∞ (r m) 32π mr 2(mr )3/4 e4π r m  d 3/2 3d = 1− + εm,r,d √ √ 4π r m (1− d1 ) 32π rm e

a mr d2

where

2 εm,r,d  .055d . mr

For d  2, 1 − 1/d  1 −

1 2

=

(10.23)

(10.24)

>0⇒

1 2

lim

m→∞

1

e

√ 4π r m(1− d1 )

=0.

(10.25)

Therefore write  1 a mr2 cm(r ) def.  1 a mr amr d2 d = + , = c∞ (r m) d c (r m) c (r m) d c (r m) ∞ ∞ ∞ d|m,d|r d|m,d|r d>0

(10.26)

d2

where each a mr /c∞ (r m) → 0 as m → ∞ for d  2, with r fixed, by (10.23), (10.24), d2 and (10.25). Also for d = 1, (10.23) and (10.24) give .055 amr 3 + εm,r,1 , εm,r,1  =1− . √ c∞ (r m) mr 32π r m

(10.27)

Thus indeed by (10.26) and (10.27) we see that cm(r ) =1 m→∞ c∞ (r m) lim

(10.28)

84

10 More on Logarithmic Corrections to Black Hole Entropy

√ √ def. for c∞ (r m) = e4π r m / 2(r m)3/4 , which proves Proposition 10.1, which in turn we can use to establish (10.1).

Theorem 10.1 For bk,m defined in (10.20) √

m→∞

bk,m ∼

1

√

ke4π km k 4 e4π km = √ √ 2(km)3/4 2m 3/4

(10.29)

for each k  1 fixed. def.

For k = 1, b1,m = cm(1) , in which case Theorem 10.1 follows by taking r = 1 in Proposition 10.1. Assume therefore that k > 1. For c∞ () defined prior to (10.23)  bk,m cm(r ) kcm(k) def. = a−r (k)r + kc∞ (km) kc∞ (km) kc∞ (km) r =1 k−1

(10.30)

where cm(k) /c∞ (km) → 1 as m → ∞ by (10.28) which is Proposition 10.1 for r = k there. Theorem 10.1 will follow by (10.30) if we can therefore check that cm(r ) =0 m→∞ c∞ (km)

(10.31)

lim

for 1  r  k − 1. If d|r , d|m, d > 0, then similar to (10.23) (by way of (10.22)) a rdm2 c∞ (km)

= =

d 3/2 ( rk )3/4 e4π e4π

√

3d + εm,r,d √ 32π r m  3d 1− + εm,r,d √ 32π r m

√ rm d

km

d 3/2 ( rk )3/4 √ √ √ e4π m( k− r /d)

 1−

(10.32)

  √ where εm,r,d  .055d 2 /mr as in (10.24), and where d  1, r < k ⇒ r /d  √ √ r< k⇒ lim

1

√ √ m→∞ e4π m( k− r /d)

Thus, by (10.18)

√

= 0 ⇒ lim

m→∞

a rdm2 c∞ (km)

 1 a r m2 cm(r ) d = =0 m→∞ c∞ (km) d c (km) ∞ d|r,d|m lim

= 0.

(10.33)

(10.34)

d>0

for 1  r < k, which is (10.31), and which completes the proof of Theorem 10.1. Our effort to establish Theorem 10.1 was motivated by the interest expressed in Chap. 3 of [123] to determine the behavior of bk,m for large k and m, with m/k fixed. Evidently towards that end, Theorem 10.1 offers a more precise result. We relied of

10 More on Logarithmic Corrections to Black Hole Entropy

85

course on the useful estimate (10.22) from [13]. As an application, again fix k and let √ √ def. def. def. 1 tn = 4π kn, An = 3/4 , C = k 1/4 / 2, n (10.35) def. Bn = bk,n , n  1 Then tn → ∞ as n → ∞, which is the statement (A1) in Chap. 9, and (10.29) is the n→∞ asymptotic statement (A2) there: Bn ∼ C An etn , which we know is equivalent to the statement (A3). That is, by (9.72), for large n (and k fixed) log bk,n  tn + log C + log An + χm (n; v) √ 1 1 3 def. = 4π kn + log k − log 2 − log n + χm (n; v), 4 2 4

(10.36)

where we are free to choose m and v. For example if we arbitrarily choose m = 4 and v = 1, then by (9.73) −(4 − 1) (4 − 1)(4 − 9) (4 − 1)(4 − 9)(4 − 25) + − 8tn 82 · 2!tn2 83 · 3!tn3  (4 − 1)(4 − 9)(4 − 25)(4 − 49) 1 + +O 5 4 4 8 · 4!tn tn  15 315 14175 1 −3 − 2 − 3 − 4 +O 5 . = 2 3 4 8tn 8 · 2!tn 8 · 3!tn 8 · 4!, tn tn

χ4 (n; 1) =

(10.37)

√ The initial term 4π kn in (10.36) is the Bekenstein-Cardy-Hawking black hole entropy S—as was pointed out in a remark before (10.1). Actually, this is the holomorphic sector BTZ black hole entropy as there is a similar contribution to entropy by way of an antiholomorphic sector. The terms 41 log k, − 21 log 2, etc. in (10.36) √ that follow S = 4π kn are correction terms to S where, for example, χ4 (n; 1) in (10.37) offers further (very small) corrections. The first 4 terms in (10.36), and no other terms (as in (10.37)), appear in (3.18) of [123]; also compare (5.15) of [75]. The derivation of (10.36) presented here, based on (9.72), is simpler and more direct than the approach presented in [117]. The BTZ black hole (after M. Ba n ados, C. Teitelboim, and J. Zanelli [7]) is described in Chap. 19, for example. The partition functions Z k (z) can be computed directly by (10.19). Another method for doing this is by way of the Faber polynomials Fn (x) = det[x1n − Mn ]

(10.38)

of degree n  1, where 1n is the identity n × n matrix, and for the an in (8.4)

86

10 More on Logarithmic Corrections to Black Hole Entropy

⎧ 0 ⎪ ⎪ ⎪ 2a ⎪ 1 ⎪ ⎪ ⎪ 3a ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ 4a3 def. Mn = ⎪ 5a4 ⎪ ⎪ ⎪ ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (n − 1)an−2 ⎪ ⎩ nan−1

1 0 a1

0 1 0

0 0 1

0 0 0

a2

a1

0

1

a3 .. .

a2 .. .

a1 .. .

0 .. .

⎤ 0 0⎥ ⎥ 0⎥ ⎥ .. ⎥ .⎥ ⎥ .. ⎥ . .⎥ ⎥ ⎥ 1 ⎥ ⎥ 0 1⎦ a1 0

··· 0 0 0 .. . .. .

an−3 an−4 an−5 an−6 an−2 an−3 an−4 an−5

(10.39)

For example, ⎡ ⎤ 

0 1 0   0 1 M1 = 0 , M2 = , M3 = ⎣ 2a1 0 1 ⎦ , 2a1 0 3a2 a1 0 ⎤ ⎡ 0 1 0 0 ⎢ 2a1 0 1 0 ⎥ ⎥ M4 = ⎢ ⎣ 3a2 a1 0 1 ⎦ ⇒ 4a3 a2 a1 0 F1 (x) = x,

F2 (x) = x 2 − 2a1 ,

(10.40)

F3 (x) = x 3 − 3a1 x − 3a2 ,

F4 (x) = x 4 − 4a1 x 2 − 4a2 x + 2a12 − 4a3 . The values a1 = 196884, a2 = 21493760, a3 = 864299970 are given following (8.4) so that F2 (x) = x 2 − 393768, F3 (x) = x 3 − 590652x − 64481280,

(10.41)

F4 (x) = x − 787536x − 85975040x + 74069419032. 4

2

One can replace x by the translate x − a0 , a0 = 744, to obtain the polynomials of degree n def. def. (10.42) φn (x) = Fn (x − 744); n  1, φ0 (x) = 1, which are also of importance. In fact if def.

jn (z) = φn ( j (z)), n  1,

def.

j0 (z) = 1

(10.43)

then it turns out [124] that jn (z), n  1, is the unique holomorphic weight zero form on π + with Fourier expansion of the form jn (z) = q −n +

∞  m=1

An (m)q m .

(10.44)

10 More on Logarithmic Corrections to Black Hole Entropy

87

By (10.40), (10.41) φ1 (x) = x − 144, φ2 (x) = x 2 − 1488x + 159768, φ3 (x) = x 3 − 2232x 2 + 1069956x − 36866976, φ4 (x) = x 4 − 2976x 3 + 2533680x 2 − 561444608x + 8507424792 ⇒ def.

j1 (z) = j (z) − 144 = Z 1 (z), j2 (z) = j (z)2 − 1488 j (z) + 159768, j3 (z) = j (z)3 − 2232 j (z)2 + 1069956 j (z) − 36866976, j4 (z) = j (z)4 − 2976 j (z)3 + 2533680 j (z)2 − 561444608 j (z) + 8507424792. (10.45) Also (10.46) jn (z) = n (T (n) j1 ) (z), n  1. That is, rephrasing (10.46), by (10.45), (10.43), and (10.42) we see that def.

def.

n (T (n)Z 1 ) (z) = jn (z) = φn ( j (z)) = Fn ( j (z) − 744) = Fn (Z 1 (z)),

(10.47)

which by (10.10) leads to the expression Z k (z) = a0 (k) +

k 

a−r (k)Fr (Z 1 (z)) ,

(10.48)

r =1

where by (10.7) def.

a−r (k) = p(k − r ) − p(k − r − 1), 0  r  k; a−k (k) = 1.

(10.49)

For k = 2, a0 (2) = p(2) − p(1) = 2 − 1 = 1, a−1 (2) = p(2 − 1) − p(2 − 2) = 1 − 1 = 0, a−2 (2) = 1, and (10.41) gives Z 2 (z) = 1 + F2 (Z 1 (z)) = Z 1 (z)2 − 393767.

(10.50)

Similarly, for k = 3, a0 (3) = p(3) − p(2) = 3 − 2 = 1, a−1 (3) = p(2) − p(1) = 1, a−2 (3) = p(1) − p(0) = 0, a−3 (3) = 1 ⇒ (by (10.41)) Z 3 (z) = 1 + F1 (Z 1 (z)) + F3 (Z 1 (z)) = 1 + Z 1 (z) + Z 1 (z)3 − 590652Z 1 (z) − 64481280 = Z 1 (z) − 590651Z 1 (z) − 64481279 3

(10.51)

88

10 More on Logarithmic Corrections to Black Hole Entropy

since F1 (x) = x. For k = 4, a0 (4) = p(4) − p(3) = 5 − 3 = 2, a−1 (4) = p(3) − p(2) = 3 − 2 = 1, a−2 (4) = p(2) − p(1) = 2 − 1 = 1, a−3 (4) = p(1) − p(0) = 1 − 1 = 0, and a−4 (4) = 1 ⇒ Z 4 (z) = 2 + F1 (Z 1 (z)) + F2 (Z 1 (z)) + F4 (Z 1 (z)) = 2 + Z 1 (z) + Z 1 (z)2 − 393768 + Z 1 (z)4 − 787536Z 1 (z)2 − 85975040Z 1 (z) + 74069419032

(10.52)

= Z 1 (z)4 − 787535Z 1 (z)2 − 85975039Z 1 (z) + 74069025266, again by (10.41). Formulas (10.50) and (10.51) for Z 2 (z) and Z 3 (z) agree with (3.8) in [123], where Z 1 is denoted by J there—notation that is avoided here because of definition (8.1). We in [117], feel, however, that formula (10.52) for Z 4 (z) corrects the one in [123]. Also √ √ n→∞ there are two formulas that need to be corrected: the assertion cn ∼ ke4π n / 2n 3/2 √ √ n→∞ should read cn ∼ e4π n / 2n 3/4 —which is really the statement (9.65) here as the cn in [117] are the an here. Secondly, the exact formula (33) in [117] should read bk,n



 √ ∞ 4π r n r  Am (n, r ) I1 = 2π a−r (k) n m=1 m m r =1 k 

(10.53)

 √   √  where I1 4πm r n here corrects the expression I1 4πmdr n there. Equation (10.53) follows by Theorem 9.4 here, where Am (n, r ) = S(n, −r ; m), given the Fourier expansion (10.19) of Z k (z), and that Z k (z) is of weight 0 (since ((T (r )Z 1 ) (z) is of weight 0 in (10.10)). Of course formula (10.48) also shows that Z k (z) is of weight 0. Another correction/misprint needed in [117] is that in the definition of Z k (z) in (10) there where the sum is over r from 1 to ∞—instead of from 1 to k as in (10.10) here.

Chapter 11

A Dedekind Type Eta Function Attached to the Hecke Group 0 (N)

Suppose now that  denotes any discrete subgroup of G = S L(2, R) with a fundamental domain having a finite G—invariant volume with respect to the measure y −2 d xd y on π + .  is called a Fuchsian group of the first kind. To each cusp κ of , L.Goldstein in [47, 48] has attached a Dedekind type eta function η(κ) (z) on π + and, among other results, he has obtained an appropriate corresponding Kronecker (κ) (z) at κ— limit formula. Such a formula of course involves an Eisenstein series E ,s defined initially for Re s > 1 and then continued meromorphically to the full plane. The main result in [47, 48] is a transformation formula for η(κ) (z) that generalizes the classical Dedekind formula (4.19) of Chap. 4, where  = S L(2, Z ) and where κ is the cusp at infinity. Such a formula therefore also involves the construction of generalized Dedekind sums S(κ) (γ), γ ∈ , attached to  and κ. In particular, these results can be quite well explicated in the case when  = 0 (N ) (γ) which was in (4.6), for example, along with a proof of the rationality of S(∞) 0 (N ) work done by my student I. Vassileva in [101]. She also derived an expression for the eta function at a general cusp κ in terms of the cusp at infinity. As was mentioned in Chap. 4, some of that work will be considered in this section. of The action of G on π + defined in (4.1) extends to an action   G on the one-point ab def. compactification R∞ = R ∪ {∞} of R: For x ∈ R, g = ∈G c d   ax+b ∈ R if cx + d = 0 def. , g · x = cx+d ∞ if cx + d = 0 (11.1)  a if c = 0 . g·∞= c ∞ if c = 0 An element γ ∈ G is called parabolic if γ = ±1 and trace γ = ±2. Assume  ⊂ S L(2, Z) and for κ ∈ R∞ set def.

κ = {γ ∈  | γ · κ = κ},

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_11

(11.2)

89

90

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

which is the stabilizer or isotropy subgroup of  at κ. Then  S L (2, Z)∞ =

ab c d



    ∈ S L(2, Z) c = 0 .

(11.3)

Definition 11.1 κ ∈ R∞ is a cusp of  if κ contains a parabolic element. If κ ∈ R∞ is a cusp of , so is γ · κ for any γ ∈ , and if κ = ∞ then one knows that κ ∈ Q = the field of rational numbers. For example, take  = 0 (N ). Then  and S L(2,  Z) have the same stabilizer at ∞ (namely S L(2, Z)∞ in (11.3)), and the 11 element ∈ ∞ is parabolic, which says that ∞ is a cusp of 0 (N ). Also let 01 n κ = m ∈ Q, where n, m ∈ Z, m = 0, and where N | m 2 . Let def.



γ =

1 + mn −n 2 m 2 1 − mn

 ∈ S L(2, Z),

(11.4)

since det γ = 1; that is γ ∈ 0 (N ) is a parabolic element: N | m 2 and trace γ = 2. By (11.1), (1 + mn) mn − n 2 n def. = = κ, (11.5) γ·κ = (m 2 ) mn + 1 − mn = 1 m which shows that n/m is a rational cusp of 0 (N ). If N = 4, for example, we could take κ = −1/2 = mn , since N | m 2 , m 2 = 4. In the argument just presented we also see that κ = 0 = N02 is also a cusp of 0 (N ). Two cusps κ1 , κ2 ∈ R∞ are equivalent if k1 = γ · k2 for some γ ∈ . Of interest is the number ν∞ () of inequivalent cusps of . It has been noted that, in particular, κ = − 21 , 0, and ∞ are cusps of  = 0 (4). These are inequivalent, and in fact ν∞ (0 (4)) = 3—a matter that we shall get back to later in this chapter. With the focus now on the cusp of 0 (N ) at infinity, we turn to some basic results in [101]. Again let Z+ denote the set of positive integers. The Möbius function μ : Z+ → {0, −1, 1} is defined as follows: For n ∈ Z+ written as a product of primes p j , say n = p1 . . . pr , def.



μ(n) =

(−1)r if the p j are distinct 0 if some p j is repeated

 ,

(11.6)

def.

μ(1) = 1. Throughout, p will denote a positive prime. The Dedekind type eta function attached to 0 (N ) at the cusp ∞ is defined by def.

η N (z) =

def. η(∞) (z) = 0 (N )

  N z a N μ(d)d −1 η d d|N d>0

(11.7)

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

on π + , where



def.

aN =

p|N

p def. , a1 = 1, p−1

91

(11.8)

and where η(z) is the classical Dedekind   eta function in (4.18). Next, the Dedekind ab sums S N (γ) of level N for γ = ∈ 0 (N ) are defined by c d def. (a + d)

S N (γ) = S∞0 (N ) (γ) = def.

12c

N

 p+1 1 c

c − − aN μ(δ)−1 s(d, |c|δ/N ) p 4 |c| |c| δ|N δ>0

p|N

(11.9) for c = 0, a N in (11.8), and for def.

s(h, k) =

   k−1

jh 1 j jh − − , k > 0, k k k 2 j=1

(11.10)

the classical Dedekind sum (as in (4.20), (4.21)). For c = 0  SN For N = 1

±1 b 0 ±1



def.

= ±

bN  p + 1 . 12 p|N p

 p + 1 def. = 1, p p|N

(11.11)

(11.12)

and clearly η N (z) = η(z), S N (γ) = χ(γ) in (4.20); also see (4.23).   ab Theorem 11.1 For γ = ∈ 0 (N ) with c = 0 and z ∈ π + c d (∞)

η(∞) (γ · z) = eπi S0 (N ) (γ) (cz + d) 2 η(∞) (z). 0 (N ) 0 (N )    p+1 bN ±1 b (z). η(∞) · z = e± 12 p|N p πi η(∞) 0 (N ) 0 (N ) 0 ±1 1

(11.13)

See definitions (11.9) and (11.11). Also every S(∞) (γ), γ ∈ 0 (N ), is a rational 0 (N ) number—in accord with other rationality results in [47, 48]. Note that if

 def.  φ(N ) =  m ∈ Z+ | m  N , (m, N ) = 1    N =N 1 − p −1 = aN p|N

(11.14)

92

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

is the Euler phi function, for a N in (11.8), then since N /d runs through the divisors N as d > 0 does, we can also express (11.7) as η∞0 (N ) (z) =

def.



η(dz) φ(N ) μ( d ) . d

N

(11.15)

d|N d>0

The eta function η(∞) (z) also satisfies an appropriate generalized Kronecker 0 (N ) limit formula, which will be the statement of Theorem 11.2 below. For this we start with the nonholomorphic Eisenstein series E N ,s (z) = E (∞) (z) = 0 (N ),s def.

def.

1 2

(m,n)∈Z2 (m N ,n)=1

ys |m N z + n|2s

(11.16)

on π + , at the cusp ∞, for Re s > 1, where y = Im z. If N = 1, for example, then ζ(2s)E 1,s (z) =

1 ys

def. 1 ∗ = E (s, z) (m,n)∈Z2 −{(0,0)} |mz + n|2s 2 2

(11.17)

(∞) where E ∗ (z, z) is the nonholomorphic Eisenstein series in (5.2); E 1,s (z) = E SL(2,Z),s (z). A nice formula of Moreno [83] expresses the Eisenstein series for 0 (N ) at ∞ in terms of that of S L(2, Z) at ∞:

(z) = N −s E (∞) 0 (N ),s

 p|N



Nz 1 −s (∞) . μ(d)d E SL(2,Z),s 1 − p −2s d|N d

(11.18)

d>0

In particular, by (11.17), 2ζ(2s)E (∞) (z) 0 (N ),s

=N

−s

 p|N



Nz 1 −s ∗ , μ(d)d E s, 1 − p −2s d|N d

(11.19)

d>0

which provides for the meromorphic continuation of E (∞) (z) as a function of 0 (N ),s s to all of C—since we have shown in (5.26) the meromorphic continuation of E ∗ (s, z) in terms of a suitable Epstein zeta function. Moreover, by way of the Fourier expansion (5.36) of E ∗ (s, z), one can use (11.19) to work out the Fourier expansion (z) at the cusp ∞. The result is that for of E (∞) 0 (N ),s

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

93

 πs 1 y 1/2 π s def. for N = 1, = ζ(2s)(s) p|N 1 − p −2s ζ(2s)(s)  ζ(2s − 1)(s − 21 )  1 − p −1 1 s 2 N 1−2s E (∞) y 1−s + (z) = y + π 0 (N ),s −2s ζ(2s)(s) 1 − p p|N def.

1

Pz (s) = (Im z) 2

p=prime>0

2Pz (s)N

−2s+1

∞



μ(d)d

−1 s− 21

n



K s− 21 (2πny)σ1−2s

n=1 d|N N d |n d>0

nd N



 2πinx  e + e−2πinx ,

(11.20) again where y = Im z, K v (r ) is the K-Bessel function in definition (6.11), ζ(s) is of course the Riemann zeta function, σk () is the divisor function in (4.11), and μ(n) is defined in (11.6). The Fourier expansion of E N ,s (z) in (11.20) also provides for its meromorphic continuation as a function of s. However, this is done more simply by way of the Moreno formula (11.19). Again by (5.26), and by the remark following √ (5.22), E ∗ (s, z) has only a simple pole at s = 1 with residue there equal to yπ/ det A(z) = π, which is therefore independent of z. The factor  p|N

1 1 − p −2s

(11.21)

in (11.19) also contributes a pole to E N ,s (z) at s = 0. s = 0 and 1 are its only poles since s → ζ(2s) is holomorphic at s = 0 and at s = 1: ζ(0) = − 21 , ζ(2) = π 2 /6. The residue of E N ,s (z) = E (∞) (z) at s = 1, for example, is by (11.19) 0 (N ),s   

Nz 1 1 ∴ −s ∗ (s − 1)E s, · N −s μ(d)d = −2s s→1 2ζ(2s) 1 − p d p|N d|N lim

d>0

⎛

⎞

μ(d) 1 6 1  ⎝ ⎠π 2 −2 2π N p|N 1 − p d d|N d>0

by Mobius inversion

=

 φ(N ) by (11.14) 3 1  p 2  p − 1 3 1  p2 = π N p|N p 2 − 1 N π N p|N p 2 − 1 p|N p =

(11.22)

3 1  p , π N p|N p + 1

which is independent of z. On the other hand, the volume of a fundamental domain for 0 (N ), again with respect to the invariant measure y −2 d xd y on π + is given by

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

94

 πN  p + 1  . vol 0 (N )\π + = 3 p|N p

(11.23)

Thus, as is well-known, this volume is the reciprocal of the residue at s = 1 in (11.22) of the Eisenstein series E (∞) (z). 0 (N ),s The Kronecker limit formula (KLF) in [101] is given as follows. Theorem 11.2 Define def.

αN =

3  p def. 3 = 2 for N = 1 π 2 N p|N p + 1 π

 1 (∞) = residue of E 0 (N ),s (z) at s = 1 in(11.22) , π ⎛ ⎞



log p  (1) ζ (2) ⎟ 6  p ⎜ − log N − − βN = 2 ⎝− ⎠ 2 π N p|N p + 1 p −1 (1) ζ(2) p|N

(11.24)

p=prime>0

def. 6 = 2 π

  (1) ζ (2) − − for N = 1. (1) ζ(2)

Then 



 2 1 1   β N − α N log 2 − α N log y 2 η(∞) (z)  . 0 (N ) s→1 2 (11.25)

(1)/ (1) is the Euler–Mascheroni constant C = lim In (11.24), − n→∞   2 1 + 21 + · · · + n1 − log n  0.577215665, ζ(2) = π6 , ζ (2)  −0.937548254. For N = 1, the KLF in Theorem 11.2 differs from the more standard version:    (1) π 1 ∗ 2 2 (11.26) = 2π − − log 2 − log y |η(z)| . lim E (s, z) − s→1 s−1 (1) lim

1 (∞) αN E (z) − 2π 0 (N ),s 2(s − 1)

=

For a positive prime p, φ( p) = p − 1 and μ( p) = −1 in (11.6). Therefore by (11.15) η p (z) =  and for γ =

η(∞) (z) 0 ( p)

ab c d

= η(z)

1 p−1 (−1)

η( pz)

p p−1

 =

η( pz) η(z)

p p−1

η(z),

(11.27)

 ∈ 0 ( p) with c = 0

S p (γ) = S(∞) (γ) = 0 ( p)

 1 c c p |c| c s(d, |c|) a+d ( p + 1) − − s d, + 12c 4 |c| |c| p − 1 p |c| p − 1 (11.28)

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

by (11.9), whereas

 Sp

±1 b 0 ±1

 =±

b( p + 1) 12

95

(11.29)

by (11.11). Since p|c, write c = c1 p, c1 ∈ Z and let def.



γ1 =

 a pb ∈ S L(2, Z) c1 d

(11.30)

since γ ∈ S L(2, Z). Then p(az + b) def. apz + pb = = p(γ · z) ⇒ c1 pz + d cz + d p  η( p(γ · z)) p−1 η(γ · z)(by (11.27)) η p (γ · z) = η(γ · z)  p  η (γ1 · ( pz)) p−1 = η(γ · z) η(γ · z) p   p−1 1 eπiχ(γ1 ) (c1 pz + d) 2 η( pz) 1 = eπiχ(γ) (cz + d) 2 η(z) 1 πiχ(γ) 2 e (cz + d) η(z) def.

γ1 · ( pz) =

(11.31)

(by (4.19) for c = 0) =e

πi



pχ(γ1 )−χ(γ) p−1



1

(cz + d) 2 η p (z).

On the other hand, by Theorem 11.1 for c = 0 η p (γ · p) = eπi S p (γ) (cz + d)1/2 η p (z) so it follows that e e

πi



pχ(γ1 )−χ(γ) p−1

 2πi



pχ(γ1 )−χ(γ) p−1



S p (γ) −

(11.32)

= eπi S p (γ) ⇒



= e2πi S p (γ) ⇒  pχ(γ1 ) − χ(γ) = n ∈ Z. p−1

(11.33)

Now for c > 0, c1 is also positive and a+d 1 − − s (d, c1 ) , 12c1 4 1 a+d − − s(d, c) χ(γ) = 12c 4

χ(γ1 ) =

(11.34)

96

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

where 1 = det γ = det γ1 = ad − bc = ad − c1 pb ⇒ ad ≡ 1(mod c), ad ≡ 1 (mod c1 ) ⇒ s(a, c) = s(d, c), s(a, c1 ) = s(d, c1 ), and where p/( p − 1) = 1 + 1/( p − 1). (11.35) Therefore pχ (γ1 ) − χ(γ) χ (γ1 ) − χ(γ) = χ (γ1 ) + = p−1 p−1    1 a+d a+d − s (a, c1 ) + s (a, c) = − χ (γ1 ) + 12c1 12c p−1 χ (γ1 ) + δ p (a, c, d)/( p − 1)

(11.36)

where for a, c, d, c1 ∈ Z with pc1 = c > 0 and ad − bc = 1 for some b ∈ Z 

 a+d a+d ⇒ − s (a, c1 ) − 12c 12c1 S p (γ) = χ (γ1 ) + δ p (a, c, d)/( p − 1) + n def.

δ p (a, c, d) =

s(a, c) −

(11.37)

by (11.33). Now the point of all of this is that by Theorem 3.11 of [4] (page 66), for example, 24 (11.38) δ p (a, c, d) ∈ 2Z for c > 0 p−1 and for p = 3, 5, 7 or 13 (ie. ( p − 1) | 24, p − 1 = 24). Also by remark (iii) following (4.30), 12χ(γ) ∈ Z for every γ ∈ S L(2, Z). By (11.37) therefore we see that 24S p (γ) ∈ 2Z for c > 0, and for p = 3, 5, 7 or 13.

(11.39)

In case c < 0, consider γ −1 =



 d −b , −c > 0. −c a

(11.40)

Now if N |c and δ|N , say δ1 δ = N for δ1 ∈ Z, then we can write |c| c N = (±b) δ1 δ, 1 = ad − bc ⇒ N N  |c| |c| δ ⇒ ad − 1 = (±b)δ1 δ ⇒ ad ≡ 1 mod N N |c| |c| s(d, δ) = s(a, δ) N N bc = b

and therefore by (11.9), for c = 0,

(11.41)

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

 p + 1 1 (−c)   def. d + a N − S N γ −1 = 12(−c) p|N p 4 |c|  |c| (−c)

aN μ(δ)δ −1 s a, δ = −S N (γ). − |c| N

97

(11.42)

δ|N δ>0

  That is, 24S N (γ) = −24S N γ −1 with −c > 0 ⇒ (11.39) also holds for c < 0( taking N = p). Of course if c = 0, then by (11.29)  24S p

±1 b 0 ±1

 = ±2b( p + 1) ∈ 2Z

(11.43)

for every prime p > 0, so we have verified   ab Theorem 11.3 Let γ = ∈ 0 ( p) where p > 0 is a prime such that p − 1 = c d 24, ( p − 1)|24: p = 3, 5, 7 or 13. Then the generalized Dedekind sums S p (γ) = (γ) in (11.28), (11.29)(which are special cases of (11.9), (11.11)) satisfy S(∞) 0 (P) 24S p (γ) ∈ 2Z for c = 0 ⇒ eπi S p (γ) = a 24th root of unity.

(11.44)

If c = 0, then (11.44) holds for any p > 0. It was remarked earlier that the number ν∞ (0 (4)) of inequivalent cusps of 0 (4) was 3—a matter which we said would be revisited. In general 0 (N ) has ν∞ (0 (N )) =

d|N d>0

φ((d,

N )) d

(11.45)

in equivalent cusps, where (n, m) is the greatest common divisor of n, m ∈ Z, and φ(n) is the Euler phi function in (11.14). Thus for N = 4, d = 1, 2, 4, N /d = 4, 2, 1, (1, 4) = 1, (2, 2) = 2, (4, 1) = 1, and φ(1) = 1, φ(2) = 1 ⇒ ν∞ (0 (4)) = 1 + 1 + 1 = 3. For another example, take N = 36: d = 1, 2, 3, 4, 6, 9, 12, 18, 36, N /d = 36, 18, 12, 9, 6, 4, 3, 2, 1, (1, 36) = 1, (2, 18) = 2, (3, 12) = 3, (4, 9) = 1, (6, 6) = 6, (9, 4) = 1, (12, 3) = 3, (18, 2) = 2, and (36, 1) = 1. Then by (11.45), the number of inequivalent cusps of 0 (36) is φ(1) + φ(2) + φ(3) + φ(1) + φ(6) + φ(1) + φ(3) + φ(2) + φ(1) = 1 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 1 = 12

(11.46)

In particular 0 ( p) has 2 inequivalent cusps: ∞ and 0. The remarks that follow, concluding this chapter, go beyond the thesis [101] and are thus independent thereof. We indicate how to construct a Maass waveform f N ,k (z)

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

98

by way of η N (z) = η(∞) (z). Here for a real number k, and for the Maass operator 0 (N ) k given by 2 ∂ ∂2 def. ∂ def. + . (11.47) k = y 2  − iky ,  = ∂x ∂x 2 ∂ y2 f N ,k (z) will satisfy



 k k f N ,k (z) = 2



k − 1 f N ,k (z) 2

(11.48)

on π + , and will also satisfy the modular property f N ,k (γ · z) = e2πik SN (γ) 

for

ab γ= c d

(cz + d)k f N ,k (z) |cz + d|k

(11.49)

 ∈ 0 (N ), c = 0

(11.50)

and for S N (γ) given in (11.9). Apart from anything having to do with any eta function, consider first an arbitrary holomorphic function F(z) = u(x, y) + iv(x, y) on π + . By the Cauchy–Riemann equations ∂v ∂u ∂v ∂u = , =− (11.51) ∂x ∂y ∂y ∂x u(x, y), v(x, y) are harmonic functions: u = 0, v = 0. Immediately, y

∂ y k/2 ∂x

k/2

(11.52)

=0⇒

k = 2



 k k k k/2−2 k/2 −1 y − 1 y k/2 . ⇒ k y = 2 2 2

(11.53)

Given two functions α(x, y), β(x, y) on π + , ∂α ∂β ∂α ∂β +2 ∂x ∂x ∂y ∂y

(11.54)

f (z) = y k/2 F(z) = y k/2 [u(x, y) + iv(x, y)],

(11.55)

(αβ) = αβ + βα + 2 by the product rule. Now define def.

and choose α(x, y) = y k/2 , β(x, y) = u(x, y) + iv(x, y). Then by (11.54), (11.52) and (11.51)

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

  ∂v ∂ y k/2 ∂u +i =  f = [u + iv]y + 2 ∂y ∂y ∂y   ∂v ∂u +i ⇒ [u + iv]y k/2 + ky k/2−1 − ∂x ∂x   ∂v ∂u k +i y 2  f = [u + iv]k y k/2 + ky 2 +1 − ∂x ∂x

99

k/2

(11.56)

since y 2 y k/2 = k y k/2 by (11.47), That is, by (11.53)    ∂v k k ∂u k k/2 +1 2 − y  f = [u + iv] − 1 y + ky +i 2 2 ∂x ∂x    ∂v k ∂u k def. k − 1 f + ky 2 +1 − +i . = 2 2 ∂x ∂x 2

(11.57)

On the other hand (again by (11.55)) −iky

∂ f def. k = −ikyy 2 ∂x



   ∂u ∂u ∂v ∂v k +i = −ky 2 +1 i − ∂x ∂x ∂x ∂x

(11.58)

so that by (11.57) def.

k f = y 2  f − iky

k k ∂f = ( − 1) f. ∂x 2 2

(11.59)

def.

In summary, the recipe f (z) = y k/2 F(z) in (11.55) for any holomorphic function F(z) on π + provides for an eigenfunction f (z) of the Maass operator k in (11.47) with eigenvalue k/2(k/2 − 1), as in (11.59). As an application, choose F(z) = η N (z)2k = e2k log η N (z)

(11.60)

where as before for the logarithm we choose arg w ∈ (−π, π] for w ∈ C, w = 0. Then, by the summary remark def.

k

f N ,k (z) = y k/2 F(z) = (Im z) 2 η N (z)2k 

ab satisfies (11.48). Also for γ = c d

 ∈ 0 (N )

(11.61)

100

11 A Dedekind Type Eta Function Attached to the Hecke Group 0 (N )

Im z (since ad − bc = 1) ⇒ |cz + d|2 k2   2k y 1 πi S N (γ) 2 η (z) e (cz + d) f N ,k (γ · z) = N |cz + d|2  cz + d k = e2πik SN (γ) f N ,k (z) |cz + d| Im(γ · z) =

(11.62)

by (11.13), say for c = 0 with S N (γ) given by (11.9). If c = 0, then f N ,k (γ · z) = e2πik S(γ) f N ,k (z)

(11.63)

for S(γ) given by (11.11). Regarding Eq. (11.48), I would like to thank Prof. Cristian David Gonzalez Avilés at the Universidad de La Serna, Chile for fruitful discussions. As best I can recall, he attended a 2001 lecture of mine at La Falda, Argentina and afterwards offered decisive, insightful comments. We did also meet in Santiago, Chile so he would pardon me if I have mis-remembered the country/occasion.

