Heat Kernel on Lie Groups and Maximally Symmetric Spaces 3031274504, 9783031274503

Table of contents :
Preface
Acknowledgments
Contents
Notation
Part I Manifolds
1 Introduction
1.1 Heat Equation
1.2 Geometric Framework
1.3 Spherical Coordinates
1.4 Hypergeometric Operators
2 Geometry of Simple Groups
2.1 Right-invariant and Left-invariant Basis
2.2 Metric and Connection
2.3 Heat Kernel on Simple Groups
2.4 Spin-tensor Bundles over Simple Groups
2.5 Geometry of SO(n+1) and SO(1,n)
3 Geometry of SU(2)
3.1 Representations of SU(2)
3.2 Right-Invariant and Left-Invariant Basis
3.3 Metric and Laplacian
3.4 Heat Kernel on SU(2)
4 Maximally Symmetric Spaces
4.1 Normal Geodesic Coordinates
4.2 Maximally Symmetric Spaces
4.3 Local Coordinates
4.4 Geodesic Spherical Coordinates
4.5 Isometries
4.6 Lie Derivatives
4.7 Laplacian
5 Three-dimensional Maximally Symmetric Spaces
5.1 Geometry of S3
5.2 Geometry of H3
Part II Heat Kernel
6 Scalar Heat Kernel
6.1 Reduction Formulas
6.2 Scalar Heat Kernel on S1 and R
6.3 Scalar Heat Kernel on S2
6.4 Scalar Heat Kernel on H2
6.5 Scalar Heat Kernel on Sn, n≥3
6.6 Scalar Heat Kernel on Hn, n≥3
7 Spinor Heat Kernel
7.1 Spinor Heat Trace
7.2 Spinor Heat Kernel
8 Heat Kernel in Two Dimensions
8.1 Heat Kernel on S2
8.2 Heat Kernel on H2
9 Heat Kernel on S3 and H3
9.1 Scalar Heat Kernel on S3 and H3
9.2 Spinor Heat Kernel on S3 and H3
9.3 Heat Trace
9.4 Heat Kernel on S3
9.5 Heat Kernel on H3
10 Algebraic Method for the Heat Kernel
10.1 Algebraic Method for Heat Kernel
10.2 Algebraic Method for S2 and H2
10.3 Heat Kernel Diagonal on Sn and Hn
A Integrals, Series, and Special Functions
A.1 Integrals
A.2 Poisson Summation Formula
A.3 Bernoulli Polynomials
A.4 Gamma Function
A.5 Hypergeometric Function
A.6 Legendre Functions
A.7 Polynomials
A.8 Dirac Matrices
References
Index

Citation preview

Frontiers in Mathematics

Ivan G. Avramidi

Heat Kernel on Lie Groups and Maximally Symmetric Spaces

Frontiers in Mathematics Advisory Editors William Y.C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany

This series is designed to be a repository for up-to-date research results which have been prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developments at the research frontiers in mathematics and at the “frontiers” between mathematics and other fields like computer science, physics, biology, economics, finance, etc. All volumes are online available at SpringerLink.

Ivan G. Avramidi

Heat Kernel on Lie Groups and Maximally Symmetric Spaces

Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM, USA

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-27450-3 ISBN 978-3-031-27451-0 (eBook) https://doi.org/10.1007/978-3-031-27451-0 Mathematics Subject Classification: 58C40, 58J35, 58J50, 58J70, 35K08 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Valentina.