Chapter 12

Elementary Particles, the E8 Root Lattice, and a Patterson–Selberg Zeta Function

In 2007, Dr. Anthony Garrett Lisi proposed a theory of everything—a theory of the unification of elementary particles and gravitation based on E 8 structure. Professor Alfred Noël has kindly agreed to offer some remarks on the mathematics involved in Lisi’s work in his lecture presented in the next chapter (Chap. 13)—remarks that will incorporate more on E 8 . The Lisi proposal however has not been free of flaws and controversy. It is best viewed then as a work in progress. The focus of this chapter is on the root lattice of E 8 —of establishing its evenness and self-duality (which imply its uniqueness). Some remarks on elementary particles (in layman’s terms) are included that lead to the role of E 8 in string theory, although at a point we briefly relate the Patterson–Selberg zeta function to the gravitino particle, and the sl(3, C) weight diagram to the baryon octet. Some background material on lattices in general is discussed before the attention and focus are shifted to the E 8 lattice. It is well understood by now, by way of work of John Dalton (1766–1844), Dimitri Mendeléev (1834–1907), John Thomson (1856–1940), Wilhelm Röntgen (1845– 1923), Marie Curie (1867–1934), Ernest Rutherford (1811–1937), James Chadwick (1871–1974), Niels Bohr (1885–1926), and of that of many others, that an atom is comprised of a central nucleus of protons and neutrons about which elections are in orbit. I would add to the above list also the names Democritus (c. 460–370 BC) and Ludwig Boltzmann (1844–1906) as great visionary atomists. Democritus postulated that all material bodies were made up extremely tiny indivisible units called “atoms”. Boltzmann, who was absolutely convinced of the reality of atoms, endured a life of sharp criticism for his brilliant ideas. I quote here the final 5 sentences of my QM book [108]: “Work on Brownian motion by Einstein and Smoluchowski, and experiments of Jean Perrin, for example, eventually confirmed once and for all the existence of atoms. Boltzmann, however, never lived to see his work find vindication. The many attacks on him wounded him deeply to the point of suicide. Peace to his spirit. And to him and other great atomists, like Democritus and John Dalton, many salutations”. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_12

101

102

12 Elementary Particles, the E 8 Root Lattice …

The nucleus of the helium atom, for example, consists of 2 protons and 2 neutrons and there are 2 orbiting elections. The lithium atom has a nucleus of 3 protons and 4 neutrons with 3 orbiting electrons—2 in an inner orbit and 1 in an outer orbit. Electrons and protons carry an electric charge, by which they attract each other, whereas the neutron has no charge. An electromagnetic force is in play for all objects that carry an electric charge. There is also the gravitational force that acts on all particles that have a mass, and thus it acts on protons, neutrons, and electrons in particular—the force be it ever so small. Particles exert a force on each other by exchanging a force carrier—an analogue of which could be exemplified by a ball tossed between two persons. As the ball (the carrier) is tossed between the two, one exerts a force on the other. The electromagnetic force carrier is the photon, and the carrier of the gravitational force is called the graviton. By (10.2), the holomorphic sector of the one-loop gravity partition function in definition (10.3) can be expressed in terms of the Dedekind eta function. There is also the one-loop graviton partigraviton tion function Z 1−loop whose product with the one-loop gravitino partition function gravitino

Z 1−loop provides for the complete one-loop partition function of N = 1 supergravity. In [118], the following formulas were established that express the graviton and gravitino partition functions in terms of a suitable zeta function: For z = x + i y ∈ π +     Z  3 + i xy Z  3 − i xy graviton    , Z 1−loop (z) = Z  2 + 2i xy Z  2 − 2i xy

(12.1)

and for the Majorana gravitino of N = 1 supergravity  gravitino

Z 1−loop (z) =

Z

3 2



Z

+ i 23 xy + 5 2

+

i x 2 y

+

iπ y iπ y

 

 Z



Z

3 2

− i 23 xy −

5 2

−

i x 2 y

−

iπ y

iπ y



 ,

(12.2)

where Z  (w) = Z (y/2,x/2) (w)

(12.3)

for w ∈ C, and where def.

Z (a,b) (w) =



  k1  −2bi k2 −(k1 +k2 +w)2a  1 − e2bi e e

(12.4)

0k1 ,k2 ∈Z

for (a, b) ∈ R2 is a Patterson–Selberg zeta function: see [85] for a more general definition. Here, for

a+ib 0 def. e , γ(a,b) = 0 e−a−ib (12.5) n def.  (a,b) = γ(a,b) | n ∈ Z

12 Elementary Particles, the E 8 Root Lattice …

103

is the Kleinian subgroup of G = S L(2, C) generated by γ(a,b) . A fundamental domain F for the action of (a,b) on real hyperbolic 3-space H 3 (R) = G/SU (2) is computed in [113], for example, where it is also shown that the hyperbolic volume of F is infinite—which is the meaning of “Kleinian” here. We have not discussed yet what a gravitino is, which we will get back to later. In addition to the electromagnetic and gravitational forces (and their carrier forces, the photon and the graviton), there are two other fundamental forces—the strong force—an attractive force between protons and neutrons, and another force with much less strength—the weak force, of which Professor Noël will comment on briefly in his lecture. Under the strong force, whose carrier is called a gluon, protons attract protons and neutrons, and neutrons attract neutrons. The gluon particle, as the name might suggest, “glues” protons and neutrons together to guard against disintegration of the nucleus. Here again the “magic number 8” (as in the closing remarks of Chap. 8) occurs in nature as, it turns out, that there are only 8 gluons. They are based on “colors and anticolors” red, anti-red, blue, anti-blue, green, anti-green. More to the point, gluons transform according to the adjoint representation of SU (3), which is 8-dimensional. Elementary particles generally fall into two groups: hadrons—particles with a sensitivity to the strong force, and leptons particles that are immune to the strong force. Protons, neutrons, anti-protons (discovered in 1955), and bosons (particles of integral spin 0, 1, 2, 3, . . .) are examples of hadrons. Electrons (with non-integral spin 21 ), muons (with mass 300 times that of the electron), the tau particle (discovered in 1975 with mass 3600 times that of the electron), and neutrinos (with mass  0) are examples of leptons. Particles with half-integer spin (like the electron, or the proton and neutron) are called fermions (after Enrico Fermi (1901–1954)). The boson particles are named in honor of Satyendra Bose (1894–1974). Thus by the above examples hadrons can contain both fermions (like protons) and bosons. However, all leptons are fermions. It follows that hadrons can be split into two groups, as a further refinement: The hadrons that are fermions are called baryons, and the ones that are bosons are called mesons. The term “spin” used here is not meant to assume its usual connotation. To say that the electron has spin 21 , for example, is not to be taken to mean that there is some actual axis about which it spins in some physical way. 21 is a certain spin quantum number of the electron. Chapter 9 of [108], for example, is devoted to the notion of spin. The chemical properties of atoms, their ability to combine and form molecules, for example, depend on this important notion. Another further refinement in atomic structure: Every proton and neutron is made up of 3 quarks. A quark also has spin 21 (and therefore it is a fermion) and its name was coined by Murray Gell-Mann (1929–2019). He, Yuval Ne’eman (1925–2006) (independently), and George Zweig (1937–)(independently), in the 1960s, set up the quark model classification of hadrons by way of SU (3) flavour symmetry—called the Eightfold Way. Flavour refers to the 3 types/“flavours” of quarks: up, down and strange. For example, of the 3 quarks of which a proton consists, two are up quarks and one is a down quark. A baryon octet based on the 8 weights of the adjoint rep-

104

12 Elementary Particles, the E 8 Root Lattice …

resentation of SU (3) (mentioned earlier with respect to the 8 gluons) was set up by Gell-Mann and Ne’eman that corresponded to the known baryons,

denoted by p (for the proton), n (for the neutron),  (for the lambda particles), (for the sigma particles) and, − (for the cascade particles):

The Baryon Octet p

n

◦

−

+

◦

−

−

−

◦

(12.6) There is also the meson octet. To add clarity to the elementary particle—weight correspondence we write out, for example, the weight diagram of SU (3) corresponding to the baryon octet for the def. adjoint representation, where the weights are simply the roots  of A2 = sl(3, C), def. The fundamental, or simple, roots are π = {α1 = α12 , α2 = α23 } with αi j (H ) = Hii − H j j for the components Hi j of an element H in the Cartan subalgebra of g = A2 . The inner product values < α1 , α1 >=< α2 , α2 >= 13 , < α1 , α2 >= − 16 lead to the values < α1 + α2 , α1 + α2 >=< α1 , α1 > +2 < α1 , α2 > + < α2 , α2 >= 1 , ||α1 || = ||α2 || = √13 (the lengths of α1 , α2 ), so that the angle θ between α1 and 3 α2 is given by < α1 , α2 > 1 (12.7) cos θ = = − ⇒ θ = 120◦ . α1  α2  2 Here  = + ∪ −+ for + = {α1 , α2 , α1 + α2 }, and we identify α1 of length √ √ √ √ 1/ 3 = 3/3 with the point ( 3/3, 0) ∈ R2 to get α2 = − 3/6, 1/2 , α1 + √  α2 = 3/6, 1/2 :

12 Elementary Particles, the E 8 Root Lattice …

105

y   √ α2 = − 63 , 21

θ = 120◦

θ 0

α1 =

(12.8)

x

√ ( 33 , 0)

In this way we see that the weight diagram of SU (3) is the following hexagon, corresponding to the baryon octet in (12.6): y   √ α2 = − 63 , 21  √  −α1 = − 33 , 0

α1 + α2 =

√

α1 =

3 1 , 6 2

√



3 ,0 3

 x

  √ −(α1 + α2 ) = − 63 , − 21

−α2 =

√

3 , − 21 6



(12.9) By the way, since we have the values < α1 , α1 >=< α2 , α2 >= 13 , < α1 , α2 >= − 16 , the Cartan matrix for A2 = sl(3, C) is given by def.

A =





2 < αi , α j > 2 −1 = −1 2 < αi , αi >

(12.10)

with det A = 3 = 2 + 1, as asserted in the first entry of Table 3.1 of Chap. 3. The gravitino particle was mentioned earlier, with the intent to assign to it some description at a later point. We will note shortly that it is the fermionic partner of the (bosonic) graviton in superstring theory. First, it is known that bosonic string theory is basically a non-realistic toy model since it consider only particles with integral spin (bosons as we have noted), and consequently no fermions—which are building

106

12 Elementary Particles, the E 8 Root Lattice …

blocks of matter. Without supersymmetry such a theory cannot describe matter. The imposition of Virasoro constraints forces the theory to live in D = 26 space-time dimensions. These constraints eliminate the existence of non-physical states—states with a negative norm that are called “ghosts”. Superstring theory, which includes fermions, can also include ghost states which (as in bosonic string theory) can be removed, in which case one has a reduction to a D = 10 space-time dimension. In this theory (whose chief architect is likely John Schwarz) each fermion has a bosonic partner. Here’s where our little gravitino comes in. As was indicated above, it is the superpartner (with spin 3/2) of the graviton with spin 2. The photon of spin l, for example, has a fermionic superpartner of spin 21 called the photino. Some people have suggested that the gravitino might be a good candidate for dark matter. Although, by way of supersymmetry, the critical D = 26 space-time dimensions theory is reduced to D = 10 space-time dimensions, there is the major effect of string theory to achieve a most realistic D = 4 dimensional theory. Is superstring theory a theory of everything? Bosonic and superstring theory has been merged to form a hybrid theory called heterotic string theory—due to David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm—the so-called “Princeton String Quartet”. Work of this string quartet has very much enhanced the superstring formulation of Michael Green and John Schwarz. In this theory the groups Spin(32)/Z2 and E 8 × E 8 (where Spin(32)/Z 2 is the double cover of S O(32)) arise in connection with the compactification of the D = 26 dimensions down to D = 10 dimensions. Like quarks, E 8 comes in different flavors. Here the reference is to the compact, simply connected Lie group E 8 —which is also the notation for the Cartan matrix in (3.3) of the complex, simple, exceptional Lie algebra E 8 . In the end, the appearance of E 8 in heteronic string theory boils down to a basic mathematical fact of life: In 8 dimensions, there is exactly one (up to isomorphism) Euclidean, even, self-dual lattice—the root lattice of E 8 . Definitions involed in this statement will be considered presently. In 16 dimensions there are only two Euclidean, even, self-dual lattices. These lead to the two types of heterotic strings: the E 8 × E 8 and the S O(32) heterotic superstrings—these groups being gauge groups of dimension 2 × 248 = 496. As final remarks along these lines, the discovery of the Princeton String Quartet (of heterotic string theory in 1985) was followed by Edward Witten, Nathan Seiberg, and others in a second superstring revolution, in the mid 1990’s, where three other string theories were described—called Type I, Type II A, and Type II B, along with 11—dimensional supergravity. The question is posed again: Is superstring theory a theory of everything? Can this theory describe all elementary particles and their interactions and gravity from a single, unified, point of view? The significant role played by the E 8 root lattice in heterotic string theory has been noted. We turn attention to the definition and properties of this lattice, starting with some remarks on lattices in general. Let (V, ) be an m—dimensional real inner product space, m  1, A lattice L ⊂ V is the Z-span of an R—basis B = {b1 , . . . , bm } of V :

12 Elementary Particles, the E 8 Root Lattice …

L =

⎧ m ⎨ ⎩

j=1

107

⎫ ⎬ n jbj | n j ∈ Z , ⎭

(12.11)

 which is an abelian subgroup of V . Let B ∗ = b1∗ , . . . , bm∗ be the dual basis of B for   def. def. the dual space V ∗ of V : bi∗ b j = δi j . Then L ∗ = { f ∈ V ∗ | f (v) ∈ Z ∀v ∈ L } ∗ is a lattice in V . Namely, L∗ = For if f ∈ L ∗ f =

m 

⎧ m ⎨ ⎩

j=1

⎫ ⎬ m j b∗j | m j ∈ Z . ⎭

(12.12)

  f i b∗j with f j = f b j ∈ Z,

(12.13)

j=1

and, conversely, for m 1 , . . . , m m ∈ Z and v =

m

n i bi ∈ L

i=1

⎛ ⎝

m 

=

⎞ m j b∗j ⎠ (v) =

m 

m j n i b∗j (bi ) =

def.

j=1

j,i=1

m 

m 

m i ni ∈ Z ⇒

i=1

m 

m j n i δ ji

j,i=1

(12.14)

m j b∗j ∈ L ∗ .

j=1

L ∗ is called the dual lattice of L . By way of the inner product on V there is the standard vector space isomorphism ε : V −→ V ∗ given by ε(v) = f v where def. f v (w) = < w, v > ∀w ∈ V . If def.

L # = {v ∈ V | < l, v >∈ Z ∀ ∈ L }, then it follows directly that

def.

ε(L # ) = L ∗

(12.15)

(12.16)

def.

and thus b#j = ε−1 b∗j ⇒ B # = {b1# , . . . , bm# } is also an R—basis of V . Note that def.

def.

def.

def.

< bi , b#j > = f b#j (bi ) = (εb#j )(bi ) = bi# (bi ) = δi j . For v ∈ L # , εv ∈ L ∗ (by (12.16)) ⇒ (by (12.12))

(12.17)

108

12 Elementary Particles, the E 8 Root Lattice …

εv =

m 

n j b∗j

⎞ ⎛ m m n      nj ∈ Z = n j εb#j = ε ⎝ n j b#j ⎠ ⇒ v = n j b#j .

j=1

Conversely,

m

i=1

j=1

j=1

j=1

(12.18) n j b#j

is clearly an element of L for the n j ∈ Z. That is, #

L# =

⎧ m ⎨ ⎩

j=1

⎫ ⎬ n j b#j | n j ∈ Z ⎭

(12.19)

which shows that L # is also a lattice in V —which by (12.16) is also called the dual lattice of L . L is called an even lattice if < v, v >∈ 2Z ∀v ∈ L , and L is an integral lattice if < v, w >∈ Z ∀v, w ∈ L . Note that an even lattice is integral: For v, w ∈ L < v + w, v + w > − < v, v > − < w, w >= 2 < v, w >⇒ 1 1 < v, w >= (< v + w, v + w > − < v, v > − < w, w >) ∈ (2Z) = Z 2 2 (12.20) if L is even. Also note that L is integral ⇔ L ⊂ L# :

(12.21)

If v ∈ L and L is integral, then < v,  >∈ Z ∀ ∈ L ⇒ v ∈ L # ⇒ L ⊂ L # . Conversely if L ⊂ L # , then ∀v, w ∈ L , w ∈ L # ⇒< v, w >∈ Z ⇒ L is integral. Definition 12.1 The lattice L is self-dual if L = L # . Next we note that the lattice L is reflexive:  # # L = L.

(12.22)

 # To see this observe first that since every bi# ∈ L # (in particular) v ∈ L # ⇒ bi# , v = m i ∈ Z so if we write v = nj=1 x j b j for x j ∈ R, then in fact by (12.17) mi =

m  j=1

xj
=

m 

x j δi j = xi ⇒ xi ∈ Z

(12.23)

j=1

m  #

n j b j ∈ L , n j ∈ Z, ⇒ v ∈ L by (12.11) ⇒ L # ⊂ L . Conversely for v =

and # =

m

i=1

j=1

m i bi# ∈ L # , m i ∈ Z (see (12.19)), we have again by (12.17) that

12 Elementary Particles, the E 8 Root Lattice … m 

v, #  =

109

m m     n j m i b j , bi# = n j m i δ ji = n jm j ∈ Z

j,i=1

j,i=1

j=1

 #  #  # ⇒ v ∈ L# ⇒ L ⊂ L# ⇒ L = L# .

(12.24)

We mention two further trivial observations: (i)

bi =

m 

< bi , b j > b#j ,

j=1 (ii) bi# =

m 

(12.25)
bj

j=1

for 1 ≤ i ≤ m: Write bi =

m

j=1

v j b#j for v j ∈ R since B # is an R-basis of V . Then,

again by (12.17). < bi , bk >=

m 

v j < b#j , bk >=

j=1

m 

v j δ jk = vk ,

(12.26)

j=1

which gives (i) in (12.25). (ii) follows similarly when bi# is expressed in terms of the R-basis B of V . As a simple example, take V = Rm with the standard inner product , on Rm def. for which B = {b1 , . . . , bm } = {e1 , . . . , em } is an orthonormal basis: < x, y >=

m 

xi yi , ei = (0, . . . , 1, . . . , 0),

(12.27)

i=1

with 1 in the ith position, xi , y j = the coordinates of x, y ∈ Rm . Then Zm = L in (12.11) is a lattice (of course) in Rm . By (i) in (12.25), ei =

m 

< ei, e j > b#j =

j=1

δi j b#j = bi# ,

(12.28)

j=1

 so that by (12.19), (Zm )# = L # =

m 

m

 n i ei | n i ∈ Z = Zm . That is, Zm is a self-

i=1

dual lattice in Rm . The Gram matrix of B(named after Jørgen P. Gram) is given by  def.  Gram(B) = < bi , b j > . Then

(12.29)

110

12 Elementary Particles, the E 8 Root Lattice … m         Gram B # Gram(B) i j = Gram B # ik (Gram(B))k j k=1 m m   def. = < bi# , bk# >< bk , b j >=< bi# , < bk , b j > bk# > k=1 =< bi# , b j



>= δi j ⇒ Gram B

#



(12.30)

k=1

Gram(B) = I

by (i) in (12.25) and by (12.17) again. Similarly by (ii) in (12.25) and by (12.17)     Gram(B) Gram B # = I ⇒ Gram B # = (Gram(B))−1 .

(12.31)

If Gram(B) is integral—that is all of its matrix entries < bi , b j >∈ Z, then by (12.11) < v, w >∈ Z ∀v, w ∈ L ⇒ L is integral. Conversely if L is integral, then in particular Gram(B) is integral: L is integral ⇔ Gram(B) is integral, also ⇔ L ⊂ L # by (12.21). For v1 , v2 ∈ L # , write by (12.19), (12.31) v1 =

m 

n i bi# ,

v2 =

i=1

< v1 , v2 >=

m 

m j b#j , n i , m j ∈ Z,

j=1 m 

def.

n j m j < bi# , b#j > =

(12.32)

i, j=1 m 

m       n i m j Gram B # i j = n i m j (Gram(B))−1 i j .

i, j=1

i, j=1

Now if L is integral so is Gram(B), and if det Gram(B) = 1, then also (Gram(B))−1 is integral so by (12.32) we see that < v1 , v2 >∈ Z. That is, L # is integral so by  # (12.21) again, and by (12.22), L # ⊂ L # = L , which shows that L # = L —i.e. L is self-dual, which proves. Proposition 12.1 If L is an integral lattice such that det(Gram(B)) = 1 (in which case L is said to be unimodular), then L is self-dual: L = L # . Let A = the Gram matrix Gram(B), which is clearly a symmetric matrix. A is also a positive definite (p.d.) matrix. This we will check later—following Theorem 12.1. In particular det A > 0. Assume, conversely, that L = L # . Then L # is integral ⇒ Gram(B # ) is integral ⇒ A−1 is integral by (12.31). But A is integral (since L is integral) which means that its inverse is integral ⇔ det A = 1 or −1. Since det A > 0, we can conclude that det A = 1. Corollary 12.1 If L is an integral lattice, then L is self-dual (L = L # ) ⇔ L is unimodular: det(Gram(B)) = 1. Let g denote the complex, simple Lie algebra E 8 , and let (V, ) be an 8—  8 dimensional real inner product space. Choose an orthonormal basis β j j=1 of V .

12 Elementary Particles, the E 8 Root Lattice …

111 def.

An example of course is V = R8 with its standard inner product, and β j = e j = (0, 0, . . . , 1, 0, . . . 0) where 1 occupies the jth position as in (12.27). A fundamental  8 system of roots (also called simple roots) π = α j j=1 of g is given by 1 βj. 2 j=1 8

def.

def.

def.

α j = β j − β j+1 , 1  j  6, α7 = β6 + β7 , α8 = − It follow directly that



 α j , α j = 2, 1  j  8,

(12.33)

(12.34)

and that (see definition 12.10) for def.

Ai j =

2 < αi , α j > =< αi , α j >, < αi , αi >

(12.35)

def.

the Cartan matrix A = [Ai j ] of g is exactly the matrix E 8 given in (3.3). For def.

example, (E 8 )75 = −1 in (3.3) and A75 =< α7 , α5 > = < β6 + β7 , β5 − β6 >= −1  8 since β j j=1 is an orthonormal set. Let ε : R8 → V denote the linear isomorphism given by def.

ε (x1 , x2 , . . . , x8 ) =

8 

xjβj.

(12.36)

j=1

The basic object of interest now is the Abelian group  def.

L = L E8 =

v=

8  i=1

xjβj ∈ V

8  j=1

x j ∈ 2Z

 1 , and either every x j ∈ Z or every x j ∈ Z + 2

(12.37)

which we show is an even, self-dual lattice—the root lattice of E 8 . We consider a linear isomorphism T : R8 → R8 such that the restriction T˜ = ε ◦ T |Z8 : Z8 → V ! : Z8 −→ L . is a group isomorphism T The linear isomorphism T : R8 → R8 is defined by T (x) = x T t , for x = (x, . . . , x8 ), where

112

12 Elementary Particles, the E 8 Root Lattice …

⎡

2 −1 0

⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ def. ⎢ 0 T = ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎣ 0

0

0

0

0

1

−1 0

0

0

0

0

1

−1 0

0

0

0

0

1

−1 0

0

0

0

0

1

−1 0

0

0

0

0

1

−1

0

0

0

0

0

1

0

0

0

0

0

0

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(12.38)

det T = 1, and T (x) = (v1 , v2 , . . . , v8 ) for v1 = 2x1 − x2 + x8 /2, v2 = x2 − x3 + x8 /2, v3 = x3 − x4 + x8 /2 v4 = x4 − x5 + x8 /2, v5 = x5 − x6 + x8 /2, v6 = x6 − x7 + x8 /2,

(12.39)

(12.40)

v7 = x7 + x8 /2, v8 = x8 /2. In particular for x = n = (n 1 , . . . , n 8 ) ∈ Z8 we write T ( n ) = (y1 , y2 , . . . , y8 ) so that y1 = 2n 1 − n 2 + n 8 /2 , y2 = n 2 − n 3 + n 8 /2, . . . , y8 = n 8 /2 by (12.40). Now either n 8 = 2 ∈ 2Z is even or n 8 = 2 + 1 ∈ 2Z + 1 is odd. That is, either n 8 /2 ∈ Z or n 8 /2 ∈ Z + 21 ⇒ every y j ∈ Z or every y j ∈ Z + 21 . Also 8 

y j = 2n 1 − n 2 + n 2 − n 3 + n 3 − n 4 + n 4 − n 5 + n 5 − n 6 + n 6 − n 7

j=1

(12.41)

+ n 7 + 8 (n 8 /2) = 2 (n 1 + 2n 8 ) ∈ 2Z, def.

which by (12.36), (12.37) says that v = ε (y1 , . . . , y8 ) = ! : Z8 → L given by have a map T def. !( T n) =

8 

8 j=1

y j β j ∈ L . Thus we

y j β j = ε (y1 , y2 , . . . , y8 ) = ε(T ( n )),

j=1

y1 = 2n 1 − n 2 + n 8 /2, y7 = n 7 + n 8 /2,

y j = n j − n j+1 + n 8 /2, 2 ≤ j ≤ 6,

(12.42)

y8 = n 8 /2, n = (n 1 , n 2 , . . . , n 8 ) .

! is surjective, take v ∈ L : To check that T v=

8  j=1

yjβj,

8  j=1

y j = 2 ∈ 2Z,

(12.43)

12 Elementary Particles, the E 8 Root Lattice …

113

with every yi ∈ Z or every y j ∈ Z + 21 . Define n 1 =  − 4y8 , n 2 = y2 + y3 + y4 + y5 + y6 + y7 − 6y8 , n 3 = y3 + y4 + y5 + y6 + y7 − 5y8 , n 4 = y4 + y5 + y6 + y7 − 4y8 , n 5 = y5 + y6 + y7 − 3y8 , n 6 = y6 + y7 − 2y8 , n 7 = y7 − y8 , n 8 = 2y8 . (12.44) Of course if every y j ∈ Z, then every n j ∈ Z. This is also true if every y j ∈ Z + 21 . For example, for y j = m j + 21 , m j ∈ Z, 1  j  8, n 5 = m 5 + 21 + m 6 + 21 + m 7 +   def. 1 − 3 m 8 + 21 = m 5 + m 6 + m 7 − 3m 8 ∈ Z. This gives n = (n 1 , n 2 , . . . , n 8 ) ∈ 2 !( Z8 for which T n ) = v. For example in (12.42) we need that y6 = n 6 − n 7 + n 8 /2. The right hand side here is (by definition of the n j ) y6 + y7 − 2y8 − (y7 − y8 ) + y8 = ! is not only surjective but is a group homomorphism also, and therefore a y6 . So T ! = ε ◦ T | Z8 . group isomorphism since T Going back to (12.39), (12.40), one has that for x = (x1 , x2 , . . . , x8 ) ∈ R8 < T (x), T (x) >=

8 

v 2j = 4x12 + 2

j=1

8 

x 2j − 4x1 x2 − 2

j=2

6 

x j x j+1 + 2x1 x8 ∈ 2R.

j=2

(12.45) In particular, < T ( n ), T ( n ) >∈ 2Z for n = (n 1 , n 2 , . . . , n 8 ) ∈ Z8 . Moreover, for !( v ∈ L,v = T n ) = ε(T ( n )) (as we have just noted) ⇒< v, v >=< T ( n ), T ( n) > (as ε preserves the inner products on R8 , V ): < v, v >∈ 2Z ∀v ∈ L .

(12.46)

There are quite other ways to prove this, of course. Given the standard orthonormal basis {e j }8j=1 of R8 above, with each e j ∈ Z8 ,  8 def. b j = (ε ◦ T )(e j ) ⇒ b j j=1 is a basis of V . By (12.39), (12.40), (12.36), (12.33) respectively T (e1 ) = (2, 0, 0, 0, 0, 0, 0, 0), T (e2 ) = (−1, 1, 0, 0, 0, 0, 0, 0), T (e3 ) = (0, −1, 1, 0, 0, 0, 0, 0), T (e4 ) = (0, 0, −1, 1, 0, 0, 0, 0), T (e5 ) = (0, 0, 0, −1, 1, 0, 0, 0), T (e6 ) = (0, 0, 0, 0, −1, 1, 0, 0), ) ( 1 1 1 1 1 1 1 1 , , , , , , , , T (e7 ) = (0, 0, 0, 0, 0, −1, 1, 0), T (e8 ) = 2 2 2 2 2 2 2 2 b1 = 2β1 , b2 = β2 − β1 = −α1 , b3 = β3 − β2 = −α2 , b4 = β4 − β3 = −α3 , b5 = β5 − β4 = −α4 , b6 = β6 − β5 = −α5 , 1 β j = −α8 . 2 j=1 8

b7 = β7 − β6 = −α6 , b8 =

(12.47)

114

12 Elementary Particles, the E 8 Root Lattice … 8

Also for v =

!( y j β j ∈ L , again v = T n ) with the coordinates y j of v given in

j=1

terms of those of n by (12.42): v = (2n 1 − n 2 + n 8 /2) β1 +

6  (n j − n j+1 + n 8 /2)β j + (n 7 + n 8 /2)β7 + n 8 /2β8 j=2

= n 1 2β1 + def.

= n 1 b1 +

7 

8   n8  n j β j − β j−1 + βj 2 j=1 j=2

7 

n j b j + n 8 b8 =

8 

j=1

n jbj,

j=1

(12.48) by (12.47). In other words, (12.48) show that L = L E8 is the Z-span of the basis {b j }8j=1 of V explicated in (12.47) and constructed via the linear isomorphism T : R8 → R8 in (12.39). That is, L is a lattice, the root lattice of the complex, simple Lie algebra E 8 , donated by g above. Also L is an even lattice: < v, v >∈ 2Z for every v ∈ by (12.46). Next we note that L is self-dual. Since L is even, L is integral as was shown in (12.20). By Proposition 12.1 it suffices therefore to check that det(Gram(B)) = 1.  8 β j j=1 is an orthonormal basis of V so by way of the expression of the bi in terms of the β j in (12.47) the matrix entries < bi , b j > are directly computed to give ⎡

4 ⎢ −2 ⎢ ⎢0 ⎢ def. ⎢ 0 Gram(B) = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 1

−2 2 −1 0 0 0 0 0

0 −1 2 −1 0 0 0 0

0 0 −1 2 −1 0 0 0

0 0 0 −1 2 −1 0 0

0 0 0 0 −1 2 −1 0

0 0 0 0 0 −1 2 0

⎤ 1 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎥ ⎥ 0⎦ 2

(12.49)

whose determinant is indeed equal to 1. In summary. Theorem 12.1 The Abelian group L E8 defined in (12.37) is the Z-span of the R 8 basis {b j }nj=1 of V given in (12.47), where π = α j j=1 is a fundamental system of roots of the complex, simple Lie algebra g = E 8 . Therefore L E8 is a lattice, the root lattice of g. Moreover, L E8 is even and self-dual. The latter two properties of L E8 , its evenness and self-duality, account for its uniqueness in the universe—and for the pivotal role that it plays in heterotic string theory, as was indicated in earlier remarks. One way to prove the uniqueness of L E8 is to consider the theta function θL (z) attached to an arbitrary even lattice L ⊂ V = a real m-dimensional vector space with an inner product , and Rbasis B = {b1 , . . . , bm } − L as before; see (12.11). Then as an initial step towards

12 Elementary Particles, the E 8 Root Lattice …

115

the proof, one shows that in particular if dim V = 8, then θL (z) coincides with the normalized Eisenstein series E 4 (z) in (4.17)—and thus with the theta function θ E8 (z) there for the matrix E 8 in (3.3). We sketch the ideas involved here. Since L is even (ie. < v, v >∈ 2Z ∀v ∈ L ) we can set def.

a = | {v ∈ L |< v, v >= 2} |

(12.50)

for  = 0, 1, 2, 3, . . ., so that a0 = 1, as < v, v >= 0 ⇔ v = 0. Then def.

θL (z) =



e

iπz

=

∞ 

a e

2πiz

=1+

=0

v∈L

∞ 

a e2πiz

(12.51)

=1

on π + . Compare (1.10), (1.11) and (1.12), where one notes that for v =

m

n jbj ∈ V

j=1

and def.

A = Gram(B), < v, v >=

m 

def.

n i n j < bi , b j > =

i, j=1

m 

def.

Ai j n i n j = f A ( n)

i, j=1

(12.52) by (1.1), for n = (n 1 , . . . , n m ) as usual. Thus def.

θL (z) =



eiπ f A (n )z = θ A (z) def.

(12.53)

n∈Z m

by (1.5) and (12.11). A is a symmetric matrix, and it is also a positive definite (p.d.) matrix—as  mwas asserted earlier but which we now check. Choose an orthonormal basis B j j=1 of V , and write bj =

m 

 def.  Ti j Bi , Ti j ∈ R, T = Ti j .

(12.54)

i=1

Then since T t T is symmetric def.

Ai j = < bi , b j >=
=

k=1

Ti Tk j < B , Bk >=

,k=1 m 

m 

 ij

m 



Tt

 jk

k=1

⇒ A = T t T.

  Tki = T t T ji =

(12.55)

116

12 Elementary Particles, the E 8 Root Lattice …

By (12.31), (det A) det(Gram(B # )) = 1 ⇒ det A = 0 ⇒ det T = 0. By the equivalent statement (ii) following (1.2), we see indeed that A is p.d. T is called a generator matrix of L . For example, the matrix T defined in (12.38) satisfies T t T = Gram(B) for the matrix Gram(B) in (12.49) so it is a generator matrix of the E 8 lattice L E8 . As L is even, it is also integral (again by (12.20)) which means that A is even integral: Ai j ∈ Z and Aii ∈ 2Z ∀i, j. If moreover L is self-dual, then we also know that det A = 1 by Corollary 12.1 and therefore by Minkowski’s theorem in Chap. 3, m must be divisible by 8. Theorem 12.2 If L ⊂ V is even  and self-dual, then m = dimR V ∈ 8Z. Also by Theorem 4.1, θ A (z) ∈ M , m2 , where A = Gram(B) (= the Gram matrix of L ),   and since θL (z) = θ A (z) (by (12.53)), θL (z) ∈ M , m2 . Thus the theta function θL (z) defined in (12.51) attached to an even, self-dual lattice L ⊂ V is a modular form with respect to  = S L(2, Z) of weight (dim V )/2 where necessarily (by Minkowski’s theorem) dim V ∈ 8Z. Now suppose that m = 8. One knows that dim M(, 4) = 1, so by Theorem 12.2 θL (z) = cE 4 (z)

(12.56)

for some c ∈ C. By (4.17) and (12.51), θL (z) and E 4 (z) have 1 as their constant term, so c = 1 ⇒ θL (z) = E 4 (z)—which completes our sketch of an initial key step towards the proof of the uniqueness of L E8 . A few more steps are required of course to establish a full proof. The set of roots of the complex Lie algebra E 8 is given by   = α ∈ L E8 |< α, α >= 2 ⊃ π (by (12.34)).

(12.57)

The Fourier expansion of E 4 (z) = θ E8 (z) = θL E8 (z) in (4.17) is explicated in (8.26): E 4 (z) = 1 + 240q + 2160q 2 + 6720q 3 + 17520q 4 + 30240q 5 + · · ·

(12.58)

In particular, by and (12.50) and (12.51), E 8 has || = a1 = 240 roots—a result that could be obtained directly of course by computing  explicitly and counting the number of its elements. We also see, for example, by (12.50) and (12.58), that L E8 has a2 = 2160 vectors of length2 = 4, and (say) a5 = 30240 vectors of length2 = 10. An example, apart from E 8 , is the hexagonal lattice Lhex ⊂ V = R2 with B = {b1 , b2 } for * √ √ + 1+ 3 1− 3 def. , b1 = (1, 1), b2 = . 2 2 (12.59) < b1 , b1 >= 2, < b1 , b2 >= 1, < b2 , b2 >= 2.

21 ⇒ Gram(B) = . 12 By (12.53)

12 Elementary Particles, the E 8 Root Lattice …

117

θL hex (z) = θ⎡ ⎣

⎤ (z)

(12.60)

2 1⎦ 12

on π + so that by (7.34) θLhex (z) = 1 + 6q + 6q 3 + 6q 4 + 12q 7 + 6q 9 + 6q 12 + 12q 13 + 6q 16 + 12q 19 + · · ·

(12.61) 21 = 1. Lhex is integral but it is not self-dual by Corollary 12.1, since det 12 As was pointed out earlier, the gravitino (of spin 3/2 ) is the fermionic superpartner of the bosonic graviton (of spin 2). Gravity has two units of spin so, necessarily, it has a little tiny baby partner—the gravitino, meaning “small gravity”. The zeta function expression (12.2) for the gravitino partition function can be completely generalized to cover higher spin fermionic particles—particles with spin

s = (2m + 1)/2, m = 1, 2, 3, . . .

(12.62)

Thus for m = 1, we get the gravitino spin s = 3/2. One of the main results in [31] is equation (3.7) there for the 1-loop determinant Z (s) F (in the notation there) for higher spin fermionic particles. A Patterson–Selberg zeta function expression for Z (s) F was found in [118]. The result is the following:  Z  s + is xy +

  Z  s − is xy − i πy    Z (s) F (z) = Z  s + 1 + i(s − 1) xy + i πy Z  s + 1 − i(s − 1) xy − iπ y



iπ y



(12.63)

for s in (12.62). Indeed for s = 3/2(m = 1) formula (12.63) reduces to formula (12.2). Similarly, formula (12.1) is generalized in [118] to cover 1-loop partition functions for bosons with integral spin s = 2, 3, 4, 5, . . .—in the context of quadratic fluctuations of fields about a thermal Ad S3 background.

Chapter 13

The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial “Theory of Everything” Alfred G. Noël

Abstract On November 6, 2007, Dr. Garrett Lisi, an independent physicist, published, on the Mathematical Archives (arXiv) of Cornell University, a manuscript entitled An Exceptionally Simple Theory of Everything [A], in which he proposed a unification of all fields of the Standard Model and gravity as an E 8 principal bundle connection. Lisi’s approach was eventually proven to be unsuccessful by J. Distler and S. Garibaldi in [DG]. This expository article which comments on the mathematical theory behind Lisi’s work, is the written rendition of a lecture that I gave at the Haitian Scientific Society Seminar Series on July 26, 2008.

13.1 Introduction to Lie Groups and Lie Algebras E 8 is the largest of the Exceptional Lie groups named after the Norwegian Mathematician Sophus Lie (pronounced Lee) [7 December 1842–18 February 1899] who developed a research program that used ideas from Évariste Galois’ theory of polynomial equations to study differential equations. This was an ambitious project to extend techniques of finite groups to continuous groups because differential equations have a continuous spectrum of solutions. Lie’s work is exposed in three volumes written with the help of Friedrich Engel, Theorie der Transformationsgruppen I, II, and III published in 1888, 1890, and 1893 respectively, all in German. An English translation of the first volume by Joël Merker can be found at https://arxiv.org/abs/ 1003.3202.

Key words and phrases. Lie Groups, Representation Theory. IN MEMORY OF MY GRANDMOTHER VILICIA AUGUSTE (1914–2008).

A. G. Noël (B) Department of Mathematics, University of Massachusetts, 100 Morrissey Boulevard, Boston, MA 02125-3393, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_13

119

120

13 The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial …

In modern mathematical parlance, see [131], a Lie group G is a differentiable manifold which is also endowed with a group structure such that the operation φ : G × G → G defined by φ((g1 , g2 )) = g1 g2−1 is smooth, that is infinitely differentiable. G L n (R), the set of n × n invertible real matrices and G L n (C), the set of nonsingular complex matrices are Lie groups under matrix multiplication. Lie’s fundamental contribution was the linearization of these groups for the purpose of facilitating their understanding through their tangent space at the identity, their Lie Algebra. A Lie algebra over a closed field K is a K-vector space g together with a bilinear operator, the bracket [,]:g×g→g such that for all A, B, C ∈ g: • [A, B] = −[B,   A];    • [A, B], C + [B, C], A + [C, A], B = 0. The Lie algebra of G L n (R) and G L n (C) are respectively gln (R), the algebra of n × n real matrices, and gln (C), the algebra of n × n complex matrices. However, it is judicious to mention that different Lie groups can share the same Lie algebra. This is indeed the case of the complex orgthogonal Lie groups, On (C) = {A ∈ G L n (C) : A−1 = A T } and SOn (C) = {A ∈ On (C) : det A = 1}, which have the same Lie algebra son (C) = {A ∈ gln (C) : A T + A = 0}. The finite dimensional complex simple Lie algebras, that is Lie algebras that are non-abelian and contain no non-zero proper ideals, were essentially classified by Wilhelm Karl Joseph Killing (10 May 1847–11 February 1923). Their real counterparts were classified by ÉLie Joseph Cartan (9 April 1869–6 May 1951). A modern classification of simple real Lie algebras via the so-called Vogan diagrams is given in [129]. A simple Lie group is a Lie group whose lie algebra is simple. The simple complex Lie algebras are usually given by their Dynkin diagrams after the work of Eugene Borisovich Dynkin (11 May 1924–14 November 2014) in [128]:

13.1 Introduction to Lie Groups and Lie Algebras

121

An Bn Cn Dn E6 E7 E8 F4 G2 According to the late Bertram Kostant (May 24, 1928–February 2, 2017): In this paper, using only elementary mathematics, and starting with almost nothing, Dynkin, brilliantly and elegantly developed the structure and machinery of semisimple Lie algebras. What he accomplished in this paper was to take a hitherto esoteric subject, and to make it into beautiful and powerful mathematics.

In this classification the Lie algebras of types An , Bn , Cn , Dn , called classical simple Lie algebras, are all Lie subalgebras of gln (C) corresponding to Lie subgroups of G L n (C). The other five are called exceptional simple Lie algebras. E 8 is the largest of the exceptional simple complex Lie groups and its dimension which is that of its Lie algebra, e8 , is 248. In this presentation, it will be in our advantage to mention that the algebra of type An is the special linear algebra, sln+1 (C) = {A ∈ gln+1 (C) : trace(A) = 0}, the Lie algebra of the special linear group: S L n+1 (C) = {A ∈ G L n+1 (C) : det(A) = 1}. Its dimension is (n + 1)2 − 1. A real form of a complex Lie algebra g is a real Lie agebra g0 such that the complexificaton g0 ⊗R C  g. Hence, gln (R) is a real form gln (C), and un , the Lie algebra of n × n skew-Hermitian matrices is also a real form of gln (C). We extend this definition so that the real Lie groups G L n (R) and Un , the real group of unitary matrices, are both real forms of G L n (C). Similarly, the special linear groups S L n (R) and SUn are real forms of S L n (C). The Lie group E 8 has three real forms.The one that is of interest to Lisi is the one that corresponds to the split real form e8(8) of the complex Lie algebra e8 . It is non-compact and its dimension is also 248. Physicists are also very interested in representations of these groups which are their preferred tools to extract information from symmetry in nature. A representation of a complex Lie group G on a complex vector space V is a homomorphism ρ : G →

122

13 The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial …

G L(V ), the group of automorphisms of V . The dimension of ρ is that of V and often V is said to be a representation of G or a G-module. A representation of a real Lie group is defined similarly on a real vector space. The irreducible representations, those which do not stabilize any proper subspace of V , are building blocks for the other reducible representations. Among the irreducible representations, the unitary ones are of primary importance to mathematicians and physicists. Likewise, a representation of a complex Lie algebra g on a complex vector space V is a homomorphism ρ : G → gl(V ), the set of endomorphisms of V .

13.2 Grand Unified Theories and the Standard Model According to theoretical physicists, four fundamental forces exist in the Universe: • The strong interaction is very strong, but very short-ranged. It acts only over ranges of order 10−13 centimeters and is responsible for holding the nuclei of atoms together. It is basically attractive, although it can be effectively repulsive in some circumstances. • The electromagnetic force causes electric and magnetic effects such as the repulsion between like electrical charges or the interaction of bar magnets. It is longranged, yet much weaker than the strong force. • The weak force is responsible for radioactive decay and neutrino interactions. It has a very short range and, as its name indicates, it is very weak. • The gravitational force is weak, but very long-ranged. In addition, mass is its source, it is always attractive, and acts between any two pieces of matter in the Universe. It is believed that at the earliest stage of development of the Universe, the first three forces were unified into a single force and then separated later as temperature dropped. A Grand Unified Theory is a vision of a theory that would combine the strong, weak, and electromagnetic forces into one single equation. At temperatures of around 1015 Kelvin, the weak and electromagnetic forces merge into a single electro-weak force. The strong and electro-weak forces should, in principle, behave as a single unified force at particle energies and temperatures that are about a trillion times higher. However, it is not clear that such temperatures are achievable on Earth. The Standard Model of particle physics which is not a Grand Unified Theory, is a theory that describes the strong, weak, and electromagnetic forces; it also classifies all known elementary particles. It combines the electroweak theory and quantum chromodynamics via a structure denoted by the gauge group SU3 × SU2 × U1 . It is a relativistic quantum field theory which is consistent with both quantum mechanics and special relativity. To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. The formulation of the unification of the electromagnetic and weak interactions in the Standard Model is due to Steven Weinberg, Abdus Salam and, subsequently, Sheldon Glashow. The unification model was initially proposed by Steven Weinberg in 1967.

13.3 A Theory of Everything

123

SU3 × SU2 × U1 has a normal subgroup that acts trivially on all known particles while being isomorphic to Z/6Z. Hence the “true” gauge group of the Standard Model should be: [see http://math.ucr.edu/home/baez/week253.html]: SU3 × SU2 × U1 )/(Z/6Z) which is isomorphic to the subgroup of SU (5) consisting of these types of matrices: 

G 0 0 H



where G is a 3 × 3 matrix and H is a 2 × 2 matrix. Baez calls this group SU3 × U2 . The 3 by 3 block is related to the strong force, and the 2 by 2 block to the electroweak force.

13.3 A Theory of Everything In current mainstream physics, a Theory of Everything, TOE, would unify all the fundamental interactions of nature, and should yield a deep understanding of the various different kinds of particles as well as the different forces. A small number of scientists claim that Gödel’s incompleteness theorem proves that any attempt to construct a TOE is bound to fail. Gödel’s theorem states that any non-trivial mathematical theory that has a finite description is either inconsistent or incomplete. In his 1966 book The Relevance of Physics, Stanley Jaki pointed out that, because any “theory of everything” will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything. Freeman Dyson and Stephen Hawking agree with that assertion. However, since most physicists would consider the statement of the underlying rules to suffice as the definition of a “theory of everything”, these researchers argue that Gödel’s Theorem does not mean that a TOE cannot exist. In this presentation, I try to clarify the nature of the mathematical objects used by Garrett Lisi in the abstract of his paper which affirms: All fields of the standard model and gravity are unified as an E 8 principal bundle connection. A non-compact real form of the E 8 Lie algebra has G 2 and F4 subalgebras which break down to strong su(3), electroweak su(2) × u(1), gravitational so(3, 1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E 8 superconnection are described by the curvature and action over a four dimensional base manifold.

In mathematics, triality is a specific property of the group Spin(8), the double cover of the 8-dimensional rotation group D4 = S O8 . In the Dynkin’s Classification the simple algebras of types Bn and Dn correspond so2n+1 and so2n , the odddimensional and the even-dimensional special orthogonal Lie algebras respectively. Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram. The

124

13 The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial …

Fig. 13.1 The Dynkin diagram of S O8

diagram has four nodes with one node located at the center, and the other three attached symmetrically (Fig. 13.1). The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spinor representations. As such, these automorphisms do not project to automorphisms of S O(8). The algebra so( p, q) is a real form of so p+q . E 8 is the largest exceptional group. Much information can be obtained from its Lie algebra e8 which has the following root decomposition: e8 = X −α120 ⊕ · · · ⊕ X −α1 ⊕ h ⊕ X α1 ⊕ · · · ⊕ X α120 where h is maximal Abelian subalgebra of dimension 8 and each X α±i is a onedimensional space contributed by the root α±i , a linear map from h to the set of complex numbers C, and there are exactly 240 roots. In Lisi’s theory, each root corresponds to one elementary particle. There are 222 known elementary particles and therefore 18 would be left to discover!! Most of what I will be saying in the next few paragraphs was learned from B. Kostant either during direct conversations at MIT in 2007 or from his talks at the MIT Lie Groups Seminar and at his 80th birthday Conference in Vancouver at the University of British Columbia in May 2008. There are also the video of a talk that he gave at Riverside CA and some notes that were taken by John Baez. http://math. ucr.edu/home/baez/kostant/. Interesting discussions can be found here also: http:// golem.ph.utexas.edu/category/2008/02/kostant_on_e8.html. Lisi is implying a possible embedding of the Standard Model in E 8 . One way to interpret this is to determine whether or not there is a natural embedding of S L 5 in E 8 . It turns out there is a way to accomplish this embedding. Indeed, E 8 a has finite subgroup called the Dempwolf group (FDemp ). Here is another important decomposition of e8 : e8 = (so8 ⊕ so8 ) ⊕ (V8 ⊗ V8 ) ⊕ (S8+ ⊗ S8+ ) ⊕ (S8− ⊗ S8− ). In this case, V8 , S8+ , S8− are 8-dimensional “Vector”, “Right-handed spinor”, “Left-handed spinor” representations of Spin(8) respectively. They are related by triality. Hence, one verifies that dim E 8 = (28 + 28) + (8 × 8) + (8 × 8) + (8 × 8) = 248

13.3 A Theory of Everything

125

Since, it is known that “Most GUT models require a threefold replication of the matter fields and as such, do not explain why there are three generations of fermions. Most GUT models also do not explain the little hierarchy between the fermion masses for different generations.”, and FDemp permutes the three 64 dimensional subspaces (V8 ⊗ V8 ), (S8+ ⊗ S8+ ) and (S8− ⊗ S8− ), Lisi used that triality result to construct the three generations of fermions. This last step seems to be very controversial from the point of view of physicists. See Distler’s blog: http://golem. ph.utexas.edu/~distler/blog/archives/001505.html. Lisi seems to accept that there is indeed a problem according to this answer he posted on November 23, 2007 11:08 AM: The so(7, 1) + so(8) acts on the (S8+ ⊗ S8+ ) as the first generation of fermions. That part works great. The structure of E 8 suggests that the second and third generations relate to the triality partners of the first, (S8− ⊗ S8− ) and (V8 ⊗ V8 ), but I don’t understand this relationship yet. As you know, and as I described in the paper, these second and third triality partners cannot literally be the second and third generation particles as the theory is currently constructed—the relationship is merely suggestive, and I suspect something more interesting is going on. I will probably end up using a slightly different (non-triality) assignment of the fermions, and may even end up using a different group for gravity. Or I might not be able to get it to work. I’ve tried to be very clear, both in the paper and to the press, that this idea is still in development. Most physicists seem to understand this theory is work in progress, and treat it accordingly—but thank you for spending the time to elucidate this fact so that others will understand.

However, one can prove that there are two copies of SU5 in E 8 and they are conjugate under an automorphism of E 8 . There is an element a11 of order 11 in E 8 such that the centralizer of a11 in E 8 , the set {ga11 g −1 = a11 , ∀g ∈ E 8 }, is: (SU3 × U2 ) × (SU3 × U2 ), a product of two copies of the gauge group of the Standard Model. Remember that SU3 × U2 is a subgroup of SU5 . Here is a way to find two conjugate copies of su5 in e8 . There is a copy of G = (Z/5Z)3 in E 8 . As a vector space over (Z/5Z), G contains exactly 31 lines. Let τ be one of such lines. Then the centralizer of τ in E 8 is C = (SU5 × SU5 )/(Z/5Z) with dimension 48. C has a 248-dimensional representation on e8 . So we need to account for a 200dimensional complement which gives: ¯ ⊕ (5¯ ⊗ 10), ¯ e8 = (su5 ⊕ su5 ) ⊕ (5 ⊗ 10) ⊕ (5¯ ⊗ 10) ⊕ (5 ⊗ 10) where 5 is the defining representation of SU5 , 10 is its second exterior power, and 5¯ ¯ are the duals of these representations. and 10

126

13 The Uncontroversial Mathematics Behind Garrett Lisi’s Controversial …

Now, consider the affine Dynkin diagram of E 8

Since C = SU5 × SU5 /(Z/5Z) is the centralizer of an element of order 5 and E 8 is adjoint, the Borel–de Siebenthal theory tells us that the conjugacy classes with semisimple centralizer are indexed by the nodes on the affine Dynkin diagram, with the centralizer given by omitting the node, and the order of the conjugacy class in the adjoint group is given by the usual label on the node. So we must remove the node labelled with 5 (Kostant’s Vancouver talk May 2008). The diagram breaks into two A4 pieces corresponding to the two copies of SU (5). Hence, the two copies of the gauge group of Standard Model S(U3 × U2 ) are given by the diagrams generated by the nodes labelled (1, 3, 4) in the first A4 piece and the ones labelled (6, 3 ,2) in the second A4 piece. An argument of Allen Knutson shows that the two copies of SU5 are conjugate by an element in E 8 . See http://golem.ph.utexas.edu/category/2008/02/kostant_on_ e8.html. Sketch of Knutson’s argument: “Let τ be such an element of order 5. Then τ 2 is another special element of order 5. Hence there exists a g such that gτ g − 1 = τ 2 . Plainly g conjugates C G (τ ) into C G (τ 2 ), which is again C G (τ ) since τ , τ 2 generate the same Z5 . We can distinguish the two factors of C G (τ ) by looking at the projection of τ into them; in one case we get the generator of SU5 ’s center (or we get its inverse, depending on identification), and in the other we get the square (or its inverse). This rule reverses when we look at the projection of τ 2 instead. Summing up, conjugating by g switches the two factors of C G (τ ).” https://golem.ph.utexas.edu/ category/2008/02/kostant_on_e8.html.

13.4 Conclusion In [127], Jacques Distler and Skip Garibaldi give a convincing argument against Lisi’s theory. Here is the content of their introductory remarks: Recently, the preprint [126] by Garrett Lisi has generated a lot of popular interest. It boldly claims to be a sketch of a “Theory of Everything”, based on the idea of combining the local Lorentz group and the gauge group of the Standard Model in a real form of E8 (necessarily not the compact form, because it contains a group isogenous to S L(2, C)). The purpose of this paper is to explain some reasons why an entire class of such models—which include the model in [126]—cannot work, using mostly mathematics with relatively little input from physics.

13.4 Conclusion

127

The first and only time I met and spoke to Garrett Lisi was in 2009 at a conference in honor of Gregg Zuckerman’s 60th birthday at Yale, http://www.liegroups. org/zuckerman/. He was one of the invited speakers, http://www.liegroups.org/ zuckerman/Lisi-Zuckerman-09.pdf. The content of his talk was later published in [130] under the title “An Explicit Embedding of Gravity and the Standard Model in E 8 ”. I write this to point out that some prominent mathematicians were interested in Lisi’s work which I did not follow. I am very happy to see that he is still very involved in physics and appears to be at peace with the Universe. Bert Kostant was also extremely interested in E 8 and spoke to me in length about it when I visited MIT during the 2006–2007 academic year. Later, he mentioned that he was thinking about the mathematical implications of Lisi’s work. Here is an excerpt of what he told Ben Wallace-Wells of THE NEW YORKER in July, 2008: A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent object in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure. It is easy to arrive at the feeling that a final understanding of the universe must somehow involve E(8), or otherwise put, (tongue in cheek) Nature would be foolish not to utilize E(8). There was a good deal of publicity about E(8) in the last few years when a team of about 25 mathematicians, using the power of present computers and a very complicated program, succeeded in determining all of the vast number of (to use a technical term) characters associated with it. Incidentally, one of the main leaders of the team was an ex-student of mine, David Vogan. It was Vogan who told me about Lisi’s paper.

Chapter 14

The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

The function dn(x, κ) will be especially useful in the next section as it will provide for the solution of certain systems of partial differential equations. It, along with the functions sn(x, κ), cn(x, κ), will also be used, for example, in the construction of a cold plasma metric. The importance of these elliptic functions in mathematics, and in other areas, is well known and no further elaboration on this point is required. Their basic definition and properties are reviewed. Quite more detail is available in standard texts of course. The theta functions θ(w | z), θ j (w | z) for j = 1, 2, 3, w ∈ C, z ∈ π + , used here are defined in Chap. 4; see (4.31), (4.32) for θ(w | z) and (4.52) for the θ j (w | z). The Legendre integrals 

x

dt    0 1 − t 2 1 − κ2 t 2   x 1 − κ2 t 2 def. E(x, κ) = dt, 1 − t2 0 def.

F(x, κ) =

(14.1)

say −1  x  1, 0  κ  1 are incomplete elliptic integrals of the first and second kind, respectively, with modulus κ. By the change of variables θ = arcsin t, t = sin θ, 0  t  1,  arcsin x dθ  , F(x, κ) = 0 1 − κ2 sin2 θ (14.2)  arcsin x  2 2 E(x, κ) = 1 − κ sin θdθ. 0

For the choice x = 1, one has the complete elliptic integrals of the first and second kind that will be denoted by K (κ) and E(κ) respectively:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_14

129

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

130

 π 2 dt dθ   , K (κ) = F(1, κ) =  = 0 0 1 − κ2 sin2 θ 1 − t 2 1 − κ2 t 2   1  π 2 1 − κ2 t 2 def. E(κ) = E(1, κ) = dt = 1 − κ2 sin2 θdθ. 2 1−t 0 0 def.



1

(14.3)

For κ = 0 in (14.1), we have of course that 

x

F(x, 0) =

√ 0

dt

= arcsin x, −1  x  1,

1 − t2

(14.4)

so that sin x is the inverse function of the function F(x, 0). This suggests the question of considering, more generally, the inverse function of an incomplete elliptic integral—a question considered by Jacobi, and also by Gauss and Abel. F(x, κ) is a strictly increasing function of x so that it has an inverse function, which is denoted by sn(x, κ):  x = F(sn(x, κ), κ) =

sn(x,κ)

0

dt    , x ∈ R, 1 − t 2 1 − κ2 t 2

(14.5)

x = sn(F(x, κ), κ), −1  x  1. sn(x, κ), whose range is [−1, 1] (= the domain of F(x, κ)), is sometimes called the Jacobi sine function, or the sine amplitude function (Jacobi’s sinus amplitudinus of 1829). A reason for this terminology is seen as follows. If def.



φ

u(φ, κ) =



0

dθ 1 − κ2 sin2 θ

, φ ∈ R,

(14.6)

then the inverse function, denoted by am(x, κ), is called the (Jacobi) amplitude function:  x = u(am(x, κ), κ) =

am(x,κ)

0



dθ 1 − κ2 sin2 θ

, x ∈R

(14.7)

φ = am(u(φ, κ), κ). Now by (14.2), (14.5), (14.6), (14.7) def.



u(arcsin(sn(x, κ)), κ) =

arcsin(sn(x,κ))



0

F(sn(x, κ), κ) = x = u(am(x, κ), κ) ⇒ arcsin(sn(x, κ)) = am(x, κ) ⇒ sn(x, κ) = sin(am(x, κ)),

dθ 1 − κ2 sin2 θ

= (14.8)

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

131

since u, which has an inverse is a 1–1 function. Equation (14.8) justifies the terminology sine amplitude function for sn(x, κ). Note that by (14.3), (14.6), (14.7) π def. , κ ⇒ am(K (κ), κ) = K (κ) = u 2   π π π , κ = ⇒ sn(K (κ), κ) = sin (by (14.8)) = 1. am u 2 2 2

(14.9)

Also u(φ, 0) = φ in (14.6) ⇒ (by (14.7)) x = u(am(x, 0), 0) = am(x, 0) ⇒ (by (14.8)) sn(x, 0) = sin(am(x, 0)) = sin x. (14.10) The change of variables t → −t in (14.1) gives 

−x

F(x, κ) = − 0

dt    = −F(−x, κ) 1 − t 2 1 − κ2 t 2

(14.11)

⇒ −F(x, κ) = F(−x, κ), −1  x  1. Then by (14.5), − x = −F(sn(x, κ), κ) = F(−sn(x, κ), κ) ⇒ sn(−x, κ) = sn (F (−sn(x, κ), κ) , κ) = −sn(x, κ),

(14.12)

which shows that sn(x, κ) is an odd function of x. By (14.8), am(x, κ) is also an odd function of x. sn(x, κ) was defined as the inverse function of the incomplete elliptic integral F(x, κ) in (14.1). The functions cn(x, κ), dn(x, κ) can be defined as the inverse functions of the incomplete elliptic integrals def.



F1 (x, κ) =

1

dt    , −1  x  1, 1 − t 2 (κ )2 + κ2 t 2

1

dt     , κ  x  1, 2 2 2  1 − t t − (κ )

x def.



F2 (x, κ) =

x

respectively, where κ =

def.

is the complementary modulus:

 1 − κ2

(14.13)

(14.14)

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

132

 x = F1 (cn(x, κ), κ) =

1

cn(x,κ)

dt   , x ∈ R 1 − t 2 (κ )2 + κ2 t 2

x = cn(F1 (x, κ), κ), −1  x  1,  1 dt  x = F2 (dn(x, κ), κ) =  , dn(x,κ) 1 − t 2 t 2 − (κ )2

(14.15)

x = dn(F2 (x, κ), κ), κ  x  1. In particular the range of dn(x, κ)(= the domain of F2 (x, κ)) is [κ , 1]: κ  dn(x, κ)  1.

(14.16)

Consider the change of variables:   −udu 1 − t 2 , t = 1 − u 2 ⇒ dt = √ ⇒ 1 − u2  √1−x 2  du def. 2, κ ,  F1 (x, κ) = = F 1 − x √   0 1 − u 2 1 − κ2 u 2 √  1 − t2 −κ2 vdv , t = 1 − κ2 v 2 ⇒ dt = √ v= ⇒ κ 1 − κ2 v 2  √1−x 2 /κ  dv def. F2 (x, κ) = = F 1 − x 2 /κ, κ . √ √ 1 − κ2 v 2 1 − v 2 0

u=

(14.17)

Consequently, by way of (14.5), (14.15), (14.17) F(sn(x, κ), κ) = x = F1 (cn(x, κ), κ) = F  ⇒ sn(x, κ) = 1 − cn(x, κ)2 ; F(sn(x, κ), κ) = x = F2 (dn(x, κ), κ) = F ⇒ sn(x, κ) =





1 − cn(x, κ)2 , κ





1 − dn 2 (x, κ) ,κ κ



(14.18)

1 − dn 2 (x, κ)/κ,

where we use that dn(x, κ) ∈ the domain of F2 (x, κ) by (14.16). That is, sn 2 (x, κ) + cn 2 (x, κ) = 1, dn 2 (x, κ) + κ2 sn 2 (x, κ) = 1.

(14.19)

  Also, in particular, cn(x, κ) = 1 − sn 2 (x, κ) and dn(x, κ) = 1 − κ2 sn 2 (x, κ) are even functions of x since sn(x, κ) is an odd function of x.

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

133

Differentiation of the equations in (14.5), (14.15) respect to x gives 1= 

dsn(x,κ) dx

 , 1 − sn 2 (x, κ) 1 − κ2 sn 2 (x, κ)

1 = − 1 = −

1−

dcn(x,κ) d x cn 2 (x, κ) (κ )2

1−

ddn(x,κ) d x 2 dn (x, κ) dn 2 (x, κ)

+ κ2 cn 2 (x, κ) − (κ )2

,

(14.20)

.

That is, Proposition 14.1 The functions sn(x, κ), cn(x, κ), dn(x, κ) satisfy the following differential equations

dy dx dy dx dy dx

2

2

2

   = 1 − y 2 1 − κ2 y 2 ,     2 = 1 − y2 κ + κ2 y 2 ,

(14.21)

   2 , = 1 − y 2 y 2 − κ

respectively, and in fact by (14.20) we can write, alternatively, that d sn(x, κ) = cn(x, κ)dn(x, κ) dx d cn(x, κ) = −sn(x, κ)dn(x, κ) dx d dn(x, κ) = −κ2 sn(x, κ)cn(x, κ) dx

(14.22)

   2 since κ + κ2 cn 2 (x, κ) = 1 − κ2 + κ2 1 − sn 2 (x, κ) =1 − κ2 sn 2 (x, κ) = dn 2 (x, κ), by (14.19). Moreover by (14.8) d sn (x, κ) d cn(x, κ)dn(x, κ) am(x, κ) =  d x = dn(x, κ). = 2 dx cn(x, κ) 1 − sn (x, κ)

By (14.2)

(14.23)

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

134 def.



F(x, 1) =

arcsin x



dθ

 =

arcsin x

sec θdθ = 0 1 − sin2 θ x log(sec θ + tan θ) |arcsin = 0 log(sec(arcsin x) + tan(arcsin x)) =

1 x 1 +√ log √ = log(1 + x) − log(1 + x)(1 − x) 2 2 2 1−x 1−x

1+x 1 = tanh−1 x ⇒ sn(x, 1) = tanh x, = log 2 1−x 0

(14.24)

since (by definition) sn(x, 1) is  the inverse function of F(x, 1). Then cn(x, 1) =  1 − sn 2 (x, 1) and dn(x, 1) = 1 − sn 2 (x, 1) ⇒ cn(x, 1) = dn(x, 1) = sech x.

(14.25)

We have seen in (14.10) that sn(x, 0) = sin x. Therefore also  1 − sn 2 (x, 0) = cos x,  dn(x, 0) = 1 − 02 sin2 x = 1. cn(x, 0) =

(14.26)

The Jacobi functions sn(x, κ), cn(x, κ), and dn(x, κ) can be expressed in terms of the Jacobi theta functions θ(w | z), θ j (w | z), j = 1, 2, 3 in (4.31), (4.52). Corresponding to the complete elliptic integral K (κ) in (14.3) and the complementary modulus κ in (14.14), we have the complementary elliptic integral    def. 1 − κ2 . K  (κ) = K κ = K Let



q(κ) = e−πK (κ)/K (κ) = eπi z(κ) , def.

z(κ) = i K  (κ)/K (κ) ∈ π + . def.

Then for 0 < κ < 1

(14.27)

 x θ | z(κ) 2 K (κ) 1  sn(x, κ) = √ x κ θ2 | z(κ) 2 K (κ)   x κ θ1 2 K (κ) | z(κ)  cn(x, κ) = x κθ 2 2 K (κ) | z(κ)  x √ θ3 2 K (κ) | z(κ) . dn(x, κ) = κ  θ2 2 Kx(κ) | z(κ)

(14.28)

(14.29)

14 The Elliptic Functions sn(x, κ), cn(x, κ), and dn(x, κ) of C. Jacobi

135

Also 2 K (κ) = πθ32 (0 | z(κ)),

(14.30)

and we can take √

θ1 (0 | z(κ)) √  θ2 (0 | z(κ)) , , κ= κ = θ3 (0 | z(κ)) θ3 (0 | z(κ))



θ2 (0 | z(κ)) κ = . κ θ1 (0 | z(κ))

(14.31)

We have restricted the discussion of the functions sn(x, κ), cn(x, κ), dn(x, κ) to a real variable x, for the applications we have in mind. These functions however can certainly can be defined on the complex domain, and in fact by way of theta functions in a way that exactly generalizes (14.29). Details of such are provided in Chap. 7 of [28], for example.

Chapter 15

The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma, and the J-T Black Hole

The four topics mentioned here might seem to be independent and entirely unrelated. In a sense they are. The genius and beauty of mathematics is often reflected in discoveries that link seemingly disparate topics/ideas. A connecting thread in the present context is, for example, a reaction-diffusion system—with elliptic function solutions. The particular Jacobi elliptic function dn(x, κ) is used to construct an explicit change of variables that transforms a certain cold plasma metric gplasma precisely to a Jackiw-Teitelboim (J-T) black hole metric—in a way that the cosmological constant  of the J-T gravitational field equations is expressed directly in terms of the constant Ricci scalar curvature of gplasma . By way of Lax pairs, the Heisenberg model and the reaction diffusion system are guage equivalent. Thus a set of intertwining, connective ideas will play out. The discussion begins with the classical continuous Heisenberg model realized by real-valued functions S j (x, t), j = 1, 2, 3, subject to the single-sheeted hyperboloid constraint (15.1) − S1 (x, t)2 + S2 (x, t)2 − S3 (x, t)2 = −1, with equations of motion St =

1 def. [S, Sx x ] for S = i 2i



 S1 − S2 S3 . S1 + S2 −S3

(15.2)

def.

The bracket here is the commutator [M1 , M2 ] = M1 M2 − M2 M1 of two matrices M1 , M2 . Let def. H (x, t) = (S1 (x, t), S2 (x, t), S3 (x, t)) , (15.3) def. def. S + = S1 + S2 , S − = S1 − S2 .

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_15

137

138

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

The equations of motion (15.2) can be written out in the form  2 + 2 − 2 ∂ S− ∂2 S− ∂ S3 1 −∂ S +∂ S − ∂ S3 , S − S − S , = = S 3 ∂t 2 ∂x 2 ∂x 2 ∂t ∂x 2 ∂x 2 ∂ S+ ∂ 2 S3 ∂2 S+ = S+ − S . 3 ∂t ∂x 2 ∂x 2

(15.4)

A parametrization of the hyperboloid in (15.1) is S1 (x, t) = sec(a(x, t)) cos(b(x, t)) S2 (x, t) = tan(a(x, t)) S3 (x, t) = sec(a(x, t)) sin(b(x, t)), π π − < a(x, t) < , −π  b(x, t)  π. 2 2

(15.5)

Consider the Minkowski inner product on R3 given by def.

< X, Y > = −x1 y1 + x2 y2 − x3 y3

(15.6)

for X = (x1 , x2 , x3 ), Y = (y1 , y2 , y3 ) ∈ R3 . The function H : R2 −→ R3 in (15.3) provides for an induced metric g H (fundamental form) on the model given by gH g11 g12 g22

 < Hx , Hx > < Hx , Ht > : = < Hx , Ht > < Ht , Ht >       ∂ S1 2 ∂ S2 2 ∂ S3 2 =− + − ∂x ∂x ∂x ∂ S2 ∂ S2 ∂ S3 ∂ S3 ∂ S1 ∂ S1 + − = g21 = − ∂x ∂t ∂x ∂t ∂x ∂t       ∂ S1 2 ∂ S2 2 ∂ S3 2 =− + − . ∂t ∂t ∂t

def.



(15.7)

The importance of this metric for our interest is that it has constant Ricci scalar curvature given by (15.8) R(g H ) = 2, as can be verified by a Maple program (tensor), for example. In practice we will work with a positive multiple g H /2B, B > 0 of g H which has Ricci scalar curvature R(g H /2B) = 4B.