Preface

The heat kernel is a universal and very powerful tool used in various areas of mathematics such as global analysis, spectral geometry, stochastic processes, and financial mathematics as well as in such areas of mathematical physics as quantum field theory, quantum gravity, string theory, and statistical physics. In particular, the main objects of interest in quantum field theory and statistical physics are the Green functions (or correlation functions) and the effective action (or the partition function) associated with elliptic partial differential operators on manifolds. The most important operators appearing in applications are of Laplace type; these are second-order partial differential operators acting on sections of vector bundles with scalar leading symbol determined by the Riemannian metric. The Green functions are determined by the resolvent and the effective action is determined by the functional determinant of these operators. The resolvent is given simply by the inverse Laplace transform of the heat kernel and the functional determinant is determined by the derivative of the spectral zeta function, which, in turn, is expressed in terms of the Mellin transform of the functional trace of the heat kernel (so-called heat trace). The heat kernel can be expressed in terms of the eigenvalues and the eigensections of these operators. That is why, the heat kernel provides an invaluable tool to study the spectral properties of such operators. Although one cannot compute the heat kernel on a generic manifold, one can develop powerful methods for computing its short-time asymptotic expansion and explicitly compute some lower order coefficients of this expansion. This short-time asymptotic expansion of the heat kernel determines the spectral asymptotics of differential operators and describes the semi-classical approximation in quantum field theory. In particular, the coefficients of this asymptotic expansion determine the spectral invariants of the operators which are used in spectral geometry to distinguish Riemannian manifolds. However, on some manifolds with a high level of symmetry, such as homogeneous spaces, Lie groups, and symmetric spaces, the spectrum of the Laplacian can be computed exactly by using purely group-theoretic methods, which enables one to obtain, in principle, the heat kernel in a closed form. This book is devoted to the study of the heat kernel for the spin-tensor Laplacians on maximally symmetric spaces, such as spheres and hyperbolic spaces. vii

viii

Preface

Roughly speaking, this book is a summary of my many papers on this subject. The book consists of two parts: Manifolds and Heat Kernel. The first part describes the geometry of simple Lie groups and maximally symmetric spaces. Chapter 1 is an Introduction. In Sect. 1.1, we introduce the resolvent and the heat kernel of an elliptic operator and describe an algebraic method for the calculation of the heat semigroup. In Sect. 1.2, we describe the geometric framework and fix the notation. In Sect. 1.3, we introduce general spherical coordinates and the corresponding differential operators. In Sect. 1.4, we introduce a special class of ordinary differential operators that we call hypergeometric operators. These operators determine the radial part of the Laplacian in any dimension. In Chap. 2, we describe the geometry of compact simple Lie groups in the form suitable for the future. In particular, we describe the groups .SO(n) and .SO(1, n) that play special role in the discussion. In Chap. 3, we consider the group .SU (2) in more detail. In Chap. 4 we describe the geometry of maximally symmetric spaces. We introduce special coordinate systems and describe the algebra of isometries and the Lie derivatives. In Chap. 5, we consider the case of three dimensions, .S 3 and .H 3 . This is a very special case since the 3-sphere .S 3 = SU (2) is, in fact, the group manifold of the group .SU (2), and, as a result, it has a basis of right-invariant vector fields. The second part is devoted to the calculation of the heat kernel for scalar, spinor, and generic Laplacian on spheres and hyperbolic spaces in various dimensions. In Chap. 6, we study the scalar heat kernel. In Sect. 6.1, we obtain reduction formulas for the scalar case which enable one to compute the heat kernel in any dimension from the heat kernels on .S 1 (and .R) and on .S 2 (and .H 2 ). We use the hypergeometric operators to obtain the resolvent and then the heat kernel for any dimension. In Chap. 7, we study the spinor heat kernel. In Sect. 7.1, we first compute the spinor heat trace by using a purely algebraic method. Next, in Sect. 7.2, we also compute the resolvent and the heat kernel by using the hypergeometric operators. In Chap. 8, we compute the resolvent and the heat kernel on .S 2 and .H 2 for a general representation. In Chap. 9, we study the heat kernel on .S 3 and .H 3 . First, in Sects. 9.1 and 9.2, we compute the scalar and the spinor heat kernel on .S 3 and .H 3 . In Sect. 9.4, we apply the group-theoretic approach to .S 3 = SU (2) to compute the heat kernel for an arbitrary representation; the heat kernel for .H 3 is obtained by the duality transformation in Sect. 9.5. In Chap. 10, we employ the method described in Sect. 1.1 to compute the heat kernel for a general spin-tensor bundle in any dimension. In Sect. 10.2, we illustrate the method on .S 2 and .H 2 in more details. As a result, we obtain some non-trivial dual integral representations of the heat kernel.