(15.9)

    Since S1 = S + + S − /2, S2 = S + − S − /2 in (15.3), the derivatives of S1 , S2 can be written in terms of those of S + , S − to provide an alternate expression of the metric components gi j in (15.7):

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

g11

∂ S+ ∂ S− − =− ∂x ∂x +

∂ S− ∂t

∂ S3 ∂x

2 +

−

S − ∂∂tS ∂∂x ∂ S3 ∂ S3 − 2 ∂x ∂t 2  + − ∂S ∂S ∂ S3 =− . − ∂t ∂t ∂t

g12 = g21 = g22



S − ∂∂x

139

(15.10)

The following is an outline of a method to construct solutions S of the equations of motion (15.2). Start with real-valued functions r (x, t), s(x, t) and constants B > 0, λ ∈ C. Define     B B def. def. def. −i 0 r, s˜ = s, τ = r˜ = + , 0 i 2 2 ⎡ ⎤ (15.11)   2 λ ∂ s˜ λ def. λ /8 − r˜ s˜ − ∂x + 2 s˜ s ˜ ⎣ ⎦ , U1 (λ) = 4 . U0 (λ) = ∂ r˜ + λ2 r˜ −λ2 /8 + r˜ s˜ ∂x r˜ − λ4 Choose g = g(x, t) ∈ G L(2, R) such that

Then

  ∂g 0 s˜ def. = gU1 (0) = g , r˜ 0 ∂x   ∂ s˜ ∂g −˜r s˜ − ∂x def. = gU0 (0) = g ∂r˜ . r˜ s˜ ∂t ∂x

(15.12)

S = gτ g −1

(15.13)

def.

solves the equations of motion (15.2) provided that r (x, t), s(x, t) are solutions of the reaction—diffusion (R-D) system ∂ 2r ∂r − 2 + Br 2 s = 0 ∂t ∂x ∂2s ∂s + 2 − Br s 2 = 0. ∂t ∂x

(15.14)

The proof involves a couple of observations: (i) The pair (U0 (λ), U1 (λ)) is a Lax pair(named after Peter D. Lax), with spectral parameter λ, for the R-D system (15.14). That is, to show that r (x, t), s(x, t) are solutions of (15.14) is equivalent to showing that U0 (λ), U1 (λ) satisfy the zero curvature condition ∂U1 (λ) ∂U0 (λ) − + [U0 (λ), U1 (λ)] = 0. ∂t ∂x

(15.15)

140

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

 g  g (ii) For g in (15.12), the gauge transformation (U0 (λ), U1 (λ)) −→ U0 (λ), U1 (λ) defined by ∂g −1 def. g g , U0 (λ) = gU0 (λ)g −1 − ∂t (15.16) ∂g −1 def. g −1 g U1 (λ) = gU1 (λ)g − ∂x is computable. For example, by (15.11), (15.12)   λ def. λ4 0 def. iτ = = U1 (λ) − U1 (0), 0 − λ4 4 ∂g ∂g −1 = gU1 (0) ⇒ gU1 (0)g −1 = g , ∂x ∂x ∂g −1 λ def.(15.13) λ iS + g ⇒ gU1 (λ)g −1 = igτ g −1 + gU1 (0)g −1 = 4 4 ∂x ∂g −1 λi g g = S. U1 (λ) = gU1 (λ)g −1 − ∂x 4

(15.17)

g

The computation of U0 (λ) in (15.16), which is also direct, is a bit more involved. The result is λ ∂S λ2 i S g + S . (15.18) U0 (λ) = 8 4 ∂x (iii) Define

 λ ∂S  λ2 i S g + S = U0 (λ) , 8 4 ∂x   def. λi g S = U1 (λ) J1 (λ) = 4 def.

J0 (λ) =

(15.19)

in (15.18), (15.17). Then (J0 (λ), J1 (λ)) is a Lax pair for the equations of motion (15.2) for S. That is, S is a solution of (15.2) ⇔ J0 (λ), J1 (λ) satisfy the zero curvature condition ∂ J1 (λ) ∂ J0 (λ) − + [J0 (λ), J1 (λ)] = 0. (15.20) ∂t ∂x Moreover, by a final, main computation, that uses ∂2g ∂2g = , ∂t∂x ∂x∂t ∂ J1 (λ) ∂ J0 (λ) − + [J0 (λ), J1 (λ)] = g ∂t ∂x



(15.21)

 ∂U1 (λ) ∂U0 (λ) − + [U0 (λ), U1 (λ)] g −1 . ∂t ∂x

(15.22) Putting the pieces together, we see that in the end if r (x, t), s(x, t) solve the R-D system (15.14), which is equivalent to U0 (λ), U1 (λ) satisfying the zero curvature condition (15.15), then by (15.22) the zero curvature condition (15.20) is also satisfied

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

141

by J0 (λ), J1 (λ), which is equivalent to S in (15.13) being a solution of the equations of motion (15.2). We should note that for g in (15.13), 

 αβ − γδ (aβ + γδ) αβ , S2 = − , g= ⇒ S1 = γ δ det g det g −(αδ + βγ) ⇒ −S12 + S22 − S32 = −1 S3 = det g

(15.23)

indeed, which is definition (15.1). A by-product of the many arguments (with various details omitted) that led to the summary just presented is, in particular, that

Then

  ∂S 0 s˜ −1 = 2ig g , −˜r 0 ∂x   ∂S 0 ∂ s˜ = −2ig ∂r˜ ∂x g −1 . ∂t ∂x

(15.24)

  ∂r˜ ∂S ∂S 0 s˜ ∂x −1 = 4g ∂ s˜ g , 0 −˜r ∂x ∂x ∂t   ∂ s˜ ∂S ∂S 0 −˜r ∂x g −1 , = 4g ∂ r˜ 0 ∂x s˜ ∂t ∂x  ∂r˜ ∂ s˜   2 ∂S 0 = −4 g ∂x ∂x ∂r˜ ∂ s˜ g −1 , 0 ∂x ∂x ∂t   2  ∂S r˜ s˜ 0 = 4g g −1 , ⇒ 0 r˜ s˜ ∂x   ∂S ∂S ∂S ∂S ∂ r˜ ∂ s˜ = trace trace = 4 s˜ − r˜ , ∂x ∂t ∂x ∂x ∂t ∂x  2 ∂S ∂ r˜ ∂ s˜ trace = −8 , ∂t ∂x ∂x  2 ∂S = 8˜r s˜ . trace ∂x

(15.25)

On the other hand, by definitions (15.2), (15.3)  S=i 

∂S ∂x

 S3 S − ⇒ S + −S3 ⎡ 

2

⎢ = −⎣ 

⇒ trace

∂S ∂x

∂ S3 ∂x

2

+

∂ S+ ∂ S− ∂x ∂x

∗ 

2 = −2

∂ S3 ∂x

2

⎤ 

∂ S3 ∂x

2

∗ +

∂ S+ ∂ S− ∂x ∂x

∂ S+ ∂ S− + ∂x ∂x



⎥ ⎦

(15.26)

142

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

so that by (15.25)

 −2

∂ S3 ∂x

Similarly, by direct computation of 



∂ S+ ∂ S− + ∂x ∂x

 ∂ S 2 ∂t

 = 8˜r s˜ .

(15.27)

and ∂∂xS ∂∂tS , and by the trace formulas (15.25)

 ∂ r˜ ∂ s˜ ∂S 2 , = −8 −2 = trace ∂t ∂x ∂x     ∂S ∂S ∂ S3 ∂ S3 ∂ S+ ∂ S− ∂ S− ∂ S+ ∂ r˜ ∂ s˜ = trace . − 2 + + = 4 s˜ − r˜ ∂x ∂t ∂x ∂t ∂x ∂t ∂x ∂t ∂x ∂x (15.28) By definition (15.11) r˜ s˜ =

∂ S3 ∂t

B r s, 2

2

2

∂ S+ ∂ S− + ∂t ∂t



∂ r˜ ∂ s˜ B ∂r ∂s ∂ r˜ ∂ s˜ B = , s˜ − r˜ = ∂x ∂x 2 ∂x ∂x ∂x ∂x 2

  ∂r ∂s s −r ∂x ∂x

(15.29)

so going back to the Eq. (15.10) for the components gi j of the Heisenberg metric g H (also see (15.7)), we can use (15.27), (15.28) to express these metric components in terms of r, s and their derivatives (using (15.29) also): g11 = 4˜r s˜ = 2Br s.

    ∂r ∂s ∂s ∂r s −r ⇒ g12 = g21 = B s −r . (15.30) ∂x ∂x ∂x ∂x ∂r ∂s ∂ r˜ ∂ s˜ = −2B . = −4 ∂x ∂x ∂x ∂x

2g12 = 2g21 = 4 g22

B 2

Before getting to the next system of partial differential equations, we remark briefly that plasma physics involves the study of matter composed of charged particles. Such particles (plasmas) are created, for example, by heating a gas until the electrons become detached from their parent atom or molecule. This ionization process can also be attained by way of microwaves, or by a powerful laser beam. The extremely high temperatures of stars, for example, provide an environment for plasmas to flourish. A hot plasma is one that is almost completely ionized. In contrast, cold plasmas, of interest here, are ones where only a tiny fraction of the gas molecules are ionized—say maybe 1%. Consider now the nonlinear system ∂ρ ∂ + (ρu) = 0, ∂t ∂x

∂ ∂u ∂u ∂ρ +u + + β2 ∂t ∂x ∂x ∂x



1 1 ∂2ρ − ρ ∂x 2 2



1 ∂ρ ρ ∂x

2  = 0,

(15.31)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

143

for a constant β > 0, which describes the uni-axial propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma of density ρ(x, t) > 0 and with speed u(x, t) across a magnetic field. MAS, for short, will also refer to this system. In [2], for example, there is a general set of equations that describe the dynamics of cold collisionless plasma in the presence of an external  magnetic field B-equations that include the Maxwell equations ∂ B ∇ · B = 0, ∇ × E = − ∂t

(15.32)

where E is the electric field; these two equations are magnetic and induction laws of Gauss and Faraday. Under some simplifying assumptions, namely B = B(x, t)k, u = u(x, t)i (=uni-axial plasma propagation, for i = (1, 0, 0), k = (0, 0, 1), u =  an initial system the velocity vector)—assumptions that in particular eliminate E, of seven dynamic equations is reduced to a system of three equations. The latter, in turn, by way of a shallow water approximation is reduced to the MAS (15.31) [70, 71]. One can construct a plane wave solution of (15.31) whose wave number k and wave frequency ω are subject to a dispersion relation of the form (see (20.44) in Chap. 20) ω 2 = k 2 ρ0 − k 4 β 2 ,

(15.33)

for a non-zero constant ρ0 . However, of much more interest for the purpose here is the traveling wave solution (ρ(x, t) > 0, u(x, t)), of Gurevich and Krylov (G-K) [51, 52] constructed as follows. Choose

and define

u 0 > 0, α3 > α2  α1  0

(15.34)

√ α3 − α1 u0 def. > 0, a = + > 0, β 2β  α3 − α2 √ def. def. κ = ∈ (0, 1), C = + α1 α2 α3  0. α3 − α1

(15.35)

ν=

Then for the Jacobi elliptic function dn(x, κ) with elliptic modulus κ (see (14.15), (14.23)) def. ρ(x, t) = α1 + 4a 2 β 2 dn 2 (a(x − βνt), κ) > 0, (15.36) C def. . u(x, t) = u 0 + ρ(x, t)

144

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

Note that def.

def.

1 − κ2 = (α2 − α1 ) / (α3 − α1 ) , 4a 2 β 2 = α3 − α1 ⇒   4a 2 β 2 1 − κ2 = α2 − α1 , 4a 2 β 2 + α1 = α3 ⇒    4a 2 β 2 (1 − κ2 ) + α1 4a 2 β 2 + α1 = α2 α3 ⇒

(15.37)

    1 1 def. 2 α1 4a 2 β 2 (1 − κ2 ) + α1 4a 2 β 2 + α1 = (α1 α2 α3 ) 2 = C.

In general if (ρ1 > 0, u 1 ) is an arbitrary solution of the system (15.31), we can construct a corresponding solution (r, s) of the R-D system (15.14) as follows: For γ < 0,

def.

B = −γ/β 2 , (x, t)

∂ def. u 1 = − , ∂x 2

def.

φ(x, t) = (x, t/β)/β,

(15.38)

def.

1 2

r (x, t) = [ρ1 (x, t/β)/(−2γ)] e

φ(x,t)

s(x, t) = − [ρ1 (x, t/β)/(−2γ)] 2 e−φ(x,t) . def.

1

We have seen that there is a corresponding solution S(x, t) therefore of the equations of motion (15.2), with Heisenberg metric g H given by (15.30). We shall refer to g H /2B as the plasma metric since it is attached to the solution (ρ1 > 0, u 1 ) of the MAS system (15.31):   def. def. gplasma = g H /2B = g H / −2γ/β 2

(15.39)

By (15.9), gplasma has constant Ricci scalar curvature given by def.

R(gplasma ) = 4B = −

4γ > 0. β2

(15.40)

By the first equation in (15.30) we have, for example, that def.(15.38)

(gplasma )11 (x, t) = r (x, t)s(x, t) ======= ρ1 (x, t/β)/2γ.

(15.41)

To compute (gplasma )12 , we consider ν + = r x /r, ν − = sx /s, ν + − ν − def.

for r x =

∂r , sx ∂x

=

∂s , ∂x

def.

as usual. We use that by (15.38)

(15.42)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma … def.

def.

φx (x, t) = x (x, t/β)/β = −u 1 (x, t/β)/2β. 2

∂ r (x, t)2 def. r x (x, t) 2r (x, t) r x (x, t) = ∂x = = 2 r (x, t) r (x, t) r (x, t)2

=

ρ1 (x,t/β) 2φ(x,t) e 2φx (x, t) −2γ r (x, t)2

+

∂ ∂x

145

(15.43)

  ρ1 (x, t/β)/(−2γ)e2φ(x,t) r (x, t)2

∂ρ1 (x, t/β)/(−2γ)e2φ(x,t) ∂x r (x, t)2

 2φ(x,t)  e 1 ∂ρ1 (x, t/β) 2γ ∂x r (x, t)2   1 ∂ρ1 −2γ u 1 (x, t/β) def.(15.38) − (x, t/β) ======= − β 2γ ∂x ρ1 (x, t/β) def.(15.38)

======= 2φx (x, t) −

∂ρ

=−

1 (x, t/β) u 1 (x, t/β) + ∂x . β ρ1 (x, t/β)

(15.44)

Similarly, using (15.43) again, ∂ρ1 (x, t/β) 2sx (x, t) u 1 (x, t/β) = + ∂x ⇒ s(x, t) β ρ1 (x, t/β)    +  1 ∂s −u 1 (x, t/β) 1 ∂r def. ν − ν − (x, t) = − (x, t) = . r ∂x s ∂x β

(15.45)

On the other hand   def. r x s − r sx = r ν + s − r ν − s = r s ν + − ν − .

(15.46)

The second equation in (15.30), with (15.45), therefore gives     1 gplasma 12 (x, t) = gplasma 21 (x, t) = (r x s − r sx ) (x, t) 2 ρ1 (x, t/β)u 1 (x, t/β) . =− 4γβ

(15.47)

The remaining component of the plasma metric is easily computed by the work already done, by noting that from (15.42) and the last equation in (15.30) r sν − ν + = r s def.

sx r x g22 def. = r x sx = − = −(gplasma )22 , s r 2B

where by (15.44), (15.45), (15.41)

(15.48)

146

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

     − + 1 ∂ρ u1 1 ∂ρ1 1 u1 1 + − + (x, t/β) ν ν (x, t) = (x, t/β) 2 β ρ1 ∂x 2 β ρ1 ∂x    1 ∂ρ1 2 u 21 1 (15.49) = − 2 (x, t/β), 4 ρ1 ∂x β (r s)(x, t) = ρ1 (x, t/β)/2γ. That is,   ρ1 (x, t/β) gplasma 22 (x, t) = −8γ



1 ∂ρ1 ρ1 ∂x

2

 u 21 − 2 (x, t/β), β

(15.50)

which establishes Theorem 15.1 Let (ρ1 (x, t), u 1 (x, t)) be any solution of the magnetoacoustic system (15.31) with ρ1 (x, t) > 0, The recipe in (15.38), where γ < 0 is arbitrary, provides for a corresponding solution (r (x, t), s(x, t)) of the reaction-diffusion system (15.14) that, in turn, leads to a solution S(x, t) of the Heisenberg equations of motion (15.2)—as we have outlined, using Lax pairs. Let def.

def.

gplasma = g H /2B = g H /(−2γ/β 2 )

(15.51)

be the cold plasma metric, where g H is the induced metric defined in (15.7) on the Heisenberg model. Then the components of gplasma are given explicitly by the following formulas: (gplasma )11 (x, t) = ρ1 (x, t/β)/2γ, ρ1 (x, t/β)u 1 (x, t/β) , (gplasma )12 (x, t) = (gplasma )21 (x, t) = −4γβ    1 ∂ρ1 2 u 21 ρ1 (x, t/β) (gplasma )22 (x, t) = − 2 (x, t/β). −8γ ρ1 ∂x β

(15.52)

gplasma has constant Ricci scalar curvature def.

R(gplasma ) = 4B = −4γ/β 2 .

(15.53)

Theorem 15.1 applies, in particular, to the G-K solution (ρ(x, t), u(x, t)) given in (15.36):   α1 + 4a 2 β 2 dn 2 (a (x − νt) , κ) (15.54) gplasma 11 (x, t) = 2γ def.

and since ν = u 0 /β in (15.35)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

  (gplasma )12 (x, t) = gplasma 21 (x, t) = u 0 ρ(x, t/β) + C ν C = ρ(x, t/β) + = −4γβ −4γ −4γβ ν C ν α1 − . − a 2 β 2 dn 2 (a(x − νt), κ) − γ 4γ 4γβ (gplasma )22 is a little messy to compute. By the differentiation formula −κ2 sn(x, κ)cn(x, κ) in (14.22)

147

(15.55)

d dn(x, κ) dx

∂ρ (x, t/β) = −8a 3 β 2 κ2 (sncndn)(a(x − νt), κ), ∂x

=

(15.56)

def.

so using ν = u 0 /β again, and (15.44) and (15.45) again C ν C u(x, t/β) def. u 0 − =− − ⇒ = − 2β 2β 2βρ(x, t/β) 2 2βρ(x, t/β) C 8a 3 β 2 κ2 (sncndn)(a(x − νt), κ) ν ν + (x, t) = − − − 2 2βρ(x, t/β) 2ρ(x, t/β) ν −4a 3 β 2 κ2 (sncndn)(a(x − νt), κ) − C/2β − , = ρ(x, t/β) 2 (15.57) 3 2 2 β κ (sncndn)(a(x − νt), κ) + C/2β −4a ν ν − (x, t) = + ⇒ ρ(x, t/β) 2 6 4 4 2 2 2 β κ (sncndn) (a(x − νt), κ) − C /4β 16a (ν + ν − )(x, t) = ρ(x, t/β)2 ν2 νC − . − 2βρ(x, t/β) 4 −

Then again by (15.48), (15.49), and (15.36) ρ(x, t/β)(ν + ν − )(x, t) 2γ 6 4 4 2 β κ (sncndn) (a(x − νt), κ) − C 2 /4β 2 16a ∴ = −2γρ(x, t/β) ν 2 α1 ν2 2 2 2 νC + + 4a β dn (a(x − νt), κ). + 4βγ 8γ 8γ

(gplasma )22 (x, t) = −(r sν + ν − )(x, t) = −

Use (15.36) this time to write

(15.58)

148

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

ρ(x, t/β) − α1 ⇒ 4a 2 β 2 16a 6 β 4 κ4 (sncndn)2 (a(x − νt), κ) = −2γρ(x, t/β)   16a 6 β 4 κ4 (sncn)2 (a(x − νt), κ) ρ(x, t/β) α1 = − −2γρ(x, t/β) 4a 2 β 2 4a 2 β 2 4a 4 β 2 κ4 (sncn)2 (a(x − νt), κ) 16a 4 β 2 κ4 α1 (sncn)2 (a(x − νt), κ) + −2γ 8γρ(x, t/β) def.

dn 2 (a(x − νt), κ) =

(15.59)

so that going back to (15.58), we see that in the end we can write  (gplasma )22 (x, t) = 4a 2 β 2



a 2 κ4 ν2 2 (sncn)2 (a(x − νt), κ) + dn (a(x − νt), κ) −2γ 8γ 2

16α1 a 4 β 2 κ4 (sncn)2 (a(x − νt), κ) + Cβ 2 νC ν 2 α1   + + . + 8γ 4γβ 8γ α1 + 4a 2 β 2 dn 2 (a(x − νt), κ) (15.60) Corollary 15.1 When Theorem 15.1 is applied to the specific Gurevich-Krylov solution (ρ(x, t), u(x, t)) of the MAS (15.31), then the formulas in (15.52) for the components of the cold plasma metric are given explicitly by Eqs. (15.54), (15.55), and (15.60)—in terms of the Jacobi elliptic functions sn(x, κ), cn(x, κ), dn(x, κ) with elliptic modulus κ. For now, we express the plasma metric as ds 2 = g11 d x 2 + 2g12 d xdt + g22 dt 2

(15.61)

where the subscript “plasma” is dropped. This metric is non-diagonal: g12 = 0. Fortunately a change of variables can be set up to transform it to a simpler diagonal form. First let   δ δ def. + νt, t . (15.62) δ = a(x − νt); x = + νt ⇒ gi j (x, t) = gi j a a For an initial change of variables (x, t) → (δ, t) dδdt dδ + νdt ⇒ d xdt = + νdt 2 , a a dδ 2 2ν dδdt + ν 2 dt 2 ⇒ dx2 = 2 + a a   g11 2 ds 2 = 2 dδ 2 + [νg11 + g12 ] dδdt + ν 2 g11 + 2νg12 + g22 dt 2 . a a dx =

(15.63)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

149

For convenience, C2 (δ), C1 (δ), and A(δ) will denote the coefficients in (15.63) of dδ 2 , dδdt, dt 2 respectively. For example, by (15.54) and (15.62), C2 (δ) =

α1 + 4a 2 β 2 dn 2 (δ, κ) . a 2 2γ

(15.64)

For A(δ) non-vanishing, a key point that we will get back to, we can choose φ(δ) such that def. C 1 (δ) (15.65) φ (δ) = 2 A(δ) and set up the main change of variables τ → t + φ(δ)

(15.66)

that diagonalizes ds 2 : dt = dτ − φ (δ)dδ ⇒ dδdt = dδdτ − φ (δ)dδ 2 , dt 2 = dτ 2 − 2φ (δ)dτ dδ + φ (δ)2 dδ 2 , def.

ds 2 = C2 (δ)dδ 2 + C1 (δ)dδdt + A(δ)dt 2 =   C2 (δ)dδ 2 + C1 (δ) dδdτ − φ (δ)dδ 2 +   A(δ) dτ 2 − 2φ (δ)dτ dδ + φ (δ)2 dδ 2 =   C2 (δ) − C1 (δ)φ (δ) + A(δ)φ (δ)2 dδ 2 +   C1 (δ) − 2 A(δ)φ (δ) dτ dδ + A(δ)dτ 2 =   C1 (δ)2 dδ 2 + A(δ)dτ 2 , C2 (δ) − 4 A(δ)

(15.67)

where the last step follows by the definition of φ (δ) in (15.65). Note also that   C1 (δ)2 /4 − A(δ)C2 (δ) def. C1 (δ)2 =− C2 (δ) − = 4 A(δ) A(δ)  2    ( a (νg11 + g12 ))2 /4 − ν 2 g11 + 2νg12 + g22 ga112 = − A(δ)  2  − g12 − g11 g22 detg(δ) = 2 ⇒ a 2 A(δ) a A(δ)   −detg(δ)/a 2 ds 2 = A(δ)dτ 2 − dδ 2 . A(δ) Moreover, using Theorem 15.1 we can compute

(15.68)

150

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

  2 (x, t) (det g)(x, t) = g11 g22 − g12

   1 ∂ρ 2 u 2 ρ(x, t/β) ρ(x, t/β) (ρu)2 (x, t/β) − 2 (x, t/β) − = 2γ (−8γ) ρ ∂x β 16γ 2 β 2 2  ∂ρ (x, t/β) /16γ 2 . =− ∂x (15.69) Therefore by (15.56) (det g)(δ) =

−4a 6 β 4 κ4 (sncndn)2 (δ, κ) , γ2

(15.70)

and by (15.68), the plasma metric assumes the diagonal form gplasma : ds 2 = A(δ)dτ 2 −

4a 4 β 4 κ4 (sncndn)2 (δ, κ)dδ 2 , A(δ)γ 2

(15.71)

where A(δ) = (ν 2 g11 + 2νg12 + g22 )( aδ + νt, t) (by (15.62)). But by (15.54) and (15.55) def.

   2  δ νC + νt, t = − = a constant, ν g11 + 2νg12 a 2γβ

(15.72)

and therefore by (15.60)  A(δ) = 4a β

2 2

 a 2 κ4 ν2 2 2 (sncn) (δ, κ) + dn (δ, κ) + −2γ 8γ 2

16α1 a 4 β 2 κ4 (sncn)2 (δ, κ) + Cβ 2 νC ν 2 α1   + − . 8γ 4γβ 8γ α1 + 4a 2 β 2 dn 2 (δ, κ)

(15.73)

As a reminder (see (15.37))   1   1 def. C = +(α1 α2 α3 ) 2 = α1 4a 2 β 2 (1 − κ2 ) + α1 4a 2 β 2 + α1 2 .

(15.74)

As was expressed prior to definition (15.65), a key point for the diagonalization of the plasma metric just presented is the non-vanishing of A(δ). This point is dealt with carefully in [119], where the notation A(ρ) is used there for A(δ) here. Also the g11 in [34, 36, 119] is the g22 here, and vice versa. In [119] it was shown that A(δ) never vanishes if ν 2 is sufficiently large. Namely, if α1 = 0 then ν 2 > 4a 2 κ4 and ν 2 

4 C2 ⇒ A(δ) = 0 ∀δ. α12 β 2

(15.75)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

151

If α1 = 0, then the single condition ν 2 > 4a 2 κ4 ⇒ A(δ) = 0 ∀δ.

(15.76)

One of the main goals of this chapter is to show that one can map the cold plasma metric to a Jackiw-Teitelboim (J-T) black hole metric by a suitable, explicit change of variables. The J-T gravitational field equations are R(g) + 2 = 0 ∇i ∇ j  + gi j = 0, 1  i, j  2

(15.77)

of which a solution consists of a triple (g, , ) where g is a pseudo-Riemannian metric,  =  (x1 , x2 ) is a dilaton field, where (x1 , x2 ) are local coordinates, and  is a cosmological constant. As in our earlier notation, the gi j are the local components of g, and R(g) is the Ricci scalar curvature of g, which is therefore constant: R(g) = −2. The Hessian operator ∇i ∇ j in (15.77) is given locally by  ∂ ∂2 − ikj ∂xi ∂x j ∂xk k=1 2

∇i ∇ j =

(15.78)

where the ikj are the Christoffel symbols of g, of the second kind. The system (15.77) has, for example, the following J-T (Lorentzian) black hole solution, expressed in the coordinates (x1 , x2 ) = (τ , r ). For fixed real number m, M with m = 0, M > 0   def. gbh : ds 2 = − m 2 r 2 − M dτ 2 + def.

def.

dr 2 , −M

m 2r 2

(15.79)

(τ , r ) = mr,  = −m < 0. 2

M is a black hole mass parameter. Not all scalar curvatures are created equal; sign conventions prevail. By our convention, which is that of [34, 36, 119], R (gbh ) = 2 m 2 , which is the negative of the scalar curvature in [61, 97], for example. Of course 2m 2 = −2, as in the first equation of (15.77). Here’s the main result. Keep in mind that throughout γ < 0 is arbitrary. β is given in (15.31); a, ν, α1 are given in (15.34), (15.35). Theorem 15.2 Suppose that ν satisfies the condition in (15.75) so that A(δ) = 0 ∀δ; see (15.73); note also (15.76). Define

152

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

 2 2   ν β − α1 a 2 β 2 dn 2 (δ, κ) a 2 β 2  2 ψ(δ) = + 2−κ − , −γ 2γ 8γ √ −2γ def. > 0, m =+ β  a2ν 2  ν4 def. 2 − κ2 + a 4 κ4 + , A =− 2   16    2 a 2 2 − κ2 3ν 2 νC 3α1 def. β M = + A − α1 + + , −2γ 2β 3 8β 2 16β 4 2β 2 def.

(15.80)

where again C is given by (15.74), and β 2 /(−2γ) = m12 . Then the change of variables (τ , δ) → (τ , r = ψ(δ)) transforms the cold plasma metric gplasma in (15.71) to the metric   dr 2 g : ds 2 = − m 2 r 2 − M dτ 2 + 2 2 , (15.81) m r −M which is exactly a J-T black hole metric, as in (15.79), if M > 0. Indeed if ν is sufficiently large, then M > 0. For example if α1 = 0, then α1 ν C



3α1 + 2a 2 β 8β



  6α1 and ν 2 > 8a 2 2 − κ2 + 2 ⇒ M > 0. β

(15.82)

If α1 = 0, the second condition alone in (15.82) implies that M > 0, and it also implies the first condition ν 2 > 4a 2 κ4 in (15.75). The cosmological constant , and the dilaton field (τ , δ) for the gravitational field equations (15.77) are given by ∴

 = −m 2 =

2γ < 0, (τ , r ) = mr. β2

(15.83)

Other dilaton fields are computed in [34, 36]. By definition (15.34), α1  0 is any choice. The choice α1 = 0 simplifies Theorem 15.2 quite a bit. Then also C = 0, by (15.35) (or (15.74)), A(δ) in (15.73) is simplified and thus so is the plasma metric in (15.71), and M in (15.80) simplifies to def.

M =

β2 A. −2γ

(15.84)

The conditions on ν in Theorem 15.2, for the non-vanishing of A(δ) and for the positivity of M reduce to the single condition ν 2 > 8a 2 (2 − κ2 ).

(15.85)

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

153

It is of some interest to consider the cold plasma ↔ black hole correspondence for non-positive M also—for a black hole vacuum with M = 0, or a naked singularity when M < 0. Consider the latter case, for example, with α1 = 0 and with the condition (15.85) suspended. Since γ < 0, M < 0 ⇔ A < 0, by (15.84). By definition (15.35), ν/a > 0 and by definition (15.80) 

 2  ν2 ν 4ν 4ν 2 2 ⇒ + 4κ + + 4κ − 16A = a a2 a a2 a ν2 ν A < 0 if 2 + 4κ2 − 4 < 0. a a 4

(15.86)

The condition for a naked singularity is therefore ν 2 − 4aν + 4κ2 a 2 < 0.

(15.87)

If the elliptic modulus κ = 21 , for example, then ν 2 − 4aν + 4κ2 a 2 = ν 2 − 4aν + √ √ a 2 = [ν − 2a + 3a][ν − 2a − 3a], and the condition √ √ 3)a > 0, ν − (2 + 3)a < 0 − ie. √ √ ν .2679 = 2 − 3 < < 2 + 3 = 3.732 a

ν − (2 −

(15.88)

is sufficient for a naked singularity. A solution (r (x, t), s(x, t)) of the reaction-diffusion system (15.14) leads not only to a solution of the equations of motion (15.2) for the Heisenberg model (as we have discussed), but in certain cases leads also to a solution ψ(x, t) of the resonance nonlinear Schrödinger equation iψt + ψx x + γ|ψ|2 ψ = δ

|ψ|x x ψ |ψ|

(15.89)

with de Broglie quantum potential |ψ|x x /|ψ|. Here γ ∈ R and δ > 1. For this, choose γ < 0 (which has been the case throughout), and in (15.14) choose def.

B = −γ/(δ − 1) > 0.

(15.90)

√ def. Assume that r (x, t) > 0 and s(x, t) < 0 for all (x, t). Then for β = + δ − 1 β

r (x,βt)

ψ(x, t) = e 2 log r (x,βt)(−s(x,βt)) e−i 2 log −s(x,βt)   r (x, βt) −iβ/2 1 2 = [r (x, βt)(−s(x, βt))] −s(x, βt) def.

1

(15.91)

154

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma …

solves Eq. (15.89). As an example, there is the 1-soliton solution 





2 − ν2 x+ a 2 + ν42 t sech a(x − νt) > 0 e B    2 ν2 x− a 2 + ν42 t def. e s(x, t) = −a sech a(x − νt) < 0 B def.

r (x, t) = a

(15.92)

of (15.14) with soliton velocity parameter ν, for any a > 0, ν ∈ R, B > 0. Then ψ(x, t) in (15.91) is given by  ψ(x, t) = a

2 −iβ/2 e B



   2 −νx+2 a 2 + ν4 βt

sech (x − νβt),

(15.93)

√ def. which therefore solves (15.89) for β = + δ − 1 and for B given by (15.90). We should note that, more generally, given any a, ν, b ∈ R with b = 0, the functions def.

r (x, t) = abe def.

  2 − ν2 x+ a 2 (2−κ2 )+ ν4 t

s(x, t) = −abe

ν 2

dn(a(x − νt), κ),

  2 x− a 2 (2−κ2 )+ ν4 t

(15.94)

dn(a(x − νt), κ)

solve the reaction-diffusion system (15.14): 2 2 r s=0 b2 2 − 2 r s 2 = 0, b

rt − r x x + st + s x x

(15.95)

with B = 2/b2 > 0. For κ = 1, dn(x, 1) = sech x, so one could refer to the solution in (15.94) as a bright soliton. In addition to the 1-soliton solution (15.92) of (15.14), there is also the Lee-Pashaev 2-soliton solution, for example. It is given as follows [70].   2 G+ 2 G− def. def. (x, t), s(x, t) = (x, t), (15.96) r (x, t) = B F B F where for a j , k j ∈ R, 1  j  4, and for

15 The Continuous Heisenberg Model, Reaction Diffusion System, Cold Plasma … def.

def.

def.

def.

155

η1 (x, t) = k1 x + k12 t + a1 , η2 (x, t) = k2 x + k22 t + a2 , η3 (x, t) = k3 x − k32 t + a3 , η4 (x, t) = k4 x − k42 t + a4 , def.

eη1 +η3 eη1 +η4 eη2 +η3 eη2 +η4 + + + + (k1 + k3 )2 (k1 + k4 )2 (k2 + k3 )2 (k2 + ku )2  2 (k1 − k2 ) (k3 − k4 ) eη1 +η2 +η3 +η4 , (k1 + k3 ) (k1 + k4 ) (k2 + k3 ) (k2 + k4 )

F = 1+

(15.97)

(k1 − k2 )2 eη1 +η2 +η3 (k1 − k2 )2 eη1 +η2 +η4 + , (k1 + k3 )2 (k2 + k3 )2 (k1 + k4 )2 (k2 + k4 )2   (k3 − k4 )2 eη1 +η3 +η4 (k3 − k4 )2 eη2 +η3 +η4 − def. η3 η4 . + G = − e +e + (k1 + k3 )2 (k1 + k4 )2 (k2 + k3 )2 (k2 + k4 )2

G + = eη1 + eη2 + def.

The author owes much to the authors of the papers [29, 69–71, 77], whose work has been stimulative and essential towards the discussion and results presented in this chapter.

Chapter 16

The Weierstrass P -Function and Some KdV Solutions

The solution of certain nonlinear differential equations in general relativity can be expressed in terms of the Weierstrass P-function P(z), z ∈ C. Of particular interest is the exact solution of the gravitational field equations with a non-zero cosmological constant, in the context of the Szekeres-Szafron model of inhomogeneous cosmology, that we take up in Chap. 19. The function P(z) can also be used to provide for solutions of the Korteweg-de Vries(KdV) equation, for example,—an equation that plays a pivotal role in plasma physics [62, 63] (which was mentioned ever so briefly in Chap. 15), along with a plethora of applications in other areas as well including nonlinear optics. This equation also has a cosmological connection as it appears in various scenarios including that of an inflationary universe [73]. J.D’ Amboise’s lecture in [33] deals with applications of elliptic and theta functions for a FRLW universe. The Weierstrass function P(z) is related to the Jacobi functions sn(x, κ), cn(x, κ), dn(x, κ) in Chap. 14, that we know are inverse functions of suitable elliptic integrals. Similarly, P(z) is an inverse function of an elliptic integral—although we shall consider the direct definition. Let ω1 , ω2 be two fixed complex numbers that are R-linearly independent. The lattice that they generate will be denoted by  =  (ω1 , ω2 ): def.

 = {mω1 + nω2 | m, n ∈ Z} =  (ω1 , ω2 ) .

(16.1)

P(z) = P(z; ω1 , ω2 ) is defined by   1 1 1 − 2 , P(z) = 2 + z (z − ω)2 ω ω∈ def.

(16.2)

ω=0

which is a meromorphic function on C with double poles at each point of ; such points are its only poles.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_16

157

16 The Weierstrass P -Function and Some KdV Solutions

158

The proof of these statements, including that P(z) is an even function having each ω ∈  as a period (P(z + ω) = P(z)), is presented nicely in chapter one of [4], for example. In particular ω1 and ω2 are periods of P(z). We collect some other basic facts from [4] which are also found elsewhere of course. Similar to the Eisenstein series G k (z) in (4.13), for k = 4, 6, 8, 10, . . ., there are the Eisenstein series def.

G k (ω1 , ω2 ) =



1 , k > 2, ωk ω∈−{0}

(16.3)

say that k = 4, 6, 8, 10, . . . again. In particular for the invariants of  def.

def.

g2 = 60G 4 (ω1 , ω2 ) , g3 = 140G 6 (ω1 , ω2 ) ,

(16.4)

there is the famous differential equation   2 P (z) = 4P(z)3 − g2 P(z) − g3

(16.5)

satisfied by P(z). Since P(z) is constructed by way of the lattice  in (16.1) that is generated by ω1 and ω2 , and since g2 and g3 depend on , (or on ω1 and ω2 ), one also writes (16.6) P(z) = P (z, ω1 , ω2 ) = P (z, g2 , g3 ) . Definition 8.1 of the modular invariant J (z) involves (similar to that of g2 ) 60 times an Eisenstein series—namely G 4 (z). Given the following notation def.

e1 = P

ω  1

2

def.

, e2 = P

ω  2

2

def.

, e3 = P



ω1 + ω2 2

 ,

(16.7)

the e j are distinct roots of the cubic equation 4z 3 − g2 z − g3 = 0

(16.8)

which means that the discriminant def.

 =  (ω1 , ω2 ) =  (g2 , g3 ) = g23 − 27g32

(16.9)

is non-zero. The roots e j and coefficients in (16.8) are related:   g2 = −4 (e1 e2 + e1 e3 + e2 e3 ) = 2 e12 + e22 + e32 , g3 = 4e1 e2 e3 .

(16.10)

16 The Weierstrass P -Function and Some KdV Solutions

159

Also e1 + e2 + e3 = 0,  (g2 , g3 ) = 16 (e1 − e2 )2 (e1 − e3 )2 (e2 − e3 )2 .

(16.11)

Of importance and of general interest are solutions ψ(x) on the nonlinear equation   2 ψ = aψ 4 + 4bψ 3 + 6cψ 2 + 4αψ + δ

(16.12)

for constants a, b, c, α, δ. In a lecture of Weierstrass (around 1860) a solution was presented, which was left to one of his students to publish. That student, G.G.A. Biermann published it in his 1865 Univ. of Berlin inaugural dissertation [11]. I thank Martin Reynolds for some personal correspondence, and for some remarks on his paper [92] on this matter. Also see [102], for example. Let def. (16.13) f (x) = ax 4 + 4bx 3 + 6cx 2 + 4αx + δ be the corresponding quartic polynomial with invariants def.

g2 ( f ) = aδ + 3c2 − 4bα, def.

g3 ( f ) = acδ + 2bcα − aα2 − δb2 − c3 , and discriminant

def.

( f ) = g2 ( f )3 − 27g3 ( f )2 .

(16.14)

(16.15)

If a = 0, then the equation f (x) = 0 has no repeated roots ⇔ ( f ) = 0. In general, suppose two complex numbers a2 and a3 are given that satisfy the condition a23 − 27a32 = 0.

(16.16)

Then there exists complex numbers ω1 , ω2 such that 

a2 = 60

(m,n)∈Z2 (m,n)=0

a3 = 140

1 , (mω1 + nω2 )4



(m,n)∈Z2 (m,n)=(0,0)

1 , (mω1 + nω2 )6

(16.17)

ω1 , ω2 = 0, Im ω2 /ω1 > 0. A. Hurwitz’s proof of this fact is given in [28], for example, on pp. 91–92, which by (16.3), (16.4) means that a2 and a3 give rise to a lattice  (as in (16.1), since ω1 , ω2 are clearly R-linearly independent), with invariants g2 = a2 , g3 = a3 , and

16 The Weierstrass P -Function and Some KdV Solutions

160

corresponding Weierstrass P-function P(z) given by (16.2). With these remarks in mind, the Biermann-Weierstrass solution ψ(z) of Eq. (16.12) is given by Theorem 16.1 Assume that a = 0, and that the discriminant ( f ) of f in (16.15) is non-zero so that, by the preceding remarks, the equation f (x) = 0 has no repeated roots and there exits a lattice (ω1 , ω2 ) with invariants g2 , g3 in (16.4) given by the invariants g2 ( f ), g3 ( f ) of f in (16.14): g2 = g2 ( f ), g3 = g3 ( f ). One can also choose ω1 , ω2 so that Im ω2 /ω1 > 0. If P(z) = P (z; ω1 , ω2 ) is the corresponding Weierstrass P-function, then for x0 ∈ C arbitrary √ ψ(z) = x0 +



 f  (x0 ) 0) P(z) − + f (x0 )P  (z) + f (x 2 24

 (x ) 2 0 2 P(z) − f 24 − a2 f (x0 )

f (x0 ) f  (x0 ) 24

;

(16.18)

here a2 = f  (x)/48. In case x0 is a root of the equation f (x) = 0, Eq. (16.18) simplifies to f  (x0 )

. ψ(z) = x0 + (16.19)  (x ) 0 4 P(z) − f 24 In Theorem 16.1, it is assumed that the equation f (x) = 0 has no repeated roots. If repeated roots exit however, then the integration of Eq. (16.12) can be achieved by way of elementary functions. For example, consider the equation

where

√   2 1 ψ = −4ψ 4 + 2ψ 2 + ψ/ 2 + 16

(16.20)

√ 1 def. f (x) = −4x 4 + 2x 2 + x/ 2 + . 16

(16.21)

By (16.14), (16.15) def.

g2 ( f ) =

1 def. 1 , g3 ( f ) = ⇒ ( f ) = 0, 12 216

(16.22)

√ and indeed x0 = − √2/4 is a repeated√root. The other two roots of the equation f (x) = 0 are 1/2 + 2/4, −1/2 + 2/4. A solution of Eq. (16.12) is ψ(x) =

√ sin(x − A) 2 √ 4 [ 2 − sin(x − A)]

(16.23)

for any constant A ∈ R; see [96], for example. Also in Theorem 16.1, it is assumed that a = 0. Of course if a = 0, then Eq. (16.12) is a generalization of Eq. (16.5) for b = 0, and it is essentially the latter equation if also c = 0. The following simpler result is available.

16 The Weierstrass P -Function and Some KdV Solutions

161

Theorem 16.2 The equation   2 ψ = 4bψ 3 + 6cψ 2 + 4αψ + δ

(16.24)

for b = 0 has as a solution ψ(z) =

P(z + A; g2 , g3 ) c − , b 2b

(16.25)

where by (16.14) (since a = 0) g2 = 3c2 − 4bα, g3 = 2bcα − δb2 − c3 ,

(16.26)

and where one assumes that g23 − 27g32 = 0, which (as in the remarks of Theorem 16.1) provides for a lattice  (ω1 , ω2 ) (with Im ω2 /ω1 > 0 from which P (z; g2 , g3 ) is constructed. A is any constant. In fact if

c def. P(z) = bψ(z) + , b = 0 2

(16.27)

then Eq. (16.24) holds ⇔ 

P

2

= 4P 3 − g2 P − g3 ,

(16.28)

which by comparison with (16.5) has the solution P(z) = P(z + A; g2 , g3 ),

(16.29)

for a constant of integration A, provided that g23 − 27g32 = 0. Then from (16.27), indeed c P (z + A; g2 , g3 ) c P(z) − = − , (16.30) ψ(z) = b 2b b 2b as claimed in (16.25). As was mentioned in initial remarks of this section, solutions of the KdV equation can be expressed in terms of the Weierstrass function P(z)—a point of particular relevance for plasma physics. This equation (with wave amplitude u) u t + au x + buu x + cu x x x = 0

(16.31)

for a function u(x, t), where a, b, c are constants with b = 0 and c = 0, was first considered by Joseph Boussinesq in 1877 and later by Diederik Korteweg and Gustav

16 The Weierstrass P -Function and Some KdV Solutions

162

de Vries in 1895 in their study of long waves in shallow water. One can check directly that for constants x0 , v > a 

v−a u(x, t) = 3 b

 sech

21

2



v−a (x − vt − x0 ) c

(16.32)

is a solution—say c > 0. Most often a = 0 and (16.31) might assume the form u t ± 6uu x + u x x x = 0

(16.33)

where the scale factor ±6 usually provides for a simpler expression of the solution. The soliton solution (16.32) is a traveling wave. Suppose more generally it is proposed that u(x, t) is a traveling wave: u(x, t) = ψ (x − vt − x0 )

(16.34)

for a function ψ(x) where v, x0 are constants. Then by (16.31) 0 = −vψ  + aψ  + bψψ  + cψ  =   d b (−v + a)ψ + ψ 2 + cψ  ⇒ dx 2 b 2 (−v + a)ψ + ψ + cψ  = A 2

(16.35)

for some constant A. It follows that

  2 d b 3 2 = (−v + a)ψ + ψ + c ψ dx 3 2(−v + a)ψψ  + bψ 2 ψ  + 2cψ  ψ  =

b 2 d   (2 Aψ) ⇒ 2ψ (−v + a)ψ + ψ + cψ = 2ψ  A = 2 dx  2 b (−v + a)ψ 2 + ψ 3 + c ψ  = 2 Aψ + B ⇒ 3     2 B b 3 v−a 2A ψ2 + ψ+ ψ =− ψ + 3c c c c

(16.36)

for constants A, B. If we write the last equation of (16.36) as   2 ψ = 4b1 ψ 3 + 6c1 ψ 2 + 4αψ + δ, then Theorem 16.2 applies:

(16.37)

16 The Weierstrass P -Function and Some KdV Solutions

A B b v−a , c1 = , α= , δ= ⇒ 12c 6c 2c c (v − a)2 + 2b A def. g2 = 3c12 − 4b1 α = 12c2   −(v − a) 3b A + (v − a)2 Bb2 2 3 g3 = 2b1 c1 α − δb1 − c1 = − 216c3 144c3 P (x, g2 , g3 ) c1 i.e. 12c v−a P (x, g2 , g3 ) + . ⇒ ψ(x) = − =− b1 2b1 b b

163

b1 = −

(16.38)

Going back to Eq. (16.34), we obtain (in summary) the solution u(x, t) = −

12c v−a P (x − vt − x0 , g2 , g3 ) + b b

(16.39)

of the KdV equation (16.31) where g2 =

  −(v − a) 3b A + (v − a)2 (v − a)2 + 2b A Bb2 , g = − 3 12c2 216c3 144c3

(16.40)

for arbitrary constants A, B, for g23 − 27g32 = 0. For a = 0, the KdV Eq. (16.31) assumes the more traditional form u t + buu x + cu x x x = 0, b = 0, c = 0,

(16.41)

which has the following cnoidal wave solution [1]   √ √ 3v +  2 1 4  cn u(x, t) = √ (x − vt − x0 ) , κ 2b 2 3c for

   3v 1 2 , v > 0; 1+ √  = 9v + 24 Ab, κ = 2  def.

def.

(16.42)

(16.43)

v, x0 , A are constants. cn(x,√κ) is of course the Jacobi function of Chap. 14. For the choice A = 0, we see that  = 3v, κ = 1 (so that cn(x, 1) = sechx by (14.25)) which means that the solution (16.42) is 3v 1 sech2 u(x, t) = b 2



v (x − vt − x0 ) , c

which agrees exactly with (16.32), since we chose a = 0.

(16.44)

164

16 The Weierstrass P -Function and Some KdV Solutions

The potential function u(x) of a particular Zakharov-Shabat system is known to satisfy Eq. (16.20). The solution ψ(x) in (16.23) admits a deformation √ sin(x + 2t) 2 ψ(x, t) = √ 4 [ 2 − sin(x + 2t)] def.

(16.45)

that, remarkably, satisfies the modified KdV equation ψt − 24ψ 2 ψx − ψx x x = 0.

(16.46)

Deformed solutions of Zakharov-Shabat systems are also considered by Varlamov [100]—with some minor corrections of the calculations there given in [115], where the conformal immersion of a surface in R3 induced by a deformed spinor associated with such a system is also computed. The modified KdV equation (16.46) also admits a cnodial wave solution ψ(x, t). In fact, proceeding as in (16.34), the proposal ψ(x, t) = φ (x − vt − x0 ) leads by way of Eq. (16.46) to the conditions −vφ − 24φ2 φ − φ = 0,    d  vφ + 8φ3 + φ = − −vφ − 24φ2 φ − φ = 0, dx φ + vφ + 8φ3 = C

(16.47)

on φ, for a constant C. For the choice C = 0, the last equation is a special case of the general Duffing equation Aφ + αφ + βφ3 = 0, say A, β > 0, α  0 of which we can select a solution of the form ⎛ ⎞ 2 α + c β 1 x + c2 , κ⎠ , φ(x) = c1 cn ⎝ A  c12 β def. 1 κ = √ 2 α + c12 β

(16.48)

(16.49)

for integration constants c1 , c2 . In particular for A = 1, α = v (say v  0), β = 8, Eq. (16.49)) provides for the solution ψ(x, t) = φ (x − vt − x0 ) = acn  a2 def. κ = 2 v + 8a 2

  v + 8a 2 (x − vt − x0 ) + b, κ ,

of the modified KdV equation (16.46), for constants a, b.

(16.50)

16 The Weierstrass P -Function and Some KdV Solutions def.

165

def.

For A = 1 again, and for α = −μ/(1 − δ), β = −1/(1 − δ), where we choose μ < 0, δ < 1, the Duffing equation (16.48) assumes the form φ −

φ3 μφ − =0 1−δ 1−δ

(16.51)

which has the solution √ φ(x) = −μ tanh x



−μ 2(1 − δ)

(16.52)

considered in Sect. 4 of [59] , for example. This solution was used to build the dark (or topological) soliton solution def.

ψ(x, t) = e

iμt

φ(x) = e

iμt √



−μ tanh x

−μ 2(1 − δ)

(16.53)

in equation (32) of [59] of the resonance nonlinear Schrödinger equation iψt + ψx x − |ψ|2 ψ = δ

|ψ|x x ψ. |ψ|

(16.54)

We considered the latter equation in Chap. 15—namely Eq. (15.89) there, and the solution (15.93) where, contrary to the present assumption with δ < 1, we assumed that δ > 1. As was pointed out in [59], for δ = 0, the solution ψ(x, t) in (16.53) reduces to the well-known dark solition solution of the defocusing nonlinear Schrödinger equation. For various practical applications, the Weierstrass function P(z) is conveniently expressed in terms of the Jacobi functions sn(z), cn(z), dn(z). Suppose for example that g2 , g3 are real number, and that the discriminant  =  (g1 , g3 ) is positive. The roots e1 , e2 , e3 of the cubic equation (16.8) are then all real, and one custom is to take e1 > e2 > e3 . For  e2 − e3 def. , (16.55) κ = e1 − e3 the Weierstrass function is expressed in terms of the Jacobi functions as follows: e1 − e3 √  e1 − e3 x, κ  2  dn √ = e2 + (e1 − e3 ) e1 − e3 x, κ sn  cn 2 √  = e1 + (e1 − e3 ) e1 − e3 x, κ . sn

P (x, g2 , g3 ) = e3 +

sn 2

(16.56)

16 The Weierstrass P -Function and Some KdV Solutions

166

We consider a concrete application of the formulas in (16.56) by way of a solution of the particular Duffing equation 3B B φ(x) + 3 φ(x)3 = 0, x0 x0

φ (x) −

(16.57)

for example. Here x0 , B ∈ R − {0}; later we take xB0 > 0; compare Eq. (16.48). By (16.57),    2 φ 3B 2 B 4 d − φ + 3φ = dx 2 2x0 4x0

3B B (16.58) φ + 3 φ3 = 0 ⇒ φ φ − x0 x0   2 φ 3B 2 B − φ + 3 φ4 = C 0 2 2x0 4x0 for some constant C0 ∈ R ⇒   2 −B 3B 2 φ = 3 φ4 + φ + 2C0 . x0 2x0 Let def.

f (x) = −

B 4 3B 2 x + x + 2C0 x0 2x03

(16.59)

(16.60)

= ax + 4bx + 6cx + 4αx + δ, 4

3

2

as in (16.13): a=−

B B , b = 0, c = , α = 0, δ = 2C0 . 2x0 2x03

(16.61)

Now, with Theorem 16.1 in mind, we choose the integration constant C0 so that x0 is a root of the equation f (x0 ) = 0: def.

0 = f (x0 ) = −

B 5 x0 + 3Bx0 + 2C0 ⇒ C0 = − Bx0 . 2 4

Also, f  (x) = − 

2B 3 B x + 6 x, 3 x0 x0

⇒ f (x0 ) = 4B,



f  (x) = −

f (x0 ) = 0.

Lastly, by (16.14), (16.61), and (16.62)

6Bx 2 6B + 3 x0 x0

(16.62)

(16.63)

16 The Weierstrass P -Function and Some KdV Solutions

g2 = g2 ( f ) = aδ + 3c2 − 4bc = aδ + 3c2 =

167

2B 2 , x02

g3 = g3 ( f ) = acδ + 2bcα − aα2 − δb2 − c3 = = acδ − c3 = def.

B3 ⇒ 2x03

 (g2 , g3 ) = g23 − 27g32 =

(16.64)

5 B6 > 0. 4 x06

Using the data just gathered, we can apply Theorem 16.1 (in particular Eq. (16.19)) to obtain the solution f  (x0 ) 4 [P (x, g2 , g3 ) − f  (x0 ) /24] B , = x0 + P (x, g2 , g3 ) 2B 2 B3 g2 = 2 , g3 = 3 x0 2x0

φ(x) = x0 +

(16.65)

of the Duffing equation (16.57). On the other hand, the roots e1 , e2 , e3 of the equation 4x 3 − g2 x − g3 = 0 in (16.8), with e1 > e2 > e3 , are given by B e1 = 2x0



 √  √  1+ 5 B 1− 5 B , , e2 = , e3 = − 2 2x0 2 2x0

(16.66)

where we now assume that B/x0 > 0. We see that √ e2 − e3 3− 5 = √ . e1 − e3 3+ 5

(16.67)

Moreover √ 4 √ =3− 5⇒ 3+ 5 √  2   2 1 3− 5 e2 − e3 4 = ⇒ √ √ √ = √ = e1 − e3 3+ 5 3 + 5 (3 + 5) 3+ 5 2 κ= √ 3+ 5

(3 +

√ √ 5)(3 − 5) = 9 − 5 = 4 ⇒

by definition (16.55). e1 − e3 =

√ B (3 + 5) 4x0

(16.68)

(16.69)

168

16 The Weierstrass P -Function and Some KdV Solutions

so that by (16.56) and (16.65) we have (in summary) a solution φ(x) of the Duffing equation (16.57) given by B P (x, g2 , g3 ) B = x0 + 3 e3 + sn 2 (√ee1 1−e −e3 x,κ)

φ(x) = x0 +

= x0 + = x0 +

where

B e2 + (e1 − e3 ) B

(16.70)

√ dn 2 ( e1 −e3 x,κ) √ sn 2 ( e1 −e3 x,κ) √

cn 2 e −e x,κ e1 + (e1 − e3 ) sn 2 (√e1 −e3 x,κ) ( 1 3 )

,

2B 2 B3 , g = , 3 x02 2x03 2 κ= √  .38197, 3+ 5 √ B e1 − e3 = (3 + 5), 4x0

g2 =

(16.71)

and where e1 , e2 , e3 are given by (16.66); we take B/x0 > 0. In the initial remarks of this chapter it was stated that “P(z) is an inverse function of an elliptic integral”. This statement in essence is the result (16.5), which one could formulate as follows. If P  (z 0 ) = 0 at some point z 0 ∈ C, then P(z) has a local inverse function f (z) on some neighborhood N of P (z 0 ) : P( f (z)) = z and f  (z) = 

1 4z 3

− g2 z − g3

(16.72)

on N . The KdV and modified KdV equations can be related by way of a Miura transformation (named after Miura [81]), which is illustrated by way of the following example. Consider the modified KdV equation ψt − 6ψ 2 ψx + ψx x x = 0,

(16.73)

which is similar to Eq. (16.46), and consider the KdV equation u t − 6uu x + u x x x = 0

(16.74)

16 The Weierstrass P -Function and Some KdV Solutions

169

in (16.33). Here the remarkable Miura transformation is given by def.

ψ → u = ψ2 +

∂ψ , ∂x

(16.75)

where regarding Eqs. (16.73) and (16.74) the following formula holds: 

   ∂ + 2ψ ψt − 6ψ 2 ψx + ψx x x = u t − 6uu x + u x x x . ∂x

(16.76)

Indeed  ∂  ψt − 6ψ 2 ψx + ψx x x = ψt x − 6ψ 2 ψx x − 12ψ (ψx )2 + ψx x x x , ∂x   2ψ ψt − 6ψ 2 ψx + ψx x x = 2ψψt − 12ψ 3 ψx + 2ψψx x x , u t = 2ψψt + ψxt , −6uu x =   − 6 ψ 2 + ψx (2ψψx + ψx x ) = −12ψ 3 ψx − 12ψ (ψx )2 − 6ψ 2 ψx x − 6ψx ψx x , u x = 2ψψx + ψx x , u x x = 2ψψx x + 2ψx2 + ψx x x , u x x x = 2ψψx x x + 2ψx ψx x + 4ψx ψx x + ψx x x x = 2ψψx x x + 6ψx ψx x + ψx x x x , (16.77) so by the equality ψxt = ψt x of mixed partials the right hand side of (16.76) is 2ψψt + ψt x − 12ψ 3 ψx − 12ψ (ψx )2 − 6ψ 2 ψx x + 2ψψx x x + ψx x x x (since the terms ∓6ψx ψx x cancel in addition), which is the sum of the first two equations in (16.77) and which therefore is the left hand side of (16.76). Consequently a solution ψ of the modified KdV equation (16.73) provides for a solution u of the KdV equation (16.74) by way of (16.75). In Chap. 15, a Lax pair with a spectral parameter λ was constructed for the Heisenberg equations of motion (15.2), and another pair was constructed for the reactiondiffusion system (15.14). There is a well-known Lax pair (L , M), similarly, for the KdV equation (16.74), for example. It is given by def.

L = −

∂2 + u, ∂x 2

def.

M = 4

∂3 ∂ − 3u x ; − 6u ∂x 3 ∂x

(16.78)

L therefore is a Sturm-Liouville operator. We will check that the commutator def. [L , M] = L ◦ M − M ◦ L is given by [L , M] f = −u x x x f + 6uu x f

(16.79)

def.

for a function f (x, t). Thus for L t f = u t f , we see that the condition L t = [L , M]

(16.80)

16 The Weierstrass P -Function and Some KdV Solutions

170

coincides with the condition that u is a solution of (16.74). Such a solution is given by √ v 2 v (x − vt), (16.81) u(x, t) = − sech 2 2 for example, which follows by (16.32) for the choices a = 0, b = −6, c = 1 in (16.31), and it represents a wave that travels with velocity v, and mainly its shape is maintained. Such a wave was observed by John Scott Russell in 1834 while on horseback at the Union Canal near Edinburgh Scotland, which he called a “wave of translation”. Whereas ordinary waves tend to flatten out or topple over, these translation (or solitary) waves remain stable as they travel large distances. u(x, t) in (16.81) (or in (16.32) more generally) is therefore called a soliton solution. To complete the discussion of the Lax pair (L , M), the Lax condition (16.80) needs to be check. Using the definition (16.78), one could compute the commutator [L , M] by brute force-but this could also be done immediately by way of Maple, for example. For the function f (x, t) above, a Maple program gives ∂5 f + 12u x x f x + 15u x f x x + 10u f x x x ∂x 5 + 3u x x x f − 6u 2 f x − 3uu x f,

(L ◦ M) f = − 4

∂5 f (M ◦ L) f = − 4 5 + 4u x x x f + 12u x x f x + 15u x f x x ∂x + 10u f x x x − 9uu x f − 6u 2 f x .

(16.82)

It follows that (L ◦ M) f − (M ◦ L) f = −u x x x f + 6uu x f indeed, as asserted in (16.79).

(16.83)

Chapter 17

The Weierstrass Sigma and Zeta Functions: Theta Function Connections

The Weierstrass sigma function with respect to the lattice  in (16.1) is defined by def.

σ(z) = σ(z; ) = z

 z  z + 1 ( z )2 eω 2 ω 1− ω ω∈

(17.1)

ω=0

for z ∈ C. There are some technicalities involved that insure σ(z) is quite welldefined, and that σ(z) is in fact an entire function. For convenience, we supply details here even though the arguments are standard in the literature—and also since we skipped the proofs of initial assertions regarding the Weierstrass function P(z) in Chap. 16. For n ∈ Z, n  1, the entire functions z2

zn

E n (z) = (1 − z)e z+ 2 +···+ n , def.

(17.2)

called elementary factors, are used in the Weierstrass factorization of an entire function over its set of zeros. These factors satisfy the following well-known estimate. Proposition 17.1 |E n (z) − 1|  |z|n+1 for |z|  1.

(17.3)

Only the case n = 2 in this Proposition is needed to establish the uniform convergence of the infinite product in (17.1) on an arbitrary closed disk—and hence to establish that σ(z) is an entire function. The basic ideas of the proof for arbitrary n are also reflected in the case n = 2, which we therefore focus on. We now write 2 def. def. (17.4) φ(z) = E 2 (z) = (1 − z)e z+z /2 , for which Proposition 17.1 asserts that |z|  1 ⇒ |φ(z) − 1|  |z|3 . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_17

(17.5) 171

172

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections def.

For f (z) = 1 − φ(z), f  (z) = −φ (z) = z 2 e z+z

2

/2

f  (0) = 0,

,

f  (0) = 0,

f  (0) = 2 = 0. (17.6)

That is, f (z) has a zero of order 3 at 0: On a neighborhood N of 0 there exists a holomorphic function g(z) such that f (z) = z 3 g(z) on N , with g(0) =

f  (0) ∴ 2 1 = . = 3! 3! 3

(17.7)

In other words, we can extend 1 − φ(z) def. f (z) = z3 z3

(17.8)

to an entire function (since clearly 1 − φ(z) is entire) by assigning to it the value g(0) = 13 at z = 0. The entire functions f (z), f  (z) have Taylor series f (z) =

∞ 

cn z n ,

f  (z) =

n=0

∞ 

ncn z n−1 ,

(17.9)

n=1

def.

since f (0) = 0 ⇒ c0 = 0. On the other hand e z+z

2

/2

=1+

∞ 

an z n

(17.10)

n=1 def.

with an > 0 for n  1. For example, for u = z + z 2 /2, u3 u4 u2 + + + ··· , 2 3 4 u2 z2 z3 z4 u3 z3 z4 z5 z6 = + + , = + + + , 2 2 2 8 3 3 2 4 24 u4 z4 z5 3 6 z7 z8 = + + z + + . 4 4 2 8 8 64

eu = 1 + u +

(17.11)

Then by (17.6), f  (z) = z 2 e z+z

2

/2

= z2 +

∞ 

an z n+2 = z 2 + a1 z 3 + a2 z 4 + a3 z 5 + a4 z 6 + · · · ,

n=1

(17.12) whereas by (17.9) f  (z) = c1 + 2c2 z + 3c3 z 2 + 4c4 z 3 + 5c5 z 4 + 6c6 z 5 + 7c7 z 6 + · · · ,

(17.13)

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

173

by which one can conclude that c1 = 0, c2 = 0, 3c3 = 1, 4c4 = a1 , 5c5 = a2 , 6c6 = a3 , 7c7 = a4 , · · · . ⇒ cn > 0, n  3

(17.14)

since an > 0 for n  1. We have noted that f (z)/z 3 is entire,so if we write its Taylor expansion as ∞  f (z)/z 3 = bn z n (17.15) n=0

we have by (17.9) ∞  n=0 b0 z 3

bn z n+3 = f (z) = + b1 z + b2 z + 4

5

∞ 

cn z n ⇒

n=0 b3 z 6

+ b4 z 7 + · · · = c0 + c1 z + c2 z 2 +

(17.16)

c3 z 3 + c4 z 4 + c5 z 5 + c6 z 6 + c7 z 2 + · · · ⇒ b0 = c3 , b1 = c4 , b2 = c5 , b3 = c6 , b4 = c7 , · · · , where (as we have seen) c0 = c1 = 0. Since cn > 0 for n  3 by (17.14), we see that bn > 0 for n  0! Then (17.5) follows. Namely, by (17.8), (17.15)     ∞  1 − φ(z)   f (z)    =  |bn | |z|n  z3   z3  n=0 =

∞ 

bn |z|n 

n=0

∞ 

bn ( if |z|  1)

(17.17)

n=0

f (1) def. = 1 − φ(1) = 1 − 0 = 1 ⇒ 13 |1 − φ(z)|  |z|3 . def.

=

To check that σ(z) is an entire function, let def.

D R = {z ∈ C | |z|  R}

(17.18)

be an arbitrary closed disk of radius R > 0. By definition (17.4) we can write  z  z + 1 ( z )2   z    z    z  eω 2 ω = = 1− φ φ φ ω ω ω ω∈ ω ω∈ ω∈ ω∈ ω =0

ω =0

ω=0 |ω|≤R

|ω|>R

(17.19)

174

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

where the finite product over the non-zero ω with |ω|  R is an entire function, since φ(z) is an entire function. Let def.

aω (z) = φ

z ω

−1

(17.20)

for ω ∈  − {0} so that 

φ

ω∈ |ω|>R

z ω

=



(1 + aω (z)),

(17.21)

ω∈ |ω|>R

which we show converges absolutely and uniformly on D R —to conclude that σ(z) is entire, as D R is arbitrary. That is, we need to show that the series 

aω (z)

(17.22)

ω∈ |ω|>R

converges absolutely and uniformly on D R . But for z ∈ D R and ω ∈   |ω| > R, we have that |ω| > R  |z| ⇒ |z/ω| < 1 so that by (17.5) and (17.20)   z |z|3 R3 def.   |aω (z)| = φ − 1   , ω |ω|3 |ω|3

(17.23)

 1  1  2, |ω|α ω∈

(17.25)

where

|ω|>R

since in general

ω =0

ω =0

a point we will get back to. Thus by (17.23) and (17.24) the absolute and uniform convergence of the series in (17.22) on D R is established, and we see indeed that σ(z) in (17.1) is a well-defined entire function. The initial step towards proving the convergence of the defining series in (16.2) for P(z) is accomplished in fact by way of the result (17.25). Equation (17.25) will also be used in the inequality (17.35) below towards establishing the convergence of the Weierstrass zeta function in (17.31). By the argument just presented the function def.

H R (z) =

 ω∈ |ω|>R

(1 + aω (z))

(17.26)

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

175

is holomorphic on |z| < R (as this product converges absolutely and uniformly on D R ). In particular, H R (z) is continuous at 0: lim H R (z) = H R (0) = 1

z→0

(17.27)

by (17.20), since φ(0) = 1 by (17.4). By (17.19)   z z  z + 1 ( z )2 eω 2 ω = H R (z) 1− φ ω ω ω∈ ω∈

(17.28)

ω=0 |ω|R

ω =0

where for the finite product here over ω ∈  − {0} with |ω|  R lim

z→1

 ω∈ ω=0 |ω|R

z φ( ) = 1, ω

(17.29)

again as φ(0) = 1. Thus by (17.1), (17.28) we see that σ(z) = 1. z→0 z lim

(17.30)

Now we check that the function   1 1 z 1  + + 2 ζ(z) = ζ (z; ) = + z z−ω ω ω ω∈ def.

(17.31)

ω =0

is a well-defined meromorphic function on C-called the Weierstrass zeta function, whose set of poles is , all of which are simple with residues equal to 1. For z = ω, ω = 0  (z − ω)ω

2

1 1 z + + 2 z−ω ω ω

 = ω 2 + (z − ω)ω + (z − ω)z

= ω 2 + (z + ω)(z − ω) = ω 2 + z 2 − ω 2 = z 2 ⇒ 1 1 z z2 z2

+ + 2 = = ⇒ z−ω ω ω (z − ω)ω 2 ω ωz − 1 ω 2    1 1 |z|2 1 z    z − ω + ω + ω 2  =  z − 1 |ω|3 ω

where

(17.32)

176

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

z  z 1 1       ⇒  − 1  1 −   ⇒  z  ω ω −1 1 −  ωz  ω    1 1 1 z  |z|2     + + . z − ω ω ω 2  1 −  z  |ω|3

(17.33)

ω

Again for R > 0, let D R be an arbitrary closed disk of radius R as in (17.18). Then for |z|  R and |ω| > 2R  2|z|, |z| < |ω|/2 ⇒ (in particular) z = ω and ω = 0, and z z 1 1 1      < ⇒1− >1− = ⇒ 2 ω 2 2 ω (17.34) 2 2  1  |z| 1 z 2R    z − ω + ω + ω 2  < 2 |ω|3  |ω|3 by (17.33), where

 ω∈ |ω|>2R

 1 1  2R

1 z 1 + + 2 z−ω ω ω

 (17.36)

converges absolutely and uniformly on every closed disk: D R : |z|  R. h R (z) is therefore a holomorphic function on |z| < R, which can be differentiated term by term:    −1 z (17.37) + h R (z) = (z − ω)2 ω2 ω∈R |ω|>2R

on |z| < R. Since ζ(z) −

     1 1 def.  1 1 z 1 z = + + 2 = + + 2 + h R (z) z z−ω ω ω z−ω ω ω ω∈ ω∈ ω =0

ω=0 |ω|≤2R

(17.38) with R > 0 arbitrary, ζ(z) − 1z is meromorphic on C with  − {0} as its set of poles, and ζ(z) is meromorphic on C with  as its set of poles, all of which are simple with residues equal to 1. Moreover, the finite sum in (17.38) with ω = 0, |ω|  2R can be differentiated term by term:

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

ζ  (z) = −

177

   −1 1 1 + h R (z) + + 2 2 z2 (z − ω) ω ω∈ ω=0 |ω|2R

     −1 1 −1 1 z =− 2 + + 2 + + 2 , z (z − ω)2 ω (z − ω)2 ω ω∈ ω∈ ω=0 |ω|2R

(17.39)

|ω|>2R

by (17.37). That is, by definition (16.2), ζ  (z) = −P(z).