Preface

ix

In Appendix, we list some well-known integrals and describe the properties of some special functions used in the book. I provide some references to the original papers and books that I found useful. However, no attempt has been made to give a more or less comprehensive review of the literature. Socorro, NM, USA October 2022

Ivan G. Avramidi

Acknowledgments

I would like to express my gratitude to many friends, collaborators, and colleagues who had a significant influence on my career and from whom I learned a lot of material in this book. Most importantly, I thank my wife, Valentina, for constant support and encouragement.

xi

Contents

Part I Manifolds 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Geometric Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hypergeometric Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 7 12 17

2

Geometry of Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Right-invariant and Left-invariant Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Metric and Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Heat Kernel on Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spin-tensor Bundles over Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Geometry of SO(n + 1) and SO(1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 31 35 38 39

3

Geometry of .SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Representations of SU (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Right-Invariant and Left-Invariant Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Metric and Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Heat Kernel on SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 52 55

4

Maximally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Normal Geodesic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Maximally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Local Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Geodesic Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59 62 65 66 69 71

xiii

xiv

5

Contents

Three-dimensional Maximally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Geometry of S 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Geometry of H 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 83

Part II Heat Kernel 6

Scalar Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.1 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Scalar Heat Kernel on S 1 and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.3 Scalar Heat Kernel on S 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 Scalar Heat Kernel on H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.5 Scalar Heat Kernel on S n , n ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.6 Scalar Heat Kernel on H n , n ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7

Spinor Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Spinor Heat Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2 Spinor Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8

Heat Kernel in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.1 Heat Kernel on S 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.2 Heat Kernel on H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9

Heat Kernel on .S 3 and .H 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Scalar Heat Kernel on S 3 and H 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Spinor Heat Kernel on S 3 and H 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Heat Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Heat Kernel on S 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Heat Kernel on H 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 138 143 147 150

10

Algebraic Method for the Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Algebraic Method for Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Algebraic Method for S 2 and H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Heat Kernel Diagonal on S n and H n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 167

A

Integrals, Series, and Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Poisson Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Hypergeometric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 176 177 178 179

Contents

xv

A.6 A.7 A.8

Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Notation

A[a1 ···ak ] γa1 ...ak .f S n .Bn (x) .Bn 2 .G  . .

.

χT a .Γ bc .A .∇ i .δ j .F .|γ | diag .f  .δ(x, x ) .γa ∗ .V i .Ai B .γ . .tr .Tr .Γ (x) .f Rn (α) .C k (x) .Gab  .d(x, x )  .F (x, x )  .U (t; x, x ) .

Anti-symmetrization over tensor indices Anti-symmetrized product of Dirac matrices Average over .S n Bernoulli polynomials Bernoulli numbers Casimir operator of the representation G Cauchy principal value integral Character of the representation T Coefficients of Levi-Civita connection Connection 1-form Covariant derivative Cronecker symbol Curvature 2-form Determinant of the matrix .γ Diagonal value of a two-point function Dirac distribution Dirac matrices Dual vector bundle Einstein summation convention over repeated indices is adopted Euler constant External tensor product Fiber trace over a vector bundle Functional trace Gamma function Gaussian average over .Rn Gegenbauer polynomials Generators of the spin algebra Geodesic distance Group multiplication map Heat kernel xvii

xviii

L2 (M, w)

.

L2 (V) .C± .

Hn .F (a, b; c; z) .ε .(u, v) (α,β) .P (x) k .Jump G(λ) .

.Pν (x) .Qν (x) .Pk (x) .Γab .εa1 ...an .[X, Y ] .Lξ ± .Ka .ψ(z) .|y| a .y  .P(x, x ) .so(n) .ea a .ω .∂r .∂t .σi r  .G(λ; x, x ) .Rab a .R bcd .dvolM a .ω± ± .Ka  .σ (x, x ) R . 0 . .

Notation

Hilbert space of square integrable functions on a manifold M with weight w Hilbert space of square integrable sections of the vector bundle .V Horizontal contour in the upper (lower) half-plane that goes just above (below) the real axis Hyperbolic space in n-dimensions Hypergeometric function Infinitesimal positive real parameter Inner product Jacobi polynomials Jump of a complex function across the branch cut Laplacian Legendre function Legendre function of the second kind Legendre polynomials Levi-Civita connection 1-forms Levi-Civita symbol Lie bracket Lie derivative Lie derivatives along the right (left)-invariant vector fields Logarithmic derivative of the gamma function Norm of vector y Normal geodesic coordinates Operator of parallel transport Orthogonal Lie algebra Orthonormal frame on the tangent bundle Orthonormal frame on the cotangent bundle Partial derivative with respect to r Partial derivative with respect to t Pauli matrices Radial geodesic coordinate Resolvent Ricci curvature tensor Riemann curvature tensor Riemannian volume element Right (left)-invariant one-forms Right (left)-invariant vector fields Ruse-Synge function Scalar curvature Scalar Laplacian Semi-direct product

Notation

Z SO(n) .SO(1, n) n .S a .θ .A .Spin(n) . 1/2 .ω, v a .C bc .A(a1 ···ak ) T .C I  . (x, x ) .ζ (s, λ) . .

xix

Set of positive integers Special orthogonal group Special pseudo-orthogonal group n-Sphere Spherical coordinates Spin connection 1-form Spin group Spinor Laplacian Standard pairing Structure constants of Lie algebra Symmetrization over tensor indices Transposition of matrix Unit matrix Van Vleck-Morette determinant Zeta function

Part I Manifolds

1

Introduction

1.1

Heat Equation

The study of spectral properties of self-adjoint elliptic partial differential operators acting on sections of a vector bundle V over a manifold M has a long history both in geometric analysis and mathematical physics. One of the most powerful tools for this analysis is the so-called heat kernel, which is the integral kernel of the heat semigroup of an elliptic partial differential operator (see, e.g., [5, 8, 18, 28, 32–35, 41]). Let C ∞ (V) be the space of smooth sections of the vector bundle V. We assume that the vector bundle V is equipped with a Hermitian metric; this defines a natural L2 inner product using the invariant Riemannian measure on the manifold M. The completion of the space C ∞ (V) in this norm defines the Hilbert space L2 (V) of square integrable sections. The Laplacian is the most important operator of this class. It is positive elliptic partial differential operator of second order Δ on the Hilbert space L2 (V) with a unique selfadjoint extension (for precise definition see next section). For compact manifolds, such as spheres S n , the spectrum of the negative Laplacian (−Δ) is discrete with positive real eigenvalues {λk }k∈Z+ with finite multiplicities and the corresponding orthonormal eigensections {ϕk }k∈Z+ (counted with multiplicities) defined by (Δ + λk )ϕk = 0,

(ϕi , ϕj )L2 = δij .

(1.1)

For noncompact manifolds, such as hyperbolic spaces H n , the spectrum of the negative Laplacian is continuous, it goes from a positive real constant c to ∞. Let V ∗ be the dual vector bundle and V  V ∗ be the external tensor product of the bundles V and V ∗ over the product manifold M × M. The resolvent G(λ; x, x  ) of the Laplacian is a section of V  V ∗ depending on a complex parameter λ defined by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_1

3

4

1 Introduction

(−Δ − λ)G(λ; x, x  ) = δ(x, x  ),

(1.2)

G(λ; x, x  ) = (−Δ − λ)−1 δ(x, x  );

(1.3)

that is,

here δ(x, x  ) is the covariant Dirac delta-distribution. Off diagonal, that is, for x = x  , the resolvent is an analytic function of λ for Re λ < c with sufficiently large negative real constant c. It has a diagonal singularity as x → x  (more on this later). The heat kernel U (t; x, x  ) is a one-parameter family of smooth sections of V  V ∗ defined by requiring it to satisfy the heat equation (∂t − Δ)U (t; x, x  ) = 0

(1.4)

for t > 0 with the initial condition U (0+ ; x, x  ) = δ(x, x  ) ,

(1.5)

U (t; x, x  ) = exp(tΔ)δ(x, x  ) .