(17.40)

Since h R (z) in (17.36) is continuous on |z| < R (in particular), h R (z) → h R (0) = 0 as z → 0. Also (again) the sum in (17.38) with ω ∈  − {0}, |ω|  2R is finite. Thus we see in (17.38) that   1 = 0. (17.41) lim ζ(z) − z→0 z Earlier it was shown that the product in (17.1) defining σ(z), namely P(z) =

 z  z + 1 ( z )2  eω 2 ω = 1− (1 + aω (z)) , ω ω∈ ω∈ ω =0

(17.42)

ω =0

converges absolutely and uniformly on every closed disk D R , R > 0. A general log derivative result therefore gives P  (z)  aω (z) = P(z) 1 + aω (z) ω∈

(17.43)

ω =0

where the series converges absolutely and uniformly on every D R in which P(z) has no zeros: on D R containing no ω ∈  − {0}. By (17.20), (17.6), (17.4), and (17.32)

2 z 1 z 2 φ ωz ω1 − z eω+2(ω) 1 aω (z) z = ω z 1 z ω2 = 1 + aω (z) φ ω 1 − ωz e ω + 2 ( ω ) 1 z z2 1

= z + + 2 ⇒ 3 z − ω ω ω ω ω −1   1 P  (z)  1 z = + + 2 . P(z) z−ω ω ω ω∈ =

ω =0

(17.44)

178

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

On the other hand 

zσ (z) − σ(z) /z 2 P  (z) σ  (z) 1 σ(z) ⇒ = = − ⇒ z P(z) σ(z)/z σ(z) z     σ (z) 1 1 1 z = + + + 2 = ζ(z) σ(z) z z − ω ω ω ω∈ def.

P(z) =

(17.45)

ω =0

by definition (17.31). Regarding the Weierstrass P-function P(z) in (16.2), and the Weierstrass sigma and zeta functions σ(z) and ζ(z) in (17.1) and (17.31), we have therefore checked in (17.30), (17.40), (17.41), (17.45), respectively, that σ(z) = 1, ζ  (z) = −P(z), z→0 z   1 σ  (z) = 0, = ζ(z). lim ζ(z) − z→0 z σ(z) lim

(17.46)

A sum or product indexed by (m, n) ∈ Z2 or Z2 − {0} is indexed by (−m, −n). In particular ω = mω1 + nω2 varies over the lattice  in (16.1) or over  − {(0, 0)}, as −ω does. For this reason P(z) in (16.2) is an even function—as was stated earlier. Similarly, by (17.1), σ(z) is an odd function, and by (17.31) ζ(z) is an odd function. The formulas (4.36) and (4.53) relate the Dedekind eta function η(z) in (4.18) to the Jacobi theta functions θ(w | z), θ1 (w | z), θ2 (w | z), θ3 (w | z) in (4.32) and (4.52); here (w, z) ∈ C × π + . There is also a formula that relates the Weierstrass sigma function σ(z) = σ(z; ) = σ (z; ω1 , ω2 ) to the Dedekind eta function and the Jacobi theta function θ(w | z); σ(z) is already related to the Weierstrass zeta function ζ(z) by way of (17.46). Before stating that formula we first define the quasi periods η1 , η2 of ζ(z) by ω  ω  def. def. 1 2 , η2 = ζ , (17.47) η1 = ζ 2 2 of which more will be said later, and where we assume the normalization def.

τ =

ω2 ∈ π+ : ω1

Im

ω2 >0 ω1

(17.48)

in (16.17). Note that the definition of η1 , η2 is analogous to that of e1 , e2 in (16.7). Here’s the formula:   2 ω1 eη1 w /ω1 θ ωw1 | τ σ (w; ω1 , ω2 ) = (17.49) 2πη(τ )3 for η1 in (17.47) and τ in (17.48), w ∈ C.

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections

179

Since ω1 , in particular, is a period of P(z) d [ζ (z + ω1 ) − ζ(z)] = −P (z + ω1 ) + P(z) = −P(z) + P(z) = 0 dz ⇒ ζ (z + ω1 ) − ζ(z) = c1

(17.50)

for some constant c1 , where (17.46) is used. Since ζ(z) is an odd function, the choice z = −ω1 /2 gives c = ζ (ω1 /2) + ζ (ω1 /2) = 2η1 , by (17.47) ⇒ ζ (z + w1 ) = ζ(z) + 2η1 .

(17.51)

Similarly, since ω2 is also a period of P(z), ζ(z + ω2 ) − ζ(z) = some constant c2 , by the argument in (17.50), and the choice z = −ω2 /2 gives c2 = 2η2 ⇒ ζ (z + ω2 ) = ζ(z) + 2η2 .

(17.52)

Equations (17.51) and (17.52) provide for the reason that η1 , η2 are called quasi¯ periods. What is the dimension of the Q-vector space generated by the elements {1, π, ω1 , ω2 , 2η1 , 2η2 }, where Q¯ is the algebraic closure of the field of rationals Q? This is a very deep question in transcendental number theory which is evidently yet unresolved. η1 and η2 can be expressed in terms of the quasi-modular form def.

G 2 (z) =

∞  π2 − 8π 2 σ(n)e2πinz , z ∈ π + . 3 n=1

(17.53)

For convergence reasons, the holomorphic Eisenstein series G k (z) in (4.13) are defined only for k > 3, say k = 4, 6, 8, 10, . . ., since a modular form of odd weight vanishes. On the other hand, the Fourier expansion of G k (z) in (4.14) can be used as a motivation to define G k (z) for k = 2, which leads exactly to the definition def. def. (17.53), where σ(n) = σ0 (n) = the sum of the positive divisors of n by (4.11), 2 and 2ζ(2) = 2π /6. Moreover, as shown on pp. 42–44 of [114] for example (i): the series in (17.53) does converge absolutely, and the convergence is uniform on compact subsets of π + so that G 2 (z) is holomorphic on π + , and (ii): in contrast to the condition f (−1/z) = z 2 f (z) (which is (M2) for k = 2, prior to condition (M1) in (4.3)), G 2 (z) satisfies the condition   1 = z 2 G 2 (z) − 2πi z G2 − z

(17.54)

so that it is not a modular form of weight 2. The expression of η1 , η2 in terms of G 2 (z) is η1 =

G 2 (τ ) τ G 2 (τ ) − 2πi ω2 G 2 (τ ) πi , η2 = = − 2ω1 2ω1 ω1 2ω12

(17.55)

180

17 The Weierstrass Sigma and Zeta Functions: Theta Function Connections def.

for τ = ω2 /ω1 ∈ π + in (17.48). In particular η1 and η2 satisfy the Legendre relation  ω2 η1 − ω1 η2

ω2 G 2 (τ ) ω2 = G 2 (τ ) − πi − ω1 2 2ω1

 = πi,

(17.56)

which could also be derived by an elementary application of the Cauchy residue theorem. Using the expression for η1 in (17.55) we can also write formula (17.49) as σ (w; ω1 , ω2 ) =

ω1 e G 2 (τ )w

2

/2ω12

θ



w |τ ω1



2πη(τ )3

.

(17.57)

Again by (17.46), ζ(w) = σ  (w)/σ(w). Direct differentiation of (17.49) gives    w 1 θ ω1 | τ 2η1 w  . ζ (w; ω1 , ω2 ) = + ω1 ω1 θ w | τ ω1

(17.58)

Note that by (4.36) again, the denominator in (17.49) and (17.57) is also given by 2πη(τ )3 = θ (0 | τ ), τ =

ω2 . ω1

(17.59)

The theta function expressions in (17.49), (17.57), (17.58) for the sigma and zeta functions are supplemented by one for the P-function: 1 P (w; ω1 , ω2 ) = e j + 2 ω1



θ (0 | τ ) θ j (0 | τ )

2

for j = 1, 2, 3, τ = ω2 /ω1 , θ, θ j in (4.32), (4.52).

θ j ( ωw1 | τ ) θ( ωw1 | τ )

2 ,

(17.60)

Chapter 18

A Finite Temperature Zeta Function

An all-important, fundamental object in quantum statistical mechanics is the partition function Z (T ) of a particular system in thermal equilibrium at temperature T . Z (T ) can be expressed in terms of a density matrix, which in turn is given as a Euclidean path integral—in the Feynman formulation. Details of these matters are presented in Chaps. 13, 14 of [108], for example. We consider here, for example, the quantized harmonic oscillator with frequency ν, where the density matrix can be expressed in terms of its normalized wave functions and quantized energy levels     1 1 νh = n + ω, n = 0, 1, 2, 3, . . . . En = n + 2 2 def.

(18.1)

def.

for ω = 2πν, and  = h/2π = the Dirac normalization of Planck’s constant h. If def. β = 1/kT is the inverse temperature, where k is Boltzmann’s constant, then the partition function is given by ∞

Z (T ) =

 1 = e−En β , ω 2 sinh 2 β n=0

(18.2)

from which other basic thermodynamic quantities are computed. For example, the Helmholtz free energy F(T ), the entropy S(T ), and the internal energy U (T ) are given by ∂F def. def. F(T ) = −kT log Z (T ), S(T ) = − . ∂T (18.3) ∂F def. . U (T ) = F(T ) + T S(T ) = F(T ) − T ∂T By (18.2) then, the following formulas hold:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_18

181

182

18 A Finite Temperature Zeta Function

  F(T ) = ω/2 + kT log 1 − e−ωβ     ωβ S(T ) = k ωβ − log 1 − e−ωβ e −1   1 1 U (T ) = ω + ωβ . 2 e −1

(18.4)

For example, by (18.2)  1 = 2 sinh ω2 β  ω 

 

ω ω − kT − log e 2 β − e− 2 β = −kT − log e 2 β 1 − e−ωβ  ω   = −kT − β − log 1 − e−ωβ 2   ω = + kT log 1 − e−ωβ , 2 

F(T ) = −kT log

(18.5)

def.

since kT β = 1. In Sect. 14.2 of [108], a finite temperature zeta function ζ(s; T ) is constructed such that the special valve ζ  (0; T ) of its derivative at the origin gives the free energy F(T ), up to a constant. Precisely.

 −kT  ω ω ζ (0; T ) = + kT log 1 − e− kT = F(T ) 2 2

(18.6)

def.

(by (18.4), again as β = 1/kT ) for def.

ζ(s; T ) =

 n∈Z

1  s 2 2 4π n + ω 2 β 2 2

(18.7)

with Re s > 21 . In this chapter, we consider the derivation of Eq. (18.6). This involves the meromorphic continuation of ζ(s; T ) (for ζ  (0; T ) to make sense in (18.6)), given that this function is defined initially in (18.7) only for Re s > 21 . By (6.4) however, ζ(s; T ) is def.

an inhomogeneous Epstein zeta function E(s; x, a) with m = 1, x = x1 = 4π 2 > 0, def. a = ω 2 β 2 2 . Therefore the meromorphic continuation of ζ(s; T ) to the domain Re s < 1 is given by Eq. (6.21):     s − 21 2 sin πs ∞ (t 2 − 1)−s 1 √ + dt. ζ(s; T ) = √ 1 1 e at − 1 a s− 2 4π (s) a s− 2 π 1

(18.8)

In particular, we see that ζ(s; T ) is holomorphic at s = 0 (as we already know since ζ(s; T ) = E(s; x, a), with m = 1).

18 A Finite Temperature Zeta Function

183

If f (s) = h(s)/ (s) where h(s) is holomorphic at s = 0, then as 1/ (s) vanishes at s = 0, h(s) h(s) f (s) − f (0) = = → h(0) s−0 s(s) (s + 1) as s → 0. That is, f  (0) = h(0) ⇒ the derivative at s = 0 of the first term in (18.8) is √ √     √ √ a a 1  1 −s+ 21 − 21 = √ (−2 π) = − a. (18.9) a (4π)  s − =√  − 2 s=0 2 4π 4π By the change of variables, u = et 

√ a

− 1, the integral

 e −1 dt du 1 √ = I (b) = =√ √ t a a u(u + 1) a e −1 −1 1 e √   eb a −1  √ 1 1 1 1 b a −1 − du = √ [log u − log(u + 1)]ee√a −1 = √ √ u u+1 a e a −1 a

√  √  √  1  1 √ log 1 − e−b a − log 1 − e− a → − √ log 1 − e− a a a def.

√ b a

b

(18.10)

as b → ∞. That is, 

∞ 1

√  dt 1 = − √ log 1 − e− a . a −1

√ et a

(18.11)

Using (18.11) and the product rule, we get that the derivative at s = 0 of the second term in (18.8) is −s    ∞2   t −1 2 d  √ dt  + (sin πs) s−1/2 at π 1 e −1 s=0 ds a s=0  −s     ∞2   t −1 2 d  √ (sin πs) dt  at s− 21 e −1 a π 1 s=0 ds s=0   

√ √  1 2 − √ log 1 − e− a π = −2 log 1 − e− a . 0 + −1/2 a π a

(18.12)

By (18.9) and (18.12) the following theorem follows: def.

Theorem 18.1 For a = ω 2 β 2 2 and Re s > 1, let def.

ζ(s; T ) =

 n∈Z

1  s 2 4π n 2 + a

(18.13)

184

18 A Finite Temperature Zeta Function

be the finite temperature zeta function in (18.7). Then Eq. (18.8) provides for the meromorphic continuation of ζ(s; T ) to the domain Re s < 1. In particular ζ(s; T ) is holomorphic at s = 0 and its derivative at this point is given by the formula

√  √ ζ  (0; T ) = − a − 2 log 1 − e− a   def. = −ωβ − 2 log 1 − e−ωβ

 ω ω def. − 2 log 1 − e− kT . =− kT

(18.14)

Thus the zeta function expression of the free energy F(T ) of the quantized harmonic oscillator asserted in Eq. (18.6) is verified. The entropy S(T ) can also be expressed in terms of ζ  (0; T ). From (18.14) kωβ ω k  ζ (0; T ) + + ωβ = 2 2T e −1   ω kωβ −ω − k log 1 − e−ωβ + + ωβ = 2T 2T e −1     ωβ = S(T ), k − log 1 − e−ωβ + ωβ e −1

(18.15)

def.

by (18.4). Moreover since F(T ) = −kT log Z (T ) for the partition function Z (T ), log Z (T ) =

1  ζ (0; T ), 2

1 

Z (T ) = e 2 ζ (0;T )

(18.16)

again by (18.6). An appropriate partition function Z (T ) exists in the broad context of quantum fields in thermal equilibrium at a finite temperature T on various curved back-ground space-times X . Similar to the path integral representation of the partition function mentioned in the initial remarks of this chapter, Z (T ) is obtained by integration over periodic fields φ(it) = φ(it + β) (again for β = 1/kT , i 2 = −1) where it is “imaginary time”. The Wick rotation t → it (named after Gian Carlo Wick) can be used to transform a Feynman integral into a Euclidean path integral. A clear mathematical meaning can be assigned to Z (T ) by way of a suitable zeta function ζ X (s) attached to X . For certain X , the formula log Z (T ) =

1  ζ (0; T ) 2 X

(18.17)

of (18.16), for example, persists. Such matters will be pursed more in Chap. 21.

Chapter 19

Lemaitre, Inhomogeneous Cosmology, and a Quick Look at the BTZ Black Hole

The Reverend Monsignor George Lemaitre (1894–1966), a Belgium Priest, made an explicit proposal of the expansion of the universe during the period from 1927 to 1931. Earlier work of the Russian mathematician (and meteorologist) Alexander Friedmann (1888–1925) in 1922 actually described a closed, expanding model. Lemaitre however connected the expansion with the redshift of galaxies (the Doppler effect), which by way of careful observations of Edwin Hubble (1889–1953) at Mount Wilson provided conclusive confirmation of his proposal. Einstein considered the universe as static, and so he inserted in his field equations an “ad hoc” cosmological constant that would imply non-expansion solutions. Afterwards he realized that to be the “biggest blunder” of his life. He would have discovered, otherwise, the expansion himself—that which was discovered by Friedmann. On the other hand, a hundred years later, scientists are beginning to realize that the cosmological constant  is possibly non-zero but assumes a very small value, and it likely plays a prominent role towards the explanation of the mysterious “dark energy” that apparently drives/accelerates the expansion. The Friedmann–Lemaitre metric is an exact solution of the Einstein field equations that not only describe an expanding universe, but also a homogeneous and isotropic universe—meaning one that is the same at all locations and in all directions, apart from some possible local irregularities. This metric is also called the FLRW metric, in honor also of Howard Robertson (1903–1961) and Arthur Walker (1909–2001). A generalization of the realistic, first approximation FLRW model is the one of Lemaitre–Tolman (after Richard Tolman, and also by Hermann Bondi later in 1947) which is isotropic but non-homogeneous. An interesting application of this model, among other applications, is to the study of black hole creation and evolution. The point is that the Schwarzschild black hole, for example, being unchanged throughout its history, is not amenable to such a study-one that would involve the imposition of initial conditions, for example. In this model, Lemaitre derives the scale factor equation

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_19

185

186

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …



∂ R(t, r ) ∂t

2

˜ ˜ ) + 2 M(r ) +  R(t, r )2 , = 2 E(r R(t, r ) 3

(19.1)

and he discusses its solution in terms of the Weierstrass P-function P(z) in (16.2) [72].  here is (again) the cosmological constant. With r fixed, this equation is mathematically a special case of Eq. (19.77) of Appendix 1 to this chapter. One could also consider cosmological models that are homogeneous but not isotropic. These are the Bianchi universes, of which there are ten types, named after Luigi Bianchi (1856–1928). The interest and focus here, however, will be on the Szekeres–Szafron model that generalizes (in particular) the Lemaitre–Tolman– Bondi model, and quite many others, as a very general exact solution of the Einstein gravitational field equations for inhomogeneous cosmology. It was remarked prior to Eq. (10.38) of Chap. 10 that the BTZ black hole would be described in this chapter, which we do in Appendix 2 here. This also provides for another concrete example of a solution of the Einstein field equations—and, moreover, to point out a Patterson–Selberg zeta function connection to this black hole. My introduction to the material presented in this chapter came by way of an inspiring paper [64] of Kraniotis and Whitehouse. Attention will be on the P. Szekeres–D. Szafron metric   def. g : ds 2 = dt 2 − e2β(t,x,y,z) d x 2 + dy 2 − e2α(t,x,y,z) dz 2

(19.2)

as an exact solution of the Einstein field equations G i j + gi j = κTi j , 1 8πG def. G i j = Ri j − gi j R, κ = − 4 2 c

(19.3)

for an inhomogeneous universe with cosmological constant , and a perfect fluid source energy momentum tensor Ti j [25, 68, 94, 95]. G i j is the Einstein tensor, and Ri j = Ri j (g), R = R(g) are the Ricci tensor and Ricci scalar curvature, respectively; G is the Newton gravitational constant and c is the speed of light. As was pointed out in Chap. 15, not all scalar curvatures are created equal: sign conventions differ for R(g). Consider for example real hyperbolic 3-space  def.  H 3 (R) = (x, y, z) ∈ R3 | z > 0

(19.4)

 a2  2 d x + dy 2 + dz 2 , a > 0 : 2 z a2 gi j = 2 δi j . z

(19.5)

with the metric def.

g : ds 2 =

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

187

For our sign convention (being the same as that of Chap. 15) R(g) = 6/a 2 . Here the non-zero components of the Ricci tensor are R11 , R22 , R33 —all of which are equal to 2/z 2 . The corresponding components of the Einstein tensor G i j in (19.3) are therefore given by def.

G ii =

2 1 a2 6 1 − = − 2 , G i j = 0, i = j, 2 2 2 z 2z a z

(19.6) def. 1 , a2

and for a zero energy momentum tensor, and cosmological constant  = Einstein equations (19.3) in this example are the vacuum equations G ii + gii = −

1 1 a2 + 2 · 2 = 0. 2 z a z

the

(19.7)

The functions α and β in (19.2) are to be determined by way of the field equations (19.3). By a Maple program (tensor) the non-zero components of the Einstein tensor for g in (19.2) are given as follows.  G 11 = − (βt )2 − 2αt βt + e−2α −2αz βz + 2βzz + 3 (βz )2  + e−2β βx x + β yy + (αx )2 + (α y )2 + αx x + α yy , G 12 = βt x + αt x + αt αx − αx βt , G 13 = βt y + αt y + αt α y − α y βt , G 14 = 2βt z + 2βt βz − 2αt βz ,  G 22 = e2β αt βt + (αt )2 + αtt + βtt + (βt )2   2 + e−2α e2β αz βz − βzz − (βz )2 − αx βx − α y − α yy + α y β y

(19.8)

G 23 = αx y + αx α y − αx β y − α y βx G 24 = βx z − αx βz  G 33 = e2β αt βt + (αt )2 + αtt + βtt + (βt )2  + e−2α e2β αz βz − βzz − (βz )2 + αx βx − α y β y − (αx )2 − αx x , G 34 = β yz − α y βz

  G 44 = − (βz )2 − e2α e−2β βx x + β yy + e2α 2βtt + 3 (βt )2 . The Szekeres–Szafron metric (19.2), which is diagonal, can be written as ⎤ 1 0 0 0 ⎢ 0 −e2β 0 0 ⎥ ⎥ g=⎢ ⎣ 0 0 −e2β 0 ⎦ . 0 0 0 −e2α ⎡

(19.9)

188

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

As the energy momentum matter tensor in (19.3) is modeled on the stress tensor of an ideal fluid, with time-dependent density ρ and pressure p, we take it to assume the form p Ti j = (ρ + 2 )gi1 g j1 + pgi j : c  p 2 T11 = ρ + 2 g11 + pg11 , Tii = pgii , i = 2, 3, 4, c Ti j = 0, i = j.

(19.10)

Later we consider a dust universe with p(t) = 0. In the steps ahead, it will be assumed that βz (t, x, y, z) > 0 everywhere; the case βz = 0 seems to be less useful for astrophysical applications. The goal is to first establish the following. For suitable functions ν(x, y, z), (t, z) eβ(t,x,y,z) = (t, z)eν(x,y,z) , eα(t,x,y,z) = (t, z)βz (t, x, y, z) = z (t, z) + (t, z)νz (x, y, z)   S. e−ν(x,y,z) = A(z) x 2 + y 2 + 2B1 (z)x + 2B2 (z)y + C(z),

(19.11)

where the functions (t, z), A(z), B1 (z), B2 (z), C(z) satisfy K (z) + 1 , 4  2M(z) κH (t, z) + (t, z)2 − , t (t, z)2 = −K (z) + (t, z) 3 3(t, z) A(z)C(z) − B1 (z)2 − B2 (z)2 =

(19.12)

for functions of integration K (z), 2M(z), and where  H (t, z) =

p(t)

 ∂  (t, z)3 dt. ∂t

(19.13)

Note that in particular for a zero pressure p(t), the second equation in (19.12) is the Lemaitre equation (19.1) for the Lemaitre–Tolman–Bondi model: z here corresponds to the variable r there and , −K correspond to R, 2E respectively. Thus, following Lemaitre, (t, z) in (19.12) would be solved for in terms of the Weierstrass P, sigma, and zeta functions. Now by (19.10), (19.9), (19.3), T14 = T24 = T34 = 0 ⇒ G 14 = 0, G 24 = 0, G 34 = 0. That is, by (19.8) βt z + βt βz − αt βz = 0, βx z = αx βz , β yz = α y βz ⇒   a.  −α  e βz y = e−α βzy − α y βz = 0,  −α  b. e βz x = e−α (βzx − αx βz ) = 0,  β−α  c. e βz t = eβ−α (βzt + βt βz − αt βz ) = 0,

(19.14)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

189

by the equality of mixed partials, which we will continue to use without comment on. a. and b. in (19.14) say that  −α  e βz (t, x, y, z) = f (t, x, z),

f x (t, x, z) = 0,

f (t, x, z) = u(t, z) (19.15)

for functions of integration f (t, x, z), u(t, z), where u(t, z) > 0 since βz (t, x, y, z) > 0 by assumption. By c. in (19.14) and (19.15)       0 = eβ−α βz t = eβ e−α βz t = eβ u t ⇒ eβ(t,x,y,z) u(t, z) = ν0 (x, y, z), (19.16) a function of integration, with ν0 (x, y, z) > 0 since u(t, z) > 0. Therefore def.

1 u(t, z) = (t, z)ν0 (x, y, z) = (t, z)eν(x,y,z) , def.

ν(x, y, z) = log ν0 (x, y, z), (t, z) = ⇒e

β(t,x,y,z)

(19.17)

which is the first equation in (19.11). The first of the second one there is Eq. (19.15): def. βz = eα u = eα / ⇒ eα = βz . Differentiation of the first equation in (19.11) with respect to z gives eν νz + z eν = eβ βz = eν βz ⇒ νz + z / = βz ⇒ νz + z = βz , which is the second part of the second equation in (19.11). We focus now on the third equation in (19.11), and on the remaining statements in (19.12) and (19.13). Differentiation of the first equation in (19.11) with respect to x, y, t gives eν νx = eβ βx = eν βx ⇒ νx = βx , and (similarly) ν y = β y , t = βt , tt = βtt + βt2 ⇒ νx x = βx x , ν yy = β yy , βtt + βt2 =

tt , βt2 = 



t 

2 .

(19.18)

Then by the formula for G 44 in (19.8) and the first two formulas in (19.11) 2   e−2α G 44 = − e−α βz − e−2β νx x + ν yy + 2βtt + 2 (βt )2 + (βt )2   2 νx x + ν yy t 1 2tt + =− 2 − + . 2 2ν   e  

(19.19)

On the other hand, by (19.8), (19.9), (19.10)    e−2α G 44 = e−2α (−g44 + κT44 ) = e−2α e2α + κ p −e2α =  − κ p. (19.20) so that by (19.19)   1 + e−2ν νx x + ν yy = 2tt + (t )2 − ( − κ p)2 ,

(19.21)

190

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

where we know that ν = ν(x, y, z) is independent of t;  = (t, z) is independent of x, y. The derivative of the right hand side of (19.21) with respect to t then has to be zero, which means that we can write 2(t, z)tt (t, z) + (t (t, z))2 − ( − κ p(t))(t, z)2 = −K (z),

(19.22)

for a function of integration −K (z). Let   ∂  (t, z)3 = 3 p(t) 2 t (t, z) : Ht (t, z) = p(t)  ∂t  ∂  def. (t, z)3 dt. H (t, z) = p(t) ∂t

(19.23)

Then by (19.22) 



 (t )2 −

 3 κH  + + K 3 3

t

κ = 2t tt + (t )3 − 2 t + 3 p2 t + K t = 0 ⇒  3  κH  (t )2 − 3 + + K  (t, z) = 2M(z) ⇒ 3 3 κH  M , (t )2 = −K + 2 + 2 − 3  3

(19.24)

which is the second equation in (19.12), for a function of integration 2M(z), with H given by (19.13). Finally, by (19.21) and (19.22)   1 + e−2ν(x,y,z) νx x + ν yy (x, y, z) = −K (z).

(19.25)

We check that a solution ν(x, y, z) of this equation can be expressed in the implicit form of the third equation S. in (19.11) provided that the functions A(z), B1 (z), B2 (z), C(z) there satisfy the condition AC − B12 − B22 =

K +1 4

(19.26)

in (19.12). Differentiate equation S. in (19.11) with respect to x and y twice: e−ν (−νx ) = 2 Ax + 2B1 ⇒ 2 A = e−ν (−νx x ) + e−ν (−νx ) (−νx ) ,        e−ν −ν y = 2 Ay + 2B2 ⇒ 2 A = e−ν −ν yy + e−ν −ν y −ν y   ⇒ 4 A = e−ν −νx x − ν yy + νx2 + ν y2   ⇒ e−ν 4 A = e−2ν −νx x − ν yy + νx2 + ν y2 .

(19.27)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

191

Also by equation S. in (19.11), and by (19.27)   e−ν 4 A = 4 A2 x 2 + y 2 + 8AB1 x + 8AB2 y + 4 AC,    2 e−2ν νx2 = e−ν (−νx ) = (2 Ax + 2B1 )2 = 4 A2 x 2 + 2 AB1 x + B12 ,   2    e−2ν ν y2 = e−ν −ν y = (2 Ay + 2B2 )2 = 4 A2 y 2 + 2 AB2 y + B22 ⇒   4 A2 x 2 + y 2 + 8AB1 x + 8AB2 y + 4 AC = e−ν 4 A = (19.28)     e−2ν −νx x − ν yy + νx2 + ν y2 = e−2ν −νx x − ν yy +     4 A2 x 2 + 2 AB1 x + B12 + 4 A2 y 2 + 2 AB2 y + B22 ⇒   4 AC = e−2ν −νx x − ν yy + 4B12 + 4B22 ⇒     e−2ν νx x + ν yy = 4 B12 + B22 − AC . Therefore indeed, if A, B1 , B2 , C satisfy the condition AC − B12 − B22 =(K + 1)/4 −2ν νx x + ν yy in (19.12),  2 then2ν givenby equation S. in (19.11) solves Eq. (19.25): e s = 4 B1 + B2 − AC = −K − 1. In case p(t) = p0 is constant, H in (19.23) is given by H = p0 3 , and  in (19.12) satisfies the differential equation t (t, z)2 = −K (z) +

2M(z) ( − κ p0 ) 2 +  , (t, z) 3

(19.29)

an equation that is mathematically of the form (19.77) in Appendix 1 to this chapter with −K (z), 2M(z), ( − κ p0 ) /3 corresponding to −K , A, B there, respectively, and with E, D both equal to zero in (19.77). Since f  (0) = A, we obtain by (19.78), (19.82), (19.88) the following parametric solution of Eq. (19.29): M(z)   2 P (w + c0 ; g2 , g3 ) + K12(z)   σ(w − w0 + c0 ) M(z) + 2 + c log ) t = β(w) = δ(z) + ζ(w (w ) 0 0 2P  (w0 ) σ(w + w0 + c0 ) (19.30) with the Weierstrass invariants given by (19.83): (t, z) = ψ(w) =

K (z)3 K (z)2 − ( − κ p0 ) M(z)2 , g3 = + , 12 12 216 3 ( − κ p0 ) M(z)2 K (z)3 − = ( − κ p0 )2 M(z)4 , 48 16 −K (z) P(w0 ) = . 12 g2 =

(19.31)

192

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

By (19.31) and by (16.5) again, note that     2 K (z) 3 K (z)2 K (z) K (z)3 ( − κ p0 ) M(z)2 P  (w0 ) = 4 − − − + − 12 12 12 12 216 ( − κ p0 ) M(z)2 = . 12 (19.32) We will henceforth take p0 = 0. Thus pressureless matter only is considered. By (19.32), (19.30), (19.31) one can also write 

  2 P (w0 ) 3 , (t, z) = ψ(w) =  [P (w + c0 ) − P (w0 )]2 2

2

(19.33)

for  = 0. From (19.11), (19.30) again (M(z)/2)2 e2ν(x,y,z) , [P (w + c0 ) − P (w0 )]2  2 = z (t, z) + (t, z)νz (x, y, z) ,

e2β(t,x,y,z) = (t, z)2 e2ν(x,y,z) = e

2α(t,x,y,z)

(19.34)

which provides for the expression ds 2 = dt 2 −

 (M(z)/2)2 e2ν(x,y,z)  2 d x + dy 2 2 [P (w + c0 ) − P (w0 )]  2 − z (t, z) + (t, z)νz (x, y, z) dz 2

(19.35)

of g in (19.2). The choice p0 = 0 is reasonable given that the pressure interaction between galaxies, for example, is negligible. Szekeres’ solution of the Einstein equations for g in (19.2) was for the case  = 0, and for an energy momentum dust source—i.e. for p0 = 0. Generalizations of his bold work was by Szafron two years later in 1977. A useful reparametrization of the Szekeres–Szafron metric (19.2) is due to Charles Hellaby [56]. It allows, for example, for a nice transition to the Lemaitre–Tolman– Bondi or Friedmann–Lemaitre–Robertson–Walker limit models. Towards a description of this reparametrization we use the equations in (19.11) again to express g in (19.2) as   g : ds 2 = dt 2 − 2 e2ν d x 2 + dy 2 − (z + νz )2 dz 2 .

(19.36)

Now suppose, somewhat abstractly, that unspecified functions S = S(z), P = P(z), Q = Q(z), = (z), and a = a(z) = 0 are given. Then where S(z) = 0 one can define the function E = E(x, y, z) by

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

193

     x−P 2 S y−Q 2 E = + + = 2 S S   y 2 − 2y Q + Q 2 S x 2 − 2x P + P 2 + + = 2 S2 S2     x 2 + y2 Q P 2 + Q2 P S +y − + +x − + . 2S S S 2S 2 def.

(19.37)

Now set e−ν(x,y,z) = a(z)E(x, y, z) :    2      P + Q2 a x 2 + y2 −a Q aP S (19.38) −ν +y +a e = +x − +a 2S S S 2S 2 by (19.37), which by comparison with the third equation in (19.11) gives aP a = A, − = 2B1 , 2S S

−a Q = 2B2 , S

  a P 2 + Q2 aS + = C, (19.39) 2S 2

where by (19.12), AC − B12 − B22 = (K + 1)/4. That is by (19.39), K +1 a = 4 2S

    a P 2 + Q2 aS a2 Q2 a2 a2 P 2 + ⇒ a 2 = K + 1. − = − 2 2 2S 2 4S 4S 4 (19.40)

Define

def.

R(t, z) = (t, z)/a(z).

(19.41)

Differentiate eν in (19.38) and R with respect to z: a E z + az E = e−ν (−νz ) = −a Eνz ⇒ az az Ez E z − ⇒ νz + z = − − + z , νz = − E a E a az − az az ⇒ Rz = ⇒ a R z = z − a2 a E z R Ez νz + z = − + a Rz = −a + a Rz E E

(19.42)

by definition (19.41). Then by (19.42)  (νz + z )2 = a 2

R Ez − Rz E



2 =

1 a2

=

R Ez E

− Rz

a 2 −K a2

2

= −

K a2

,

(19.43)

194

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

and since 2 e2ν = a 2 R 2 /a 2 E 2 by (19.38), (19.41), we see that Szekeres–Szafron metric in (19.36) can be written as   R  g : ds = dt − 2 d x 2 + dy 2 − E 2

2

2

Rz −

R Ez E

2

dz 2

− K /a 2

,

(19.44)

which is the Hellaby parameterization of g in (19.2). By definition (19.41), and by (19.29) t (t, z)2 K (z) 2M(z) (t, z)2 = − + + a(z)2 a(z)2 a(z)2 (t, z) 3a(z)2 K (z) 2M(z)  =− + + R(t, z)2 , a(z)2 a(z)3 R(t, z) 3

Rt (t, z)2 =

(19.45)

since we chose p0 = 0. Let K (z) def. ˜ 2 E(z) = − , a(z)2 R2 g : ds 2 = dt 2 − 2 E

def. M(z) ˜ M(z) = : a(z)3  2  (Rz − R E z /E)2 , d x + dy 2 −

+ 2 E˜ ˜  2 M(z) ˜ + R(t, z)2 , Rt (t, z)2 = 2 E(z) + R(t, z) 3

(19.46)

which is the Lemaitre scale factor Eq. (19.1). We proceed more concretely by computing, for example, the metric  R 2 /E 2 d x 2 + dy 2 in (19.44). For this, we start first with the unit 2-sphere  def.  S 2 = (x, y, z) ∈ R3 | x 2 + y 2 + z 2 = 1 .