(1.6)

that is,

Contrary to the resolvent, the heat kernel is regular at the diagonal with a well defined diagonal value U diag (t; x) = U (t; x, x),

(1.7)

which is a section of the endomorphism bundle End(V) = V ⊗ V ∗ . The heat kernel diagonal is well defined on both compact and noncompact manifolds, which makes it well suited for the duality transformation of spheres and hyperbolic spaces. The heat kernel is related to the resolvent by the inverse Laplace transform 1 U (t; x, x ) = 2π i 

c+i∞ 

dλ e−tλ G(λ; x, x  )

c−i∞

=

e−ct



πi

dν e−tν ν G(c + ν 2 ; x, x  ), 2

(1.8)

C+

where c is a sufficiently large negative real constant, such that the resolvent is analytic for Re λ < c, and C+ is the contour in the upper half plane that goes just above the real axis

1.1 Heat Equation

5

from −∞ + iε to ∞ + iε, with ε > 0, such that the resolvent is analytic function of ν for Im ν > 0. For compact manifolds, the resolvent has simple poles at the eigenvalues 

G(λ; x, x ) =

∞  k=1

1 ϕk (x)ϕk∗ (x  ), λk − λ

(1.9)

with the residues determining the eigensections. The heat kernel can be computed in terms of the residues of the resolvent U (t; x, x  ) =

∞ 

e−tλk ϕk (x)ϕk∗ (x  ).

(1.10)

k=1

For compact manifolds the L2 - heat trace is well defined Tr exp(tΔ) =

∞ 

e

−tλk

 =

k=1

dvol tr U diag (t);

(1.11)

M

here and below tr denotes the fiber trace and Tr denotes the L2 -trace. This also enables one to define the spectral zeta function by the Mellin-Laplace transform of the heat trace ζ (s, λ) =

∞  k=1

1 1 = (λk − λ)s Γ (s)

∞ t s−1 etλ Tr exp(tΔ),

(1.12)

0

where λ is a complex parameter with a sufficiently large negative real part. On homogeneous manifolds, such as spheres and hyperbolic spaces, the fiber trace of the heat kernel diagonal is constant, therefore, for compact manifolds tr U diag (t) =

1 Tr exp(tΔ). vol (M)

(1.13)

For large negative λ the zeta function is a meromorphic function of s analytic at s = 0; this enables one to define the zeta-regularized determinant    Det (−Δ − λ) = exp −∂s ζ (s, λ)

 s=0

.

(1.14)

For noncompact manifolds, the resolvent has a branch cut singularity along the real axis from some real constant c to ∞ with the jump across the cut Jump G(λ; x, x  ) = G(λ + iε; x, x  ) − G(λ − iε; x, x  ).

(1.15)

6

1 Introduction

The heat kernel can be computed in terms of the jump of the resolvent 1 U (t; x, x ) = 2π i 

∞

dλ e−tλ Jump G(λ; x, x  )

c

=

e−ct 2π i



dν e−tν ν Jump G(c + ν 2 ; x, x  ). 2

(1.16)

R

It is well known [5, 27] that as t → 0 the heat kernel has the asymptotics   σ (x, x  ) , U (t; x, x  ) ∼ (4π t)−n/2 Δ1/2 (x, x  )P(x, x  ) exp − 2t

(1.17)

where n = dim M is the dimension of the manifold, σ (x, x  ) is the Ruse–Synge function [39, 40] (equal to one half of the square of the geodesic distance between the points x and x  ), Δ(x, x  ) is the Van Vleck–Morette determinant and P(x, x  ) is the operator of parallel transport of sections of the vector bundle V from the point x  to the point x along the geodesic (see Sect. 3.1 for details, also [5, 8]). This means that the resolvent has the leading diagonal singularity as x → x  of the form 

G(λ; x, x ) ∼

Γ

n 2

− 1 Δ1/2 (x, x  )P(x, x  )

4π n/2

n

[2σ (x, x  )] 2 −1

.

(1.18)

On a general manifold the heat kernel cannot be computed exactly, and that is why the short-time asymptotic expansion as t → 0 of the heat kernel (or of its L2 trace on compact manifolds) is studied; it has the form [28] Tr exp(tΔ) ∼ (4π t)−n/2

∞ 

t k Ak (M, V),

(1.19)

k=0

with the coefficients Ak being some spectral invariants; this is the subject of spectral geometry (see, e.g., [2, 5, 7, 27, 35, 41]). However, on manifolds with a high level of symmetry, such as Lie groups and homogeneous spaces, in particular, symmetric spaces, one can solve the Eqs. (1.2), (1.4) exactly and obtain closed formulas for the heat kernel in terms of the root vectors and the weight vectors of the corresponding algebras (see, e.g., [1,14–17,26,38]); similar problems were also studied in the physical literature [13, 19, 20, 22–24, 30, 31]). In our papers [3, 4] we developed an algebraic method for the computation of the heat kernel on symmetric spaces for scalar Laplacian , which was generalized in [6] for arbitrary vector bundles. The idea is to represent the Laplacian in the form Δ = GAB LA LB ,