(19.47)

For the standard metric g0 on R3 , ⎡

⎤ 100 2 def. g0 = ⎣ 0 1 0 ⎦ : ds 2 = d x 2 + dy 2 + dz 2 , g S = i ∗ g0 , 001

(19.48)

2

the pull-back of g0 by way of the inclusion map i: S 2 → R3 . g S is described more concretely as follows. Define h : (0, 2π) × (0, π) → S 2 by def.

h(φ, θ) = (sin θ cos φ, sin θ sin φ, cos θ), φ ∈ (0, 2π), θ ∈ (0, π). Then

 sin2 θ 0 : ds 2 = sin2 θdφ2 + dθ2 . g (h(φ, θ)) = 0 1

(19.49)



S2

(19.50)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

195

Consider now the transformation of variables (x, y) → (φ, θ) given by  θ cos φ, x = P(z) + S(z) cot 2 



 θ y = Q(z) + S(z) cot sin φ 2

(19.51)

for z fixed. We will make use of the formulas 1 d 1 θ II I = 1 + cot 2 x, . cot = − 2 dθ 2 sin x 2 sin2 2θ

(19.52)

By II, 

   θ cos φ dθ (− sin φ)dφ + − d x = S(z) cot ⇒ 2 2 sin2 2θ      θ sin φ cos φdφdθ cos2 φ 2 2 2 2 θ 2 2 cot sin φdφ + cot + dθ . d x = S(z) 2 2 sin2 2θ 4 sin4 2θ (19.53) Similarly 

    θ θ cos φ sin φ sin2 φ 2 2 2 cot cos φdφ − cot dφdθ + dθ dy = S(z) 2 2 sin2 2θ 4 sin4 2θ   dθ2 2 2 2 2 θ 2 ⇒ d x + dy = S(z) cot dφ + . 2 4 sin4 2θ (19.54) On the other hand, by (19.51), (19.37), and I in (19.52) 2

2

2

  θ θ y − Q(z) 2 2 cos φ, = cot = cot 2 sin2 φ ⇒ 2 S(z) 2   S(z) 1 S(z) 2 θ cot + (z) = , E(x, y, z) = 2 2 2 sin2 2θ 

x − P(z) S(z)

2

2

(19.55)

for the choice (z) = 1 that we now make, for which it follows from (19.54) that   2 4 sin4 2θ d x 2 + dy 2 θ dθ = S(z)2 cot 2 dφ2 + = dθ2 + sin2 θdφ2 E(x, y, z)2 S(z)2 2 4 sin4 2θ (19.56) since   cos2 2θ θ 2 θ θ 2 θ 2 θ 2 θ 4 sin cot = 4 sin sin = 2 sin cos = sin2 θ. 2 2 2 2 sin2 2θ 2 2 4

(19.57)

196

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

By (19.50), we see that (d x 2 + dy 2 )/E(x, y, z)2 is the standard metric on the unit sphere S 2 , for (z) = 1. Also for (z) = 1 and for E z = 0, the Szekeres–Szafron metric in (19.46) reduces to the Lemaitre–Tolman–Bondi (LTB) metric   Rz (t, z)2 dz 2 , g : ds 2 = dt 2 − R(t, z)2 dθ2 + sin2 θdφ2 − ˜ 1 + 2 E(z) 0 < θ < π, 0 < φ < 2π

(19.58)

since by (19.40), a 2 = a 2 = K + 1. Here we have assumed that E z = 0, which we note is the case if Sz , Pz , Q z = 0, for example. Namely, by (19.37)  Ez =

      x 2 + y2 S Pz − P Sz S Q z − Q Sz Sz − y − 2 −x 2 S S2 S2    1 S (2P Pz + 2Q Q z ) − P 2 + Q 2 Sz + 2 S2  1 2 S z + Sz = 0 + 2

(19.59)

since also (z) = 1 ⇒ z = 0. A further specialization of the LTB metric in (19.58) is by way of the choices ˜ ˜ = M0 z 3 : R(t, z) = z A(t), 2 E(z) = −kz 2 , M(z)   A(t)2 dz 2 g : ds 2 = dt 2 − A(t)2 z 2 dθ2 + sin2 θdφ2 − , 1 − kz 2

(19.60)

which is the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric with spatial curvature parameter k = 1, 0, or −1. A(t) here is not the function A(z) in (19.11). By (19.46), 2   dA 2M0 (t) = −k + + A(t)2 . (19.61) dt A(t) 3 In the LTB model the Hubble expansion rate of the universe is given by   Rt (t, z) Rt z (t, z) 1 2 + . H (t, z) = 3 R(t, z) Rz (t, z) def.

(19.62)

Therefore in the FLRW model, with R(t, z) = z A(t) in (19.60), this rate is given by the familiar expression  ˙  ˙ 1 A(t) A(t) def. 3 = = H (t). 3 A(t) A(t)

(19.63)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

197

Some details so far have been provided for the quasi-spherical case with (z) = 1 in (19.44), with the metric there transformed to the LTB metric (19.58) by the change of variables (x, y) → (φ, θ) in (19.51). There is also the quasi-hyperbolic case, with

(z) = −1, and the quasi-plane case with (z) = 0. In the quasi-hyperbolic case, for example, the change of variables, (x, y) → (φ, θ) is given by  θ cos φ, x = P(z) + S(z) coth 2 



 θ y = Q(z) + S(z) coth sin θ, 2

(19.64)

again for z fixed, and in place of the formulas I, II in (19.52), one can use 1 I = coth x − 1, 2 sinh x

θ II d −1 coth = dθ 2 2 sinh2

θ 2

(19.65)

to mimic the arguments presented in the quasi-spherical case to compute that d x 2 + dy 2 = sinh2 θdφ2 + dθ2 , E(x, y, z)

(19.66)

and that the Hellaby parameterization in (19.44) is transformed to   Rz (t, z)2 dz 2 g : ds 2 = dt 2 − R(t, z)2 dθ2 + sinh2 θdφ2 − K (z)

(z) − a(z) 2   Rz (t, z)2 dz 2 = dt − R(t, z) dθ2 + sinh2 θdφ2 − ˜ −1 + 2 E(z) 2

(19.67)

2

by (19.46). Here (again) it is assumed that E z = 0 which follows by (19.59), for example, if Sz , Pz , Q z = 0. Also −a(z)2 = 1 + K (z) by (19.40) for (z) = −1. The metric dθ2 + sinh2 θdφ2 in (19.67) (or in (19.66)) is an induced metric on the two-sheeted hyperboloid z2 − x 2 − y2 = r 2

(19.68)

for r = 1. A surface with this metric cannot be embedded in R3 with its usual Euclidean inner product. Similar to the Minkowski inner product  , in (15.6) for the continuous Heisenberg model, we now choose def.

X, Y = x1 y1 + x2 y2 − x3 y3

(19.69)

for X = (x1 , x2 , x3 ), Y = (y1 , y2 , y3 ) ∈ R3 . Define h : R2 → R3 by def.

h(φ, θ) = (x1 = r sinh θ cos φ, x2 = r sinh θ sin φ, x3 = r cosh θ) ⇒ x32 − x12 − x22 = r 2 ,

(19.70)

198

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

as in (19.68). Then, using (19.69) h φ (φ, θ) = (−r sinh θ sin φ, r sinh θ cos φ, 0), h θ (φ, θ) = (r cosh θ cos φ, r cosh θ sin φ, r sinh θ) ⇒ h φ , h φ = r 2 sinh2 θ, h θ , h θ = r 2 ,

(19.71)

h φ , h φ = 0 ⇒   2   r sinh2 θ 0 def. h φ , h φ h φ , h θ = , gh (φ, θ) = h φ , h θ h θ , h θ 0 r2 similar to the fundamental form in (15.7). That is,   gh : ds 2 = r 2 sinh2 θdφ2 + r 2 dθ2 = r 2 dθ2 + sinh2 θdφ2

(19.72)

is the induced metric on the two-sheeted hyperboloid (19.68), as was asserted. We assumed that βz > 0; others assume that βz = 0. However, for the first two equation in (19.11) to be valid, with α, β and ν real-valued, necessarily  > 0 by the first one, and then necessarily βz > 0 by the second one there. One can consider electromagnetic generalizations of the preceding discussion. The Krasi´nski book [67], for example, contains a more extended discussion and list of references; see Sect. 2.14 there, for example. The bottom line for the interest here is that the Lemaitre equation (19.46) is supplemented by a term −Q 2 /R(t, z)2 where Q is a constant electric charge: ˜ +2 Rt (t, z)2 = 2 E(z)

˜  M(z) Q2 + R(t, z)2 − . R(t, z) 3 R(t, z)2

(19.73)

Mathematically, this equation like the one in (19.29) is of the form in (19.77) of Appendix 1 to this chapter with the correspondence B, E, −K , A, D ↔ /3, 0, ˜ ˜ 2 E(z), 2 M(z), −Q 2 , respectively. By (19.81) then (taking E = 0 there) ˜ 2  2 E(z) Q + , 3 3 2 ˜ 2 ˜ 3 ˜  M(z) E(z)  E(z)Q g3 = − − − , 12 27  92 6 2 ˜2 4 ˜4 2 ˜ 2 − 27Q − 2Q9 E − Q 3E − Q 2M(z) − = 4 ˜ − 3 M(z) 16 g2 = −

˜ 3 M(z) ˜ 2 E(z) 6

(19.74)

 .

Equations (19.82) and (19.88) of Appendix 1 provide for the parametric solution

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

R(t, z) = ψ(w) = x0 +

x03 3

199

˜

M(z) ˜ + E(z)x 0+ 2  2 ˜ , x0 + E(z) P (w + c0 ; g2 , g3 ) − 6

t = β(w) = δ(z) + x0 w+   ˜ M(z)  3   ˜ x + E(z)x 0+ 2 3 0 σ (w − w0 + c0 ) log + 2 + c ) ζ(w (w ) 0 0 P  (w0 ) σ (w + w0 + c0 ) (19.75) def. 2 ˜ ˜ of Eq. (19.73) where again f (x0 ) = 0 for f (x) = 3 x 4 + 2 E(z)x + 2 M(z)x − 2 Q , and where σ(z), ζ(z) are the Weierstrass sigma and zeta functions in the definitions (17.1), (17.31). δ(z) is a function of integration, and the constant c0 = c0 (z) depends on z, with z fixed. As before, w0 is chosen to satisfy P (w0 ) = f  (x0 ) /24 and it is assumed that P (w0 ) = the roots e1 , e2 , e2 of Eq. (16.8) (so that P  (w0 ) = 0). x0 also depends on z, and for  = 0 (where /3 is the leading coefficient of f (x)), we observe from Chap. 16 that the (complicated) expression for  in (19.74) is indeed non-zero ⇔ the equation f (x) = 0 has no repeated roots. If  = 0 however, Eq. (19.73) has parametric solutions in terms of elementary, non-elliptic functions. Such were found by Shikin [93], for example. In the paper [35], part II with Jennie, some solutions not found by Shikin (again for  = 0) are presented. One of these is the direct, non-parametric solution R(t, z) =

   √ Q2 ˜ ˜ 2 E(z) t/Q 2 M(z) ˜ 1 + W B(z)e ˜ M(z)

(19.76)

˜ for a function of integration B(z), where W (z) is the Lambert W-function—also called the product logarithm, or the Omega function. Also in [35], part II we present an infinite family of new solutions of Einstein equations for non-isotropic Bianchi IX cosmological models—solutions (again) in terms of the Weierstrass P, sigma, and zeta functions. We have barely touched on the vast subject of inhomogeneous cosmology— having offered a few initial remarks with a few mathematical details. The Krasi´nski book [67] provides for a comprehensive review of this subject, including a coherent overview regarding the physics of the exact solutions. The textbook [68] of Pleba´nski and Krasi´nski offers an extended introduction to general relativity and cosmology, including background material on differential geometry, Lemaitre–Tolman geometry, the Szekeres–Szafron solution, and a wide array of material which supplements that in [67]. Some of the notation in [68], however, differs of course from that employed here. For example, for the Einstein tensor components in (19.8), our G 11 corresponds to—G 00 in [68]. We have the following correspondence of notation: Our notation in (19.8): G 11 , G 12 , G 13 , G 14 , G 22 , G 23 , G 33 , G 34 , G 44 , G 24

The notation on p. 387 of [68] −G 00 , −eβ G 02 , −eβ G 03 , −eα G 01 , −e2β G 22 , −e2β G 23 , −e2β G 33 , −eα+β G 13 , −e2α G 11 , −eα+β G 12 , respectively

200

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

Appendix 1 With Eq. (19.73) in mind, we consider more generally a nonlinear equation of the form A D (19.77) [Y˙ (t)]2 = BY (t)2 + EY (t) − K + + Y (t) Y (t)2 for constants K , E, B, A, D; Y˙ =

dY dt

. Suppose we have a parametrization

Y = ψ(w), t = β(w), β  (w) = ψ(w),   2 ψ (w) = Bψ(w)4 + Eψ(w)3 − K ψ(w)2 + Aψ(w) + D.

(19.78)

Then 

ψ  (w) β  (w)

2

Bψ(w)4 + Eψ(w)3 − K ψ(w)2 + Aψ(w) + D ψ(w)2 D A + = Bψ(w)2 + Eψ(w) − K + , ψ(w) ψ(w)2 =

(19.79)

and thus Y = ψ(w), t = β(w) in (19.78) would provide for a parametric soludef. tion of Eq. (19.77). Now if f (x) = Bx 4 + E x 3 − K x 2 + Ax + D = ax 4 + 4bx 3 + 2 6cx + 4αx + δ for a = B, 4b = E, 6c = −K , 4α = A, δ = D,

(19.80)

then by (16.14), (16.15) AE K2 − , 12 4 EK A B A2 DE2 K3 BK D def. − − − + , g3 ( f ) = − 6 48 16 16 216 B DK 4 K 2 A2 F 2 def. + ( f ) = g2 ( f )3 − 27g3 ( f )2 = 256 16 DE2K 3 B 2 D2 K 2 27B 2 A4 B A2 K 3 + − − + 64 64 2 256 A3 E 3 3B 2 D 2 AE 3B D A2 E 2 5B D K 2 AE − − − − 16 64 4 128 9B K D 2 E 2 9E K A3 B −9B 2 K D A2 − − 16 16 128 2 4 E 9E 3 K AD 27D + B 3 D3 − . − 128 256 def.

g2 ( f ) = B D +

(19.81)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

201

f  (x) = 4Bx 3 + 3E x 2 − 2K x + A and f  (x) = 12Bx 2 + 6E x − 2K . In particular if x0 is a root of the equation f (x) = 0, then by (16.19) ψ(w) = x0 +

f  (x0 )



4 P (w + c0 ; g2 , g3 ) −

f  (x0 ) 24



(19.82)

is a solution of the nonlinear equation in (19.78) for  = ( f ) = 0, g2 = g2 ( f ), g3 = g3 ( f ), and for c0 = any constant. Life is a bit simpler in case D = 0, for example. Then x0 = 0 is a root of the equation f (x) = 0, AE B A2 K3 K2 EK A − , g3 = − − + , 12 4 48 16 216 B A2 K 3 27B 2 A4 A3 E 3 9E K A3 B K 2 A2 E 2 + − − − = 256 64 256 64 128 g2 =

(19.83)

in (19.81), and (19.82) reduces to ψ(w) =

A  4 P (w + c0 ; g2 , g3 ) +

K 12

.

(19.84)

The equation β  (w) = ψ(w) in (19.78) also needs to be solved for β(w). For this, we make use of the result P  (w0 ) = ζ(w − w0 ) − ζ(w + w0 ) + 2ζ(w0 ). P(w) − P(w0 )

(19.85)

For any constant c and for any branch of the logarithm, let F(w) = log so that F  (w) =

d dw





σ (w − w0 + c) σ (w + w0 + c)

σ(w−w0 +c) σ(w+w0 +c)

σ(w−w0 +c) σ(w+w0 +c)





σ  (w − w0 + c) σ  (w + w0 + c) − σ (w − w0 + c) σ (w + w0 + c) = ζ (w − w0 + c) − ζ (w + w0 + c) P  (w0 ) − 2ζ(w0 ) = P(w + c) − P(w0 )

=

by (17.46) and (19.85). Therefore if we set

(19.86)

(19.87)

202

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

  σ (w − w0 + c0 ) f  (x0 ) +δ log + 2 + c ζ (w ) (w ) 0 0 4P  (w0 ) σ (w + w0 + c0 ) (19.88) for a choice of w0 for which def.

β(w) = x0 w +

P (w0 ) =

f  (x0 ) , 24

f (x0 ) = 0,

(19.89)

we see by (19.82) and (19.87) that for c = c0 f  (x0 )   F (w) + 2ζ(w0 )  4P (w0 )   P  (w0 ) f  (x0 ) = x0 + 4P  (w0 ) P (w + c0 ) − P (w0 ) f  (x0 )  = ψ(w), = x0 +   (x ) 0 4 P(w + c0 ) − f 24

β  (w) = x0 +

(19.90)

as desired. Here we assume that P  (w0 ) = 0; that is, by (16.5), we assume that P (w0 ) = the roots e1 , e2 , e3 of Eq. (16.8). δ is any constant.

Appendix 2 The BTZ metric g BT Z is a three-dimensional solution of the Einstein field equations for a zero energy momentum tensor and a negative cosmological constant  [7]. It is named after Maximo Ba´nados, Claudio Teitelboim, and Jorge Zanelli. I had the pleasure of speaking with Prof. Zanelli during one of my visits to Chile and to learn first hand from him some of the interesting facets that accompanied the discovery of this solution. Particularly, the feature  < 0 renders the solution surprisingly useful, in contrast to a merely flat vacuum solution for  = 0. Recall that the J-T cosmological constant in (15.79) is also negative. In coordinates τ , r , φ, g = gBTZ is given (in Euclidean form, with τ = Euclidean time) as follows: ⎡ ⎢ ⎢ g=⎢ ⎣



− M + r

2



0

0

−4r 2 4r 4 +4Mr 2 +J 2

− 2J

0

−J 2

⎤

⎥ ⎥ 0 ⎥: ⎦ r2

  ds 2 = − M + r 2 dτ 2 − J dτ dφ −

4r 2 dr 2 + r 2 dφ2 , 4r 4 + 4Mr 2 + J 2

(19.91)

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

203

for mass and angular momentum parameters M > 0, J  0, respectively, and where (importantly) periodicity of the Schwarzschild coordinate φ is imposed: φ and φ + 2πn for n ∈ Z are identified. J > 0 corresponds to the charged, spinning black hole. g has constant Ricci scalar curvature R(g) = 6(−). The non-zero components of the Einstein tensor are   −J  , G 33 = r 2 G 11 = − M + r 2 , G 13 = 2 −4r 2 G 22 = 4 4r + 4Mr 2 + J 2

(19.92)

by Maple, so immediately by (19.91) the Einstein equations Gi j +

1 gi j = 0 2

are satisfied by gBTZ for  = −1/2 , R(gBTZ ) = 62 . By way of a suitable change of coordinates (τ , r, φ) → (x, y, z > 0) due to Carlip and Teitelboim [24], one can show that the Euclidean BTZ black hole (using the aforementioned periodicity of φ) can be realized as a quotient space n   def. B =  \ H 3 (R),  = (a,b) = γ(a,b) | n ∈ Z ,   a+ib 0 e ∈ S L(2, C) γ(a,b) = 0 e−a−ib

(19.93)

for suitable a, b ∈ R, as in (12.5), where H 3 (R) is real hyperbolic 3-space, as in (19.4). In fact, gBTZ is transformed to the metric g=

 2  2 1 d x + dy 2 + dz 2 , 2 = − , 2 z 

(19.94)

as in (19.5), where a, b in (19.93) are given in terms of the outer and inner event horizon radii r+ , r− :   21   π|r− | πr+ M2 J2 2 , b= , r± = a= . 1± 1+ 2 2   2 M 

(19.95)

This means that, in particular, one can attached to the BTZ black hole B a PattersonSelberg zeta function Z  (w), as defined in (12.4), (12.5); also see the second lecture in [114]. This was done initially in [109]. In some follow-up work [112], we also attached to the BTZ black hole with a conical singularity a zeta function obtained by way of a suitable deformation of Z  (w), which as an application provided for a zeta function expression of quantum corrected black hole entropy. We saw in formula (18.15) a zeta function expression of entropy also—but there in a completely different

204

19 Lemaitre, Inhomogeneous Cosmology, and a Quick Look …

context. A Selberg type trace formula in the presence of a conical singularity is developed in [113]. With Peter Perry [87] scattering theory on B is considered, and a Poisson type trace formula is presented. As was mentioned in Chap. 12, a fundamental domain F for the action of (a,b) on H 3 (R) has infinite hyperbolic volume, for any a, b ∈ R:  d xd ydz = ∞. (19.96) z2 F The traditional role played by eigenvalues of the Laplacian in a Selberg trace formula, for example, is played by scattering resonances in this infinite volume context— where (a,b) is called a Kleinian subgroup of S L(z, C).

Chapter 20

A Cold Plasma-Sine-Gordon Connection

The nonlinear system ∂ ∂ρ + (ρu) = 0 ∂t ∂x     ∂u 1 ∂ 2 ρ 1 1 ∂ρ 2 ∂u ∂ρ 2 ∂ − =0 +u + +β ∂t ∂x ∂x ∂x ρ ∂x 2 2 ρ ∂x

(20.1)

for a constant β > 0 was considered in Chap. 15. It describes the uni-axial propagation of a magnetoacoustic wave in a cold plasma with density ρ(x, t) > 0, and with speed u(x, t) across a magnetic field. Of particular interest was the Gurevich-Krylov solution ρ(x, t) = α1 + 4a 2 β 2 dn 2 (a(x − βνt), κ) > 0, (20.2) C , u(x, t) = u 0 + ρ(x, t) for any choice u 0 > 0, α3 > α2  α1  0 where  α3 − α1 α3 − α2 u0 a =+ , ν= , , κ= 2β α3 − α1 β



1   2 2 √ 2 C = + α1 α2 α3 = α1 4a β 1 − κ + α1 4a 2 β 2 + α1 2 . √

(20.3)

Using this solution we were able to construct a solution Sρ,u (x, t) of the Heisenberg equations of motion (15.2), and therefore a Heisenberg metric g H as in (15.7) on the Heisenberg model (15.1). In turn, given any γ < 0 (for example, the γ in the nonlinear Schrodinger ¨ equation (15.89), we defined the (non-diagonal) plasma metric def. gρ,u = g H /(−2γ/β) with constant Ricci scalar curvature −4γ/β 2 > 0 : g12 = 0.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_20

205

206

20 A Cold Plasma-Sine-Gordon Connection

The point, however, was to establish that: (i) the change of variables τ → t + φ(δ) set up in (15.66) led to a diagonalization gplasma : ds 2 = A(δ)dτ 2 −

4a 4 β 4 κ4 (sncndn)2 (δ, κ)dδ 2 A(δ)γ 2

(20.4)

of gρ,u , for A(δ) given by (15.73), and for ν sufficiently large to insure that A(δ) = 0 (ii) by way of the further change of variables (τ , δ) → (τ , r ) for  2 2 ν β − α1 a 2 β 2 dn 2 (δ, κ) a 2 β 2  2 2−κ − r = ψ(δ) = + −γ 2γ 8γ def.

(20.5)

gplasma in (20.4) is transformed exactly to the Jackiw-Teitelboim (J-T) black hole metric  dr 2 (20.6) gbh : ds 2 = − m 2 r 2 − M dτ 2 + 2 2 m r −M —this being the main goal, where m and M are given by definition (15.80). Also the J-T gravitational field equations R (gbh ) + 2 = 0 ∇i ∇ j  +  (gbh )i j = 0

(20.7)

of (15.77) are satisfied for the (negative) cosmological constant  and dilaton field  given by def. 2γ (20.8)  = −m 2 = 2 , (τ , r ) = mr. β Thus we have a cold plasma-continuous Heisenberg model-J-T black hole connection. On the other hand, a connection between J and T black holes and solitons in sineGordon field theory was found by Gegenberg and Kunstatter in 1997; see [41–43]. This connection was made quite explicit in [110]; also see [9, 111, 116]. It follows that, in particular, an explicit cold plasma-sine-Gordon connection can be set up, which is illustrated in this chapter. In the appendix here, we also indicate how the dispersion relation (15.33) for the system (20.1) is derived, and how it compares with the dispersion relation (20.9) ω 2 ≈ k 2 gh of Lagrange for shallow water waves of depth h with kh 0 ,

(21.2)

as in (19.4), (although M d could also be complex or quaternionic), and for  a discrete group of isometries of H d (R). H d (R) also has the group-theoretic realization def.

H d (R) = S O1 (d, 1)/S O(d) = G/K

(21.3)

for which  ⊂ G = S O1 (d, 1) is a discrete subgroup that is torsion free; i.e. only 1 ∈  has finite order. The calculations in [107] were carried out in the presence of a non-vanishing chemical potential, they were easily applied to obtain low and high temperature asymptotics of the effective potential, and they extended results found in [12] for the case p = 1. See also [16, 17] for the cases p = 1, d = 3 and p = d − n, d = n, respectively—and the books [15, 16], for example, for general background material and references. It is likely no surprise that, given the Kaluza-Klein factor Md = \G/K , the Selberg trace formula serves as a crucial mathematical tool for the calculations, details of which are elaborated on in Chap. 16 of [108]; some general background information on the trace formula is presented in [103]. The starting point there, and the main definition of Chap. 16, is the zeta function ζ (s; b, β, μ) =



1

def.

(4π)

p−1 2

(s)

∞

θβ,μ (t) (t; b)t [s−(

p−1 2 )]−1

dt

(21.4)

0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3_21

215

216

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

for

p,d

Re s >

dim X  2

=

p+d , 2

p,d

X  = R p × \G/K ,

(21.5)

for suitable theta functions θβ,μ (t),  (t; b) (described later), where for a quantum p,d field over X  in thermal equilibrium at a finite temperature T , β = 1/kT (with k = Boltzmann’s constant as in Chap. 18) μ is the chemical potential of the field, and b is a positive parameter. In practice one could take b = m 2 for the scalar field mass m. Regarding some general remarks in the final paragraph of Chap. 18, this zeta function provides for a clear mathematical definition of the partition function Z (T ) (or Z (β)) in the present context. Namely, for a suitable constant c def.

log Z (β) =

 1  ζ (0; b, β, μ) + cζ (0; b, β, μ) , 2

(21.6)

which presupposes the meromorphic continuation of zeta with holomorphicity at s = 0—a point to return to shortly. Actually when p is odd (for example p = 1, the case of interest in [12] as mentioned earlier), it turns out that ζ (0; b, β, μ) = 0 so that 1 (21.7) log Z (β) = ζ (0; b, β, μ), 2 which compares exactly with (18.17)—the case of the quantized harmonic oscillator. See Theorem 21.1 below. θβ,μ (t) and  (t; b) in (21.4) are spectral theta functions, for differential operators Dμ , − where def.



Dμ = − =

xd2

2 d d2 d − μ = − 2 + 2μ − μ2 , dt dt dt

d ∂2 ∂ + (2 − d)xd , 2 ∂xd ∂x j j=1

(21.8)

 being the Laplace-Beltrami operator on H d (R) in (21.2) corresponding to the metric (as in (19.5)) given by def.

gi j (x1 , . . . , xd ) = δi j /xd2 ,

(21.9)

where  projects to a differential operator  on \H d (R) = Md . Dμ acts on functions on R, where we write 

p,q X  = R × R p−1 × Md , p,q

(21.10)

which illustrates why we call X  a Kaluza-Klein spacetime (with spatial sector R p−1 × Md ). Dμ has spectrum, or eigenvalues,

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space … def.



λn (β, μ) =

2 2πn + iμ , n ∈ Z, β

217

(21.11)

with periodic eigenfunctions on R given by def.

f n (t) = f n (t + β) = e

2πin β t

, n∈Z

(21.12)

and − has spectrum {λ j , n j }∞ j=0 with n j = the (finite) multiplicity of the eigenvalue λ j of − , and with 0 = λ0 < λ1 < λ2 < λ3 < · · · , λ j → ∞

(21.13)

by the minus sign attached to  . Here of course the compactness of Md is used, and of course the eigenvalues λ j are unknown in value, even for d = 2. Now we can say that θβ,μ and  (λ; b) in (21.4) are given by def.

θβ,μ (t) =



e−tλn (β,μ) ,

n∈Z ∞ def.

 (t; b) =

n j e−t (λ j +b)

(21.14)

j=0

for t > 0, b > 0, given the notation just established. The zeta function ζ (s; b, β, μ) in (21.4) indeed has a meromorphic continuation to C that is holomorphic at s = 0, which in particular means that the partition function Z (β) in (21.7) is well-defined. The continuation is obtained by starting with the Jacobi inversion formula β − n2 β2 −nμβ e 4t , t >0 θβ,μ (t) = √ 4πt n∈Z

(21.15)

for θβ,μ (t), which follows from formula (2.5): n∈Z

eπt1 (w−ni/t1 ) = 2

√ −πn 2 t1 −2πnwt1 t1 e ,

(21.16)

n∈Z

t1 > 0, w ∈ C. Namely, choose t1 = β 2 /4πt, w = 2tμ/β so that the right hand side of (20.16) is the right hand side of (20.15). Also  2tμ ni4πt 2 − β β2 (21.17)     2 πβ 2t 2πn 2 2πn 2 = = −t iμ + = −tλn (β, μ) i −iμ − 4πt β β β

πt1 (w − ni/t1 )2 =

πβ 2 4πt



by definition (21.11), so that the left hand side of (21.16) is

218

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …



e−tλn (β,μ) = θβ,μ (t)

(21.18)

n∈Z

by definition (21.14), which proves the inversion formula (21.15) that can also be written as   ∞ ∞ 2 2 2 2 β − n 4tβ −nμβ − n 4tβ +nμβ e + e θβ,μ (t) = √ 1+ 4πt n=1 n=1   (21.19) ∞ 2 2 β − n 4tβ =√ e cosh nμβ . 1+2 4πt n=1 The next step in the meromorphic continuation of ζ (s; b, β, μ) is to use (21.19) to write the integrand in (21.4) as 

θβ,μ (t) (t; b)t

  s− p−1 −1 2

p β 2β = √  (t; b)t s− 2 −1 + √ 4πt 4π

∞     − n2 β2 s− p−1 −1 2 e 4t cosh nμβ  (t; b)t

(21.20)

n=1

and then set  ∞ 1  (t; b)t s−1 dt, Re s > d/2, ζ (s; b) = (s) 0  ∞ ∞ β − n2 β2 def. e 4t (cosh nμβ) (t; b)t s−1 dt, s ∈ C Mβ,μ (s; b) = √ πt n=1 0 def.

to conclude that for Re s >

p 2

+

(21.21)

d 2

 

 Mβ,μ s − ( p−1)  s − 2p β 2 ;b ζ (s − p/2; b) + p−1 (s) (4π) p/2 (4π) 2 (s)

  ∞ p  s − 2p 1 β def. 

=  (t; b)t s− 2 −1 dt p p √ 1 (s) −  s − 0 2 (4π) 2 (4π) 2 4π  ∞ ∞ 2 β2 n ( p−1) 2β 1 e− 4t (cosh nμβ)  (t; b)t s− 2 −1 dt + √ p−1 4πt n=1 (4π) 2 (s) 0  ∞

p β 1 = √  (t; b)t s− 2 −1 p−1 4π (4π) 2 (s) 0     ∞ 2β − n 2 β 2 s− p−1 −1 2 +√ e 4t (cosh nμβ) (t; b)t dt 4πt n=1     ∞ 1 s− p−1 −1 2 θ (t) (t; b)t dt = ζ (s; b, β, μ), =  β,μ p−1 0 (4π) 2 (s)

(21.22)

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

219

by (21.20) and definition (21.4)! We get: Theorem 21.1 Mβ,μ (s; b) in (21.21) is an entire function of s, and the zeta function ζ (s; b) there has an explicit meromorphic continuation to C given in terms of  structure and the Harish-Chandra spherical harmonic analysis on the non-compact, rank one symmetric space G/K = S O1 (d, 1)/S O(d) in (21.3)—by the result in [106] which holds, in fact, for the zeta function of any non-compact, rank one symmetric space G/K . It follows then that, by (21.22), ζ (s; b, β, μ) defined in (21.4) (for Re s > ( p + d)/2) has a meromorphic continuation to C given by  

  ( p−1)  s − 2p p  Mβ,μ s − 2 ; b ζ s − ; b + . ζ (s; b, β, μ) = p p−1 (s) 2 (4π) 2 (4π) 2 (s) (21.23) In particular if d(=dim G/K ) is even, ζ (s; b) has at most simple poles at s = 1, 2, 3, . . . , d2 . Moreover, for d even and p odd β

ζ (s; b, β, μ) = 0 for s = 0, −1, −2, −3, . . .. The Selberg trace formula, which replaces the Jacobi inversion formula in nonabelian harmonic analysis, plays a key role in the meromorphic continuation of the zeta function ζ (s; b) in [106] for an arbitrary non-compact, rank one symmetric spaces G/K . In the case at hand, for G/K = S O1 (d, 1)/S O(d), this trace formula is explicated in Theorem 16 of [108]. As in (18.3) one can use (21.6) (or (21.7) in particular), and Theorem 21.1, to compute the Helmholtz free energy F(T ) (or the entropy S(T ) and internal energy U (T )) in case p is odd and d is even, for example, d even being the more difficult, but more interesting case: def.

F(T ) = −kT log Z (T ) =

−1  ζ (0; b, β, μ). 2β 

(21.24)

By Theorem 16.3 of [108] (based on Theorem 21.1 here), one finds that F(T ) consists of a temperature independent term

 − − 2p 2(4π)

p 2

 p  ζ − ; b = m 2 , 2

(21.25)

the Casimir energy of the spacetime R p × Md [15–17, 104, 105], and a temperature dependent term that is the one-loop effective potential—the finite temperature thermodynamic potential computed in [107] that was referenced in the initial remarks of this chapter. In more recent work [122], we focused on the one-loop effective potential for a non-compact symmetric space X = G/K of higher rank > 1—work that we consider now in this final chapter, which involves no discrete subgroup  of G.

220

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

Here G is a connected non-compact real semisimple Lie group with finite center, and K ⊂ G is a maximal compact subgroup. First, of independent interest, we attach to X = G/K a theta function  X (t; b) and a zeta function ζ X (s; b). Some standard notation and definitions, from [54] for example, follow. Let g0 , k0 denote the Lie algebras of G, K , let ( , ) denote the Killing form of g0 , let g0 = k0 + p0 be a Cartan + decomposition of g0 , let a p ⊂ p0 be a maximal abelian subspace of p0 , and let  be a choice  of positive restricted roots contained in the full set of restricted roots of g0 , a p . ( , ) is positive definite on p0 and negative definite on k0 . For n0 =



(g0 )α ,

 α∈ +

N = exp n 0 ,

A = exp a p ,

(21.26)

 where (g0 )α is the root space corresponding to α ∈ , G = K A p N is an Iwasawa decomposition of G. The function H : G → a p given by g ∈ K exp H (g)N for g ∈ G is the corresponding  projection. W will denote the Weyl group of the

Iwasawa  root system (or of g0 , a p ). W can be realized as the quotient M  /M with the def.

 centralizer M of a p in K and the normalizer M of a p in K . As usual, if m α = dim(g0 )α for α ∈ , then def. 1 ρ = m α α. (21.27) 2 + α∈

Finally, for λ ∈ a ∗p (the real dual space of a p ) let Hλ ∈ a p be the unique element that satisfies the condition λ(H ) = (H, Hλ ) for every H ∈ a p . ( , ) is therefore extended  def.

to a ∗p , namely (λ1 , λ2 ) = Hλ1 , Hλ2 for λ1 , λ2 ∈ a ∗p , and moreover ( , ) extends Cdef. C bilinearly to the complexification  = a ∗p of a ∗p , which we regard as the space of R—linear maps from a p to C. One also writes |λ|2 for (λ, λ), λ ∈ . By definition, dimR a p is the (real) rank of G/K . In [45, 46], theta and zeta functions θ X (t), ζ X◦ (s; b) are set up and studied; the asymptotic behavior of θ X (t) as t → 0+ , and the meromorphic continuation of ζ X◦ (s; b) to C are considered: def.