(1.20)

1.2 Geometric Framework

7

where GAB is a positive symmetric matrix and LA are some first-order partial differential operators satisfying a Lie algebra [LA , LB ] = C C AB LC + FAB ,

(1.21)

with some constants C C AB and FAB ; here the capital Latin indices run over 1 to some N. Then one can show that the heat semigroup can be represented as an integral  exp(tΔ) =

RN

dξ Φ(t, ξ ) exp ξ, L ,

(1.22)

where ξ, L = ξ A LA and Φ(t, ξ ) is a function depending on the constants C C AB and FAB . This immediately gives an integral representation of the heat kernel 



U (t; x, x ) =

RN

dξ Φ(t, ξ )V (ξ ; x, x  ),

(1.23)

where V (ξ, x, x  ) = exp ξ, L δ(x, x  ),

(1.24)

and its trace  Tr exp(tΔ) =

 dvol (x)

M

RN

dξ Φ(t, ξ )tr V (ξ, x, x).

(1.25)

These methods were applied to the study of the effective action in quantum field theory and quantum gravity in our papers with S. Collopy [9–11]. In the present book we study a very special case of maximally symmetric spaces, that is, spheres S n and the hyperbolic spaces H n .

1.2

Geometric Framework

Let (M, g) be a n-dimensional Riemannian orientable spin manifold without boundary with a metric g. Let ea be a local orthonormal frame on the tangent bundle T M and ωa be the dual frame on the cotangent bundle T ∗ M. We denote the frame indices by the low case Latin indices from the beginning of the alphabet, which also run over 1, . . . , n. The frame indices are raised by the Euclidean matric δab , so that the position of the frame indices (up or down) does not matter. Then (ωa , ωb ) = δ ab ,

(ea , eb ) = δab ,

 a ω , eb = δ a b ,

(1.26)

8

1 Introduction

where the parenthesis denote the inner product and the brackets denote the standard pairing between the tangent and the cotangent space. The Riemannian volume element has the form dvol M = ω1 ∧ · · · ∧ ωn .

(1.27)

The Levi-Civita connection is defined by ∇ea eb = Γ c ba ec ,

(1.28)

where Γabc =

1 (γabc + γbca + γcba ) , 2

(1.29)

with  γ a bc = − ωa , [eb , ec ] .

(1.30)

dωa = Γ a bc ωb ∧ ωc ,

(1.31)

γ a bc = (dωa )(eb , ec ).

(1.32)

Then

and, therefore,

The curvature 2-form of the Levi-Civita connection is 1 a R bcd ωc ∧ ωd 2 = dΓ a b + Γ a c ∧ Γ c b ,

Ra b =

(1.33)

where Γab = Γabc ωc . The Ricci tensor and the scalar curvature are defined by R a b = R ca cb ,

R = R a a = R ab ab .

(1.34)

Let V be a spin-tensor vector bundle realizing an N -dimensional representation G of the spin group Spin(n). Let EA , A = 1, . . . , N, be the basis in the fiber of the vector bundle V. The generators of this representation are matrices Gab = (GA B ab ) satisfying the algebra [Gab , Gcd ] = −4δ [a [c Gb] d] .

(1.35)

1.2 Geometric Framework

9

The matrix G2 =

1 Gab Gab 2

(1.36)

commutes with all generators and determines the Casimir operator of the representation G. The spin connection induces a connection on the bundle V by ∇ea EA = AB Aa EB ,

(1.37)

where the matrices Ac = (AA B c ) are defined by Ac =

1 ab Γ c Gab . 2

(1.38)

The curvature of this connection is F = dA + A ∧ A =

1 ab R Gab , 2

(1.39)

where A = Ac ωc is the connection one-form. By slightly abusing notation we will denote a section of the bundle V simply by the column-vector of its components ϕ = (ϕ A ) in the basis EA ; one should keep in mind that these components do depend on the choice of the basis. The covariant derivative of a smooth section of the bundle V along a vector field ξ is then ∇ξ ϕ = (ξ + A, ξ ) ϕ .

(1.40)

The commutator of covariant derivatives of a section is

[∇ξ , ∇η ]ϕ = F(ξ, η) + ∇[ξ,η] ϕ,

(1.41)

in particular,

[∇ea , ∇eb ]ϕ =

 1 cd c R ab Gcd − γ ab ∇ec ϕ. 2

(1.42)

In particular, we will also consider the spinor representation of the spin group of n dimension N = 2[ 2 ] . Let γa be the Dirac matrices satisfying the Clifford algebra γa γb + γb γa = 2δab I,

(1.43)

10

1 Introduction

where I is the unit matrix, and γa1 ...ak = γ[a1 · · · γak ]

(1.44)

be their anti-symmetrized products (these matrices should not be confused with the commutation coefficients γ a bc ). By using the algebra of Dirac matrices [42] γ ab γcd = γ ab cd − 4δ [a [c γ b] d] − 2δ [a c δ b] d I,

(1.45)

one easily sees that the matrices Σab =

1 γab 2

(1.46)

satisfy (1.35) and, therefore, are the generators of the spin group Spin(n) in spinor representation. Furthermore, one can show that these matrices satisfy the important equation Σ c a Σcb = −

n−2 n−1 Σab − δab I, 2 4

(1.47)

and, therefore, the Casimir operator of the spinor representation is Σ2 = −

n(n − 1) I. 8

(1.48)

The Eq. (1.47) leads to a very important property of the spinor representation. Namely, for any unit vector θ a the norm of the vector Wa = Σab θ b

(1.49)

is constant, that is, W 2 = Wa W a = −

n−1 I. 4

(1.50)

The algebra of the Dirac matrices (1.45) also leads to a very important equation R ab cd γab γ cd = −2RI.

(1.51)

The Lie derivative of a section ϕ along a vector field ξ is defined by

Lξ ϕ = ∇ξ + Sξ ϕ ,

(1.52)

1.2 Geometric Framework

11

where Sξ = −

1 a ω , ∇eb ξ Gab . 2

(1.53)

By using eqs. (1.29), (1.30), it is easy to show that the Lie derivative has the form 

1 Lξ ϕ = ξ + ωa , [ξ, eb ] Gab ϕ. 2

(1.54)

In particular, 

1 Lea ϕ = ∇ea + Γabc Gbc ϕ 2 

1 = ea + γbca Gbc ϕ. 2

(1.55)

The Laplacian is a second-order partial differential operator (that we will call the spintensor Laplacian) acting on sections of the vector bundle V defined by Δ = ∇ea ∇ea ;

(1.56)

of course, it has the form

Δ = ea − γ

b

ba

1 + Γbca Gbc 2

  1 pq ea + Γpqa G . 2

(1.57)

The Dirac operator is a first-order partial differential operator acting on smooth sections of the spinor bundle defined by D = γ c ∇e c

 1 ab c = γ ec + Γ c γab . 4

(1.58)

By using the algebra of the Dirac matrices (1.43) and eq. (1.51) it is easy to see that the square of the Dirac operator is determined by the spinor Laplacian Δ1/2 , 1 D 2 = Δ1/2 − RI, 4

(1.59)

this formula is called the Lichnerowicz formula. Therefore, for manifolds with constant scalar curvature, in particular, maximally symmetric spaces, the square of the Dirac operator is just a shift of the Laplacian. That is why, we restrict ourselves in the present book to the study of Laplacians only.

12

1 Introduction

1.3

Spherical Coordinates

We introduce spherical coordinates on Rn as follows. Let y a , a = 1, . . . , n, be the Cartesian coordinates on Rn . We define the radial coordinate r = |y| =

 ya y a

(1.60)

and the angular coordinates θa =

ya , r

(1.61)

so that θ a θa = 1,

y a = rθ a .

and

(1.62)

That is, these are the coordinates in Rn restricted to the unit sphere S n−1 centered at the origin in the North pole. Of course, only (n − 1) of the angular coordinates θ a are independent. We define the Gaussian average over Rn

f (y)

Rn

=π

−n/2



  dy exp −r 2 f (y).