θ X (t) =

 a ∗p

def. ζ X◦ (s; b) =

e−t|λ| |c(λ)|−2 dλ, t > 0 2

1 |W |(s)

∞

e−t (|ρ|

2

+b2 )

θ X (t)t s−1 dt,

(21.28)

0

d def. Re s > , b > 0, d = dim X, 2 ¯ b) in [46], and where c(λ) is where ζ X◦ (s; b) corresponds to the notation ζ X (s; 1, the Harish-Chandra c-function on . In [122], we found it very convenient to tweak

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

221

these definitions, especially that of θ X (t). Thus, henceforth we attach to the symmetric space X = G/K the theta and zeta functions  X (t; b) and ζ X (s; b) given by  X (t; b) = e−t (|ρ| def.

def.

ζ X (s; b) =

2

+b2 )

∴



θ X (t) =

|W |ζ X0 (s; b)

1 = (s)

e−t (|ρ|

2

+|λ|2 +b2 )

a ∗p

|c(λ)|−2 dλ, t, b > 0,

∞  X (t; b)t s−1 dt, 0

d Re s > , b > 0, d = dim G/K , 2

(21.29) by (21.28). The meromorphic continuation of ζ X (s; b) to C was worked out in [122], in a simpler form compared with that in [46]. The only possible points for its poles, def. all of which are simple, are the points s = d2 − ,  ∈ {0, 1, 2, 3, . . .} = N◦ with residue at such a point equal to d (4π)− 2

d  C (21.30)  2 − where the C are the Minakshisundaram-Pleijel coefficients in the assumed asymptotic expansion ∞ d  X (t; b) ∼ (4πt)− 2 C t  as t → 0+ (21.31) =0

of  X (t; b). If d is even, then in (21.30) 0    d2 − 1; i.e. ζ X (s; b) has finitely many poles. Also for d even, the special values formula ζ X (−; b) = (−1) !(4π)− 2 C d2 + d

(21.32)

is established for  ∈ N◦ . The results (21.30), (21.32) are extensions to the noncompact manifolds G/K classical results of Minakshisundaram-Pleijel (M-P) [80] for compact manifolds. They were also obtained by Camporesi [20, 22] for G/K of rank 1 : dim a p = 1. The brilliant thesis of Tomás Godoy [45], directed by Prof. Roberto Miatello, served as an impetus for the papers [46, 122]. In particular, by results in [45], (21.31) is valid for G of classical type. Equation (21.31) is also valid for G of complex type, as we shall discuss shortly. At this point, we get back to the consideration of the one-loop effective potential (sometimes, denoted by W (1) ) for the higher rank symmetric spaces X = G/K of non-compact type. Camporesi computes W (1) for X = S O1 (4, 1)/S O(4) (a rank 1 example) in [21]—the Euclidean sector of anti-de Sitter space. Also see [23], for example, where, W (1) is computed for fields of arbitrary spin. Given some general second-order elliptic differential operator D and anarbitrary “renormalization” parameter μ R , an expression like ± 21 log det D/μ2R was

222

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

regularized by Steven Hawking, by his introduction of a suitable zeta function ζ(s) into quantum field theory [53]: −



 def. 1    1 log det D/μ2R = ζ (0) + ζ(0) log μ2R , 2 2

(21.33)

which compares with (21.6). Here we would like to choose ζ(s) = ζ X (s; b) in (21.29). However, some brief remarks regarding the motivation of definition (21.33) would be in order. We side-step the issue of looking for a suitable differential operator D in the non-compact picture here for us. Let’s go back to the case of D = − ,  = the Laplace-Beltrami operator on the compact manifold \H d (R), for example. D   has the spectrum λ j , n j with finite multiplicities of the eigenvalues λ j in (21.13). The corresponding spectral zeta function [80] is given by def.

ζ (s) =

∞ nj d , s , Re s > λj 2 j=1

(21.34)

and the zeta regularized determinant is given by det D =

∞ 



λnj = e−ζ (0) , def.

(21.35)

j=1

since formally (illegally) ζ (s) =

∞

 n j e−s log λ j − log λ j ⇒

j=1

− ζ (0) =

∞

n j log λ j ⇒

(21.36)

j=1 

e−ζ (0) =

∞

en j log λ j =

j=1

∞ 

nj

λj ,

j=1

with ζ (0) well-defined by the meromorphic continuation of ζ (s). Now for c > 0, cD = −c has spectral zeta function ζ(c) (s) =

def.

∞ j=1

nj

s = c−s ζ (s) cλ j

which by formal differentiation leads to

(21.37)

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space … Table 21.1 E. Cartan’s list of irreducible X of complex type/type IV X = G/K d = dimX rank of X  S L(n, C)/SU (n) S O(2n + 1, C)/S O(2n + 1) S P(n, C)/S P(n) S O(2n, C)/S O(2n) GC 2 /G 2 F4C /F4 E 6C /E 6 E 7C /E 7 E 8C /E 8

223

|W |

n2

−1 (2n + 1)n

n−1 n

An−1 Bn

n! 2n n!

(2n + 1)n (2n − 1)n 14 52 78 133 248

n n 2 4 6 7 8

Cn Dn G2 F4 E6 E7 E8

2n n! 2n−1 n! 12 1152 51840 2903040 696729600



ζ(c) (s) = c−s ζ (s) + ζ (s)c−s (− log c) ⇒ (c)

e−ζ

(0)



= e−ζ (0)+ζ (0) log c ⇒ (c)

log(det cD) = log e−ζ (0) = −ζ (0) + ζ (0) log c ⇒



  1 1 ∓ log det D/μ2R = ± ζ (0) + ζ (0) log μ2R , 2 2 def.

(21.38)

for the choice c = 1/μ2R , by which one arrives at a definition like that of (21.33). For  a non-compact space like X = G/K , apart from any discrete spectrum λ j , n j to think about, it is therefore yet meaningful to define the one-loop effective potential W (1) by  1 def. (21.39) W (1) = − ζ X (0; b) + ζ X (0; b) log μ2R . 2 We assume now that X is of complex type. That is, G is a complex semisimple Lie group as illustrated in the following table that includes the classical groups with complex simple Lie algebras given in Table 3.1 of Chap. 3. For X of complex type, result of R. Gangolli [40] can be applied to provide for explicit formulas for the theta and zeta functions in (21.29), and for the MP coefficients C in (21.31). Specifically, the following formulas are developed in [122]. For π −(rank of G/K )/2 def. P0 = d=dim G/K   , (21.40) 2 α∈ + (ρ, α) d

 X (t; b) =

P0 |W |(4π) 2 (4πt)

d 2

e−t (|ρ|

2

+b2 )

, t > 0, b > 0

(21.41)

224

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …



P0 |W | s − d2 ζ X (s; b) =

s− d , s ∈ C (s) |ρ|2 + b2 2  P0 |W |(−1) 2 C = |ρ| + b2 , l  0. d (4π) 2 !

(21.42)

In fact, the asymptotic result t→0+

 X (t; b) ∼ (4πt)− 2 d

∞

C t 

(21.43)

=0

in (21.31) is exact for X of complex type: We have the equation  X (t; b) = (4πt)− 2 d

∞

C t  , t > 0

(21.44)

=0

for the C in (21.42). Also |ρ|2 in (21.41), (21.42) (for ρ defined in (21.27)) is given by the Freudenthal-de Vries formula |ρ|2 =

d , 12

(21.45)

for X of complex type. A proof of this formula is provided on pp. 474–475 of [55], for example. The cardinality |W | in (21.41), (21.42) of the Weyl group W of the  restricted root system , and the rank of G/K in (21.40) are provided  in Table I of this section in case G/K is irreducible. Of course so are d and . Note also 

that for d = 2m even, the quotient  s − d2 / (s) of gamma functions in (21.42) simplifies: P0 |W | ζ X (s; b) = m , s ∈ C. (21.46)

 2 2 s−m j=1 (s − j) |ρ| + b

 In particular, the special value 21 ζ X − 21 ; 0 which was computed in the rank 1 cases in [104] is immediately obtained—as a local contribution to the Casimir energy of X . The results that we have presented for X of complex type can be extended to X of split-rank type—meaning that the ranks of X , G, and K are related by rank X = rank G—rank K . Equivalently, this means that every restricted root α ∈ has even def. multiplicity: m α = dim (g0 )α , as defined  prior to (21.27), is an even integer. If X is ⇒ X is of split-rank type. Other examples of complex type, then m α = 2 ∀α ∈ include SU ∗ (2n)/S P(n), n  2 with m α = 4, S O1 (2n + 1, 1)/S O(2n + 1) (= real hyperbolic space of odd dimension), n  3 with m α = 2n, and E 6(−26) /F4(−20) with m α = 8. We have not worked out the details yet for all the split-rank types.

21 A Theta and Zeta Function Attached to a Non-compact Symmetric Space …

225

Consider X = SU ∗ (2n)/S P(n), n  2, of rank  = n − 1, and dimension d = (2n + 1)(n − 1) = (2 + 3), with = A . G = SU ∗ (2n) = S L( + 1, H), where H denotes the quaternions. X is type II on E. Cartan’s list [26, 54]. The computation of the theta function θ X (t) in (21.28) was made in [45], and is given (in English) in Eq. (4.4) of [46], from whence the zeta function ζ X (s; b) in (21.29) is computed for Re s > d2 = (2n + 1)(n − 1)/2, and is meromorphically continued. If d is even, for example, the poles of ζ X (s; b) ( all of which are simple) are finite in number, being located at the points s = 1, 2, 3, . . . , d/2. This statement, for X = SU ∗ (2n)/S P(n) of split-rank type, we know also holds for the general X of complex type; see (21.46). We saw in Theorem 21.1 that such a statement also holds for the zeta function ζ (s; b) in (21.21). The effective potential W (1) = −

 1  ζ X (0; b) + ζ X (0; b) log μ2R 2

(21.47)

in (21.39), now restricted to X of complex type, can therefore be computed using formula (21.42). The result (with details of the proof in [122]) is that ⎡

√ m+ 1 P0 |W |(−1)m+1 2m+1 π (|ρ|2 +b2 ) 2 m (2 j+1) j=1

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ for d = 2 m + 1 odd,  ⎥, ⎢ ζ X (0; b) = ⎢ ⎥ 



m  m ⎢ P0 |W | (−1) Hm − log |ρ|2 + b2 |ρ|2 + b2 , ⎥ m! ⎦ ⎣ def.  for d = 2 m even, Hm = mj=1 1j ⎡  ⎤ m

2 2 m P0 |W | (−1) + b , |ρ| m! ⎥ ⎢ ⎥ ⎢ ζ X (0; b) = ⎢ ⎥. for d = 2 m even, ⎦ ⎣ 0, for d = 2m + 1 odd

(21.48)

Again |ρ|2 = d/12, d = dimX by (21.45). The harmonic numbers Hm in (21.48) also occur in a zeta function calculation of W (1) in [18]; the context there is strikingly quite different from the present one here. W (1) is also calculated in [19] for a zeta function attached to a product of Laplace type operators—in the general rank 1 setting. The Refs. [15, 16, 20] provide for further useful computational examples with applications to finite temperature quantum field theory. The Selberg trace formula is also used in [16] to study the one-loop effective potential on R × M3 ; see (21.1), or (21.5) with p = 1. There symmetry breaking is also discussed. It is known that quantum effects in anti-de Sitter space are a relevant issue regarding the possible breaking of supersymmetry. Such effects are also studied in a brane-world model [30], for example, where a brane universe is embedded in a higher-dimensional bulk anti-de Sitter black hole.

References

Added in proof The following paper supplements references [30, 107], and it is of interest therefore for Chap. 21: G. Cognola, K.Kirsten, and S. Zerbini, One loop effective potential on hyperbolic manifolds, Physical Rev. D 48 (1993) no. 2, 790–799. The following recent survey paper provides (in particular) for additional results and perspective regarding the explicit connection of the cold plasma metric assigned to the magnetoacoustic system (15.31) and the J-T black hole metric (15.81) of Chap. 15. The paper also presents explicit elliptic function dilation solutions of the J-T gravitational field equations in (15.78): F.L. Williams. From a magnetoacoustic system to a J-T black hole: A little trip down memory lane. Commun. Anal. Mech. 15(3) (2023) 342–361. In Chap. 15, the Jackiw—Teitelboim black hole of positive mass was considered— along with the case of a negative mass that corresponds to a naked singularity. One can also consider the mass zero case—i.e. a black hole vacuum. The following two papers deal with the latter case, where the first reference has a connection to Chaps. 8 and 12: F.L. Williams, Remarks on the Patterson-Selberg Zeta function, black hole vacua and extremal CFT partition functions, Special issue in honour of Stuart Dowker’s 75th birthday, Guest Editors F. Dowker, E. Elizalde, and K. Kirsten, J. Phys. A: Math. and Theor. 45(2012) 374008 (19 p), and A tractroid realization of a 2d black hole vacuum, Annals of Math. Phys. 5(2) (2022) 097–099.

References 1. Adams, R., Mancas, S.C.: Stability of solitary and cnoidal traveling wave solutions for a fifth order Korteweg-de Vries equation. Appl. Math. Comput. 321(C), 745–751 (2018) 2. Akhiezer, A.I.: Plasma Electrodynamics. Pergamon, Oxford (1975) 3. Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, vol. 2. Cambridge University Press (1998) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 F. L. Williams, Some Musings on Theta, Eta, and Zeta, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-5336-3

227

228

References

4. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, No. 41, 2nd edn. Springer, New York (1990) 5. Ashraf, M.: Three dimensional pure gravity and generalized Hecke operators. J. High Energy Phys. 09, 083 (2020) 6. Avramis, S.D., Kehagias, A., Mattheopoulou, C.: Three-dimensional AdS gravity and extremal CFTs at c = 8 m. J. High Energy Phys. 11, 022 (2007) 7. Bañados, M., Teitelboim, C., Zanelli, J.: Black hole in three dimensional spacetime. Phys. Rev. Lett. 69(13), 1849–1851 (1992) 8. Beheshti, S.: Solutions of the dilaton field equations with applications to the soliton-black hole correspondence in generalized JT gravity. Ph.D. dissertation, University of Massachusetts, Amherst (2008) 9. Beheshti, S.: Integrable systems and 2D gravitation: how a soliton illuminates a black hole. In: Kirsten, K., Williams, F.L. (eds.) A Window into Zeta and Modular Physics. Mathematical Sciences Research Institute Publications, Berkeley, CA, vol. 57. Cambridge University Press, pp. 295–306 (2010) 10. Berndt, B.C., Rankin, R.A.: Ramanujan Letters and Commentary, History of Mathematics. American Mathematical Society, vol. 9 (1995) 11. Biermann, G.G.A.: Problemata Quaedam Mechanica Functionum Ellipticarum Ope Soluta. Inaugural Dissertation, Berlin, Germany (1865) 12. Brevik, I., Bytsenko, A., Goncalves, A., Williams, F.L.: Zeta function regularization and the thermodynamic potential for quantum fields in symmetric spaces. J. Phys. A 31(19), 4437– 4448 (1998) 13. Brisebarre, N., Philibert, G.: Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Ramanujan Math. Soc. 20(4), 255–282 (2005) 14. Bump, D.: Automorphic Forms and Representations. In: Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press (1997) 15. Bytsenko, A.A., Elizalde, E., Odintsov, S.D., Romeo, A., Zerbini, S.: Zeta regularization techniques with applications. World Scientific, Singapore (1994) 16. Bytsenko, A.A., Cognola, G., Vanzo, L., Zerbini, S.: Quantum fields and extended objects in space-times with constant curvature spatial section. Phys. Rep. 266(1 and 2) (1996) 17. Bytsenko, A.A., Vanzo, L., Zerbini, S.: Massless scalar Casimir effect in a class of hyperbolic Kaluza-Klein space-times. Mod. Phys. Lett. A 7(5), 397–409 (1992) 18. Bytsenko, A.A., Goncalves, A., Zerbini, S.: One-loop effective potential for scalar and vector fields on higher dimensional noncommutative flat manifolds. Mod. Phys. Lett. A 16, 1479– 1486 (2001) 19. Bytsenko, A.A., Goncalves, A., Williams, F.L.: The conformal anomaly in general rank 1 symmetric spaces and associated operator product. Mod. Phys. Lett. A 13, 99–108 (1998) 20. Camporesi, R.: Harmonic analysis and propagators on homogeneous spaces. Phys. Rep. 196(1 and 2) (1990) 21. Camporesi, R.: ζ -function regularization of one-loop effective potentials in anti-de Sitter spacetime. Phys. Rev. D 43(12), 3958–3985 (1991) 22. Camporesi, R.: On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact symmetric spaces of rank one. J. Math. Anal. Appl. 214(2), 524–549 (1997) 23. Camporesi, R., Higuchi, A.: Arbitrary spin effective potentials in anti-de Sitter spacetime. Phys. Rev. D 7(8), 3339–3344 (1993) 24. Carlip, S., Teitelboim, C.: Aspects of black hole quantum mechanics and thermodynamics in 2+1 dimensions. Phys. Rev. D 51(2), 622–631 (1995) 25. Carroll, S.: Spacetime and geometry: an introduction to general relativity. Addison Wesley, San Francisco (2004) 26. Cartan, E.: Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France 54, 214–264 (1926) 27. Cartan, E.: Oeuvres Complètes, vols. 1 and 2. Gauthier-Villars, Paris (1952) 28. Chandrasekharan, K.: Elliptic Functions, Grundlehren der Mathematischen Wissenschaften, vol. 281. Springer, Berlin Heidelberg New York Tokyo (1985)

References

229

29. Clarkson, P.A., Rogers, C.: Ermakov-Painlevé II Reduction in cold plasma physics: application of a Bäcklund transformation. J. Nonlinear Math. Phys. 25(2), 247–261 (2018) 30. Cognola, G., Elizalde, E., Zerbini, S.: One-loop effective potential from higher-dimensional AdS black holes. Phys. Lett. B 585, 155–162 (2004) 31. Creutzig, T., Hikida, Y., Rønne, P.: Higher spin AdS3 supergravity and its dual CFT. J. High Energy Phys. 109 (2012) 32. Dabholkar, A.: Going beyond Bekenstein and Hawking (exact and asymptotic degeneracies of small black holes). Int. J. Mod. Phys. D15(10), 1561–1572 (2006) 33. D’Ambroise, J.: Applications of elliptic and theta functions to Friedmann-RobertsonLemaitre-Walker cosmology with cosmological constant. In: Kirsten, K., Williams, F.L. (eds.) A Window into Zeta and Modular Physics. Mathematical Sciences Research Institute Publications, vol. 57. Cambridge University Press, Berkeley, CA, pp. 279–293 (2010) 34. D’Ambroise, J., Williams, F.L.: Elliptic function solutions in Jackiw-Teitelboim dilaton gravity. Adv. Math. Phys. 2017 (2017). Hindawi, id.2154784 35. D’Ambroise, J., Williams, F.L.: Parametric solution of certain nonlinear differential equations in cosmology. J. Nonlinear Math. Phys. 18, 269–278 (2011); part II of this paper: Proceedings of Science, 7th Conference on Mathematical Methods in Physics, Rio de Janeiro, Brazil (2012). arXiv:1208.4812 [gr-qc] 36. D’Ambroise, J., Williams, F.L.: Relating some nonlinear systems to a cold plasma magnetoacoustic system. J. Modern Phys. 11, 886–906 (2020) 37. Epstein, P.: Theorie, Zur, allgemeiner Zetafunktionen, I, II, Math. Ann. 56, 614–644; Math. Ann. 63(1907), 205–216 (1903) 38. Frenkel, I.B., Lepowsky, J., Meurman, A.: A moonshine module for the Monster. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.) Vertex Operators in Mathematics and Physics. Mathematical Sciences Research Institute Publications, vol. 3. Springer, Berkeley, CA, pp. 231–273 (1985) 39. Gaiotto, D.: Monster symmetry and extremal CFTs. J. High Energy Phys. 11 (2012) 40. Gangolli, R.: Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces. Acta Math. 121, 151–192 (1968) 41. Gegenberg, J., Kunstatter, G.: Solitons and black holes. Phys. Lett. B 413, 274–280 (1997) 42. Gegenberg, J., Kunstatter, G.: Geometrodynamics of sine-Gordon solitons. Phys. Rev. D 58(12) (1998), id.124010 43. Gegenberg, J., Kunstatter, G.: From two-dimensional black holes to sine-Gordon solitons. In: MacKenzie, R., Paranjape, M., Zakrzewski, W. (eds.) Solitons: Properties, Dynamics, Interactions. Springer, New York, 99–106 (2000) 44. Glaisher, J.W.L.: On the numbers of representations of a number as a sum of 2r squares, where 2r does not exceed eighteen. Proc. London Math. Soc. 5(2), 479–490 (1907) 45. Godoy, T.: Minakshisundaram-Pleijel coefficients for compact locally symmetric spaces of classical type with non-positive sectional curvature. Ph.D. Dissertation, National University of Córdoba, Argentina (1987) 46. Godoy, T., Miatello, R., Williams, F.L.: The local zeta function for symmetric spaces of non-compact type. J. Geom. Phys. 61(1), 125–136 (2011) 47. Goldstein, L.J.: Dedekind sums for a Fuchsian group, I. Nagoya Math. J. 50, 21–47 (1973) 48. Goldstein, L.J.: Errata for Dedekind sums for a Fuchsian group, I. Nagoya Math. J. 53, 235– 237 (1974) 49. Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products: Corrected and Enlarged Edition. Academic Press (1980) 50. Gunning, R.C.: Lectures on modular forms. In: Annals of Mathematics Studies, vol. 48. Princeton University Press (1962) 51. Gurevich, A., Krylov, A.: A shock wave in dispersive hydrodynamics. Sov. Phys. Dokl. 32, 73–74 (1988) 52. Gurevich, A., Krylov, A.: (Russian version) Dokl. Akad. Nauk SSSR 298(3), 608–611 (1988) 53. Hawking, S.: Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys. 55(2), 133–148 (1977)

230

References

54. Helgason, S.: Differential Geometry and Symmetric Spaces. In: Pure and Applied Mathematics Series, vol. 12. Academic Press, Boca Raton (1962) 55. Helgason, S.: Geometric Analysis on Symmetric Spaces. In: Mathematical Surveys and Monographs, vol. 39, 2nd ed. American Mathematical Society (2008) 56. Hellaby, C.: The nonsimultaneous nature of the Schwarzschild r = 0 singularity. J. Math. Phys. 37(6), 2892–2905 (1996) 57. Hirschhorn, M.D.: A letter from Fitzroy house. Am. Math Mon. 115(6), 563–566 (2008) 58. Höhn, G.: Selbstduale Vertexoperatorsuperalgebren und das Babymonster. In: Bonner Mathematische Schriften. Universität Bonn Mathematical Institute , vol. 286. Dissertation (1996), Rheinische Friedrich-Wilhelms Universität Bonn (1995) 59. Horikis, T.P., Kevrekidis, P.G., Tsitoura, F., Williams, F.L.: Solitary waves in the resonant nonlinear Schrödinger equation: stability and dynamical properties. Phys. Lett. A 384(22)(2020) 60. Iwaniec, H.: Topics in Classical Automorphic Forms. In: Graduate Studies in Mathematics, vol. 17. American Mathematics Society (1997) 61. Jackiw, R.: A two-dimensional model of gravity. In: Christensen, S. (ed.) Quantum Theory of Gravity. Adam Hilger Ltd., pp. 403–420 (1984) 62. Jeffrey, A.: The role of Korteweg-de Vries equation in plasma physics. Q. J. R. Astron. Soc. 14, 183–189 (1973) 63. Jeffrey, A., Kakutani, T.: Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation. SIAM Rev. 14, 582–642 (1972) 64. Knopp, M.: Automorphic forms of nonnegative dimension and exponential sums. Mich. Math. J. 7(3), 257–287 (1960) 65. Knopp, M.: Rademacher on J(τ ), Poincare series of nonpositive weights, and the Eichler cohomology. Not. Am. Math. Soc. 31(4), 385–393 (1990) 66. Kraniotis, G.V., Whitehouse, S.B.: General relativity, the cosmological constant and modular forms. Class. Quantum Gravity 19, 5073–5100 (2002) 67. Krasi´nski, A.: Inhomogeneous Cosmological Models. Cambridge University Press (1997) 68. Krasi´nski, A., Pleba´nski, J.: An Introduction to General Relativity and Cosmology. Cambridge University Press (2006) 69. Lee, J.-H., Pashaev, O.K.: Resonance NLS solitons as black holes in Madelung fluid. Modern Phys. Lett. A 17(24), 1601–1619 (2002) 70. Lee, J.-H., Pashaev, O.K.: Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions and Hirota bilinear method. Theor. Math. Phys. 152(1), 991–1003 (2007) 71. Lee, J.-H., Pashaev, O.K., Rogers, C., Schief, W.: The resonant nonlinear Schrödinger equation in cold plasma physics: application of Bäcklund-Darboux transformations and superposition principles. J. Plasma Phys. 73(2), 257–272 (2007) 72. Lemaitre, G.: The expanding universe. Ann. Soc. Sci. Brux. A 53, 51–85; Gen. Relativ. Gravit. 29(1997), 641–680 (1933) 73. Lidsey, J.E.: Cosmology and the Korteweg-de Vries equation. Phys. Rev. D 86(12) (2012). id.123523 74. Lorenz, L.: Contributions à la théorie des nombres. Tidsskrift for Mathematik 3, 97–114 (1871) 75. Maloney, A., Witten, E.: Quantum gravity partition functions in three dimensions. J. High Energy Phys. 02, 029 (2010) 76. Manschot, J., Moore, G.W.: A modern Farey tail. Commun. Number Theory Phys. 4, 103–159 (2010) 77. Martina, L., Pashaev, O.K., Soliani, G.: Bright solitons as black holes. Phys. Rev. D 58(8) (1998). id.084025 78. Mason, G., Tuite, M.: Vertex operators and modular forms. In: Kirsten, K., Williams, F.L. (eds.) A Window into Zeta and Modular Physics. Mathematical Sciences Research Institute Publications, vol. 57. Cambridge University Press, Berkeley, CA, pp. 183–278 (2010) 79. Meinardus, G.: Über partitionen mit differenzenbedingungen. Math. Z. 61(1), 289–302 (1954)

References

231

80. Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949) 81. Miura, R.: Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202–1204 (1968) 82. Molk, J., Tannery, J.: Eléments de la Théorie des Fonctions Elliptiques, 4 vols. GauthiersVillars, Paris (1896) 83. Moreno, C.J.: Explicit formulas in the theory of automorphic forms. In: Lecture Notes in Mathematics, vol. 627. Springer, Berlin, pp. 73–216 (1977) 84. Ram Murty, M., Sampath, K.: On the asymptotic formula for the Fourier coefficients of the j-function. Kyushu J. Math. 70, 83–91 (2016) 85. Patterson, S.J.: The Selberg zeta-function of a Kleinian group. In: Aubert, K.E., Bombieri, E., Goldfeld, D. (eds.) Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, vol. 1989. Academic Press, pp. 409–441 (1987) 86. Perry, P.: Heat trace and zeta function for the hyperbolic cylinder in three dimensions, four page fax based in part on notes of F. L. Williams (2001) 87. Perry, P., Williams, F.L.: Selberg zeta function and trace formula for the BTZ black hole. Int. J. Pure Appl. Math. 9(1), 1–21 (2003) 88. Petersson, H.: Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58(1), 169–215 (1932) 89. Rademacher, H.: The Fourier coefficients of the modular invariant J(τ ). Am. J. Math. 60(2), 501–512 (1938) 90. Rademacher, H.: Fourier expansions of modular forms and problems of partition. Bull. Am. Math. Soc. 46, 59–73 (1940) 91. Rademacher, H., Zuckerman, H.S.: On the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. 39(2), 433–462 (1938) 92. Reynolds, M.J.: An exact solution in non-linear oscillations, Letter to the Editor. J. Phys. A: Math. Gen. 22, L723–L726 (1989) 93. Shikin, I.: Gravitational fields with groups of motions on two-dimensional transitivity hypersurfaces in a model with matter and a magnetic field. Comm. Math. Phys. 26, 24–38 (1972) 94. Szafron, D.A.: Inhomogeneous cosmologies: New exact solutions and their evolution. J. Math. Phys. 18, 1637–1677 (1977) 95. Szekeres, P.: A class of inhomogeneous cosmological models. Comm. Math. Phys. 41, 55–64 (1975) 96. Taimanov, I.A.: Modified Novikov-Veselov equation and differential geometry of surfaces. Am. Math. Soc. Transl. Sec. 2(179), 131–151 (1997) 97. Teitelboim, C.: The Hamiltonian structure of two-dimensional spacetime and its relation with the conformal anomaly. In: Christensen, S. (ed.) Quantum Theory of Gravity. Adam Hilger Ltd., pp. 327–344 (1984) 98. Terras, A.: Harmonic analysis on symmetric spaces and applications I. Springer, Berlin Heidelberg New York Tokyo (1985) 99. Uspensky, J.V.: Asymptotic formulae for numerical functions which occur in the theory of partitions[Russian]. Bull. Acad. Sci. URSS 14(6), 199–218 (1920) 100. Varlamov, V.: Spinor fields on the surface of revolution and their integrable deformations via the mKdV-hierarchy (1999). arXiv:9906140 [math.DG] 101. Vassileva, I.N.: Dedekind eta function, Kronecker limit formula, and Dedekind sum for the Hecke group, Ph.D. Dissertation, University of Massachusetts, Amherst (1996) 102. Watson, G.N., Whittaker, E.T.: A Course in Modern Analysis, 2nd ed. Merchant Books (1915) 103. Williams, F.L.: Lectures on the Spectrum of L 2 (\G). In: Pitman Research Notes in Mathematics Series, vol. 242. Longman House, Harlow, Essex, UK (1991) 104. Williams, F.L.: Topological Casimir energy for a general class of Clifford-Klein space-times. J. Math. Phys. 38(2), 796–808 (1997) 105. Williams, F.L.: The role of Selberg’s trace formula in the computation of Casimir energy for certain Clifford-Klein space-times. In: Dean, D. (ed.) African Americans in Mathematics. Discrete Mathematics & Theoretical Computer Science, vol. 34. The American Mathematical Society, pp. 69–82 (1997)

232

References

106. Williams, F.L.: Meromorphic continuation of Minakshisundaram-Pleijel series for semisimple Lie groups. Pac. J. Math. 182(1), pp. 137–156 (1998) 107. Williams, F.L.: One-loop effective potential for quantum fields on hyperbolic Kaluza-Klein space-times. In: Gnedin, Y.N., Grib, A.A., Mostepanenko, V.M., Rodrigues Jr., W.A. (eds.) Proceedings of the Fourth Alexander Friedmann International Seminar on Gravitation and Cosmology, St. Petersburg, Russia, pp. 351–358 (1999) 108. Williams, F.L.: Topics in Quantum Mechanics. In: Progress in Mathematical Physics, vol. 27. Birkhäuser, Boston Basel Berlin (2003) 109. Williams, F.L.: A zeta function for the BTZ black hole. Int. J. Mod. Phys. A18(12), 2205–2209 (2003) 110. Williams, F.L.: Remarks on harmonic maps, solitons, and dilaton gravity. In: Milton, K.A. (ed.)Quantum Field Theory Under the Influence of External Conditions. Rinton Press. Inc., Princeton, N.J., pp. 370–372 (2004) 111. Williams, F.L.: Further thoughts on first generation solitons and J-T gravity. In: Chen, L.V. (ed.) Trends in Soliton Research. Nova Science Publishers, pp. 1–14 (2006) 112. Williams, F.L.: Conical defect zeta function for the BTZ black hole. In: One Hundred Years of Relativity: Proceedings of the Einstein Symposium, Iais, Romania, Scientific Annals of Alexandru Ioan Cuza University, pp, 54–58 (2006) 113. Williams, F.L.: A resolvent trace formula for the BTZ black hole with conical singularity. In: Noël, A.G., King, D.R., N’Guérékata, G.M., Goins, E.H. (eds.) Council for African American Researchers in the Mathematical Sciences, vol. V, Contemporary Mathematics, vol. 467, The American Mathematical Society, pp. 49–62 (2008) 114. Williams, F.L.: Lectures on zeta functions, L-functions and modular forms with some physical applications. In: Kirsten, K., Williams, F.L. (eds.) A Window into Zeta and Modular Physics, Mathematical Sciences Research Institute Publications, vol. 57. Cambridge University Press, Berkeley, CA, pp. 7–100 (2010); also: The role of the Patterson-Selberg zeta function of a hyperbolic cylinder in three-dimensional gravity with a negative cosmological constant, pp. 329–351 115. Williams, F.L.: Zakharov-Shabat systems and conformal immersions induced by Dirac spinors. In: Proceedings of Science, 3rd International Satellite Conference on Math. Methods in Physics, Londrina, Brazil (2013) 116. Williams, F.L.: Some selected thoughts old and new on soliton-black hole connections in 2D dilaton gravity. In: Maraver, J.C., Kevrekidis, P.G., Williams, F.L. (eds.) The Sine-Gordon Model and its Applications. Springer Publishing, pp. 177–205 117. Williams, F.L.: Remainder formula and zeta expression for external CFT partition functions. In: Howe, R., Hunziker, M., Willenbring, J.F. (eds.) Symmetry: Representation Theory and Its Applications, In Honor of Nolan R. Wallach, Progress in Mathematics, vol. 257. Birkhäuser, pp. 505–518 (2014) 118. Williams, F.L.: Zeta function expression of spin partition functions on thermal AdS3 . Mathematics 3, 653–665 (2015) 119. Williams, F.L.: Exploring a cold plasma-2D black hole connection. Adv. Math. Phys. 2019 (2019). Hindawi, id.4810904 120. Williams, F.L.: Remarks on the dispersion relation for long magnetoacoustic waves in a cold plasma: a 2D black-hole connection. In: Price, L.S. (ed.) The Wave Equation: An Overview, Nova Science, Chap. 6, pp. 171–182 (2020) 121. Williams, F.L.: Negative weight modular forms and black hole entropy: Logarithmic corrections. In: New Insights into Physical Science, vol.11, Book publishing International, India United Kingdom, Chap. 9, pp. 109–124 (2021) 122. Williams, F.L.: Minakshisundaram-Pleijel coefficients for non-compact higher rank symmetric spaces. Analysis and Mathematical Physics, vol. 10, issue 4, (2020), id.52 123. Witten, E.: Three-dimensional gravity revisited (2007). arXiv:0706.3359 [hep-th] 124. Zagier, D.: Traces of singular moduli. In: Bogomolov, F., Katzarkov, L. (eds.) Motives, Polylogarithms and Hodge Theory, Part I, International Press Lecture Series. International Press, pp. 211–244 (2002)

References

233

125. Zwiebach, B.: A First Course in String Theory. Cambridge University Press (2004) 126. Lisi, A.G.: An Exceptionally Simple Theory of Everything. arXiv: 0711.0770 [hep-th] 127. Distler, J., Garibaldi, S.: There is no “Theory of Everything” inside E 8 . Comm. Math. Phys. 298, 419 (2010). arXiv:0905.2658 [math.RT] 128. Dynkin, E.B.: The structure of semi-simple algebras. Uspekhi Mat. Nauk. 24(20), 59–127 (1947) 129. Knapp, A.W.: Lie Groups Beyond an Introduction, 2nd edn. Birkhäuser, Boston (1996) 130. Representation Theory and Mathematical Physics: Conference in Honor of Gregg Zuckerman’s 60th Birthday October 24–27, 2009 Yale University by Adams, J., Lian, B., Sahi, S. (eds.), American Mathematical Society, Contemporary Mathematics (2011) 131. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, New York (1983)