(1.63)

Rn

By using the Gaussian average of the exponential 

1

exp (i p, y)Rn = exp − |p|2 , 4

(1.64)

where p, y = pa y a and |p|2 = p, p = pa pa , we obtain the Gaussian averages of the polynomials (2k)! (a1 a2 δ · · · δ a2n−1 a2k ) , k!22k = 0.

y a1 · · · y a2k Rn =

y a1 · · · y a2k+1 Rn

(1.65) (1.66)

We compute the average of the constant function f = 1, 1 = 1Rn = vol (S

n−1

)π

−n/2

∞ 0

  dr exp −r 2 r n−1 .

(1.67)

1.3 Spherical Coordinates

13

By using the integral ∞

 1  dr r 2s−1 exp −r 2 = Γ (s) 2

(1.68)

0

we obtain the volume of the unit sphere S n−1  2π n/2  vol S n−1 = n . Γ 2

(1.69)

Next, we define the average of a function f over the sphere S n−1 by

f (θ )S n−1

1 = vol (S n−1 )

 dvol (θ )f (θ ),

(1.70)

S n−1

where vol (S n−1 ) =



dvol (θ ).

S n−1

Proposition 1.1 The averages over the sphere S n−1 of homogeneous polynomials are

θ a1 · · · θ a2k S n−1 =

Γ ( n2 ) (2k)! (a1 a2

δ · · · δ a2k−1 a2k ) , Γ k + n2 k!22k

θ a1 · · · θ a2k+1 S n−1 = 0.

(1.71)

(1.72)

Proof First, we reduce the Gaussian average to the integral of the spherical average

f (y)Rn

∞     n−1 −n/2 π = vol S dr exp −r 2 r n−1 f (rθ )S n−1 .

(1.73)

0

Furthermore, by using (1.73) we have

y

a1

· · · y Rn = vol (S ak

n−1

)π

−n/2

∞

  dr r n+k−1 exp −r 2 θ a1 · · · θ ak S n−1 .

0

Finally, we compute the integral over r by using (1.68) to get the result.

(1.74)

14

1 Introduction

Proposition 1.2 The average over the sphere S n−1 of the exponential is

exp (i p, y)S n−1

 n  2  n2 −1 =Γ J n2 −1 (rρ), 2 rρ

(1.75)

where ρ = |p|, r = |y| and Jν (z) is the Bessel function of the first kind [25]. Proof We use Eq. (1.71) to prove this. We have

exp (i p, y)S n−1 =

∞  (−1)k r 2k k=0

=

(2k)!

∞  (−1)k k=0

k!22k

pa1 · · · pa2k θ a1 · · · θ a2k S n−1



Γ n2

(rρ)2k . Γ k + n2

(1.76)

This gives the result by using the expansion of the Bessel function (see, e.g., [25]). We will be using differential operators in spherical coordinates. First of all, we have dy a = θ a dr + rdθ a .

(1.77)

ya dy a = rdr ,

(1.78)

θa dθ a = 0 ,

(1.79)

dya dy a = dr 2 + r 2 dθa dθ a .

(1.80)

Therefore,

hence,

and, further,

We compute the derivatives in the spherical coordinates. It is easy to see that ∂r = θa , ∂y a  ∂θ b 1 b b δ . = − θ θ a a ∂y a r We define the radial operator

(1.81) (1.82)

1.3 Spherical Coordinates

15

∂r =

1 a ∂ y , r ∂y a

(1.83)

and the angular operators

 ∂ y b ya Na = r δ b a − 2 , ∂y b r

(1.84)

∂ ∂ − yb a . ∂y b ∂y

(1.85)

∂ 1 = θa ∂r + Na . a ∂y r

(1.86)

Lab = y a Then

It is easy to see that these operators are related by Nb = θ a Lab .

(1.87)

By using (1.86) one can also show that Lab = θa Nb − θb Na .

(1.88)

Therefore, the operators Lab satisfy a non-trivial identity Lab = θ c (θa Lcb − θb Lca ) .

(1.89)

The operator ∂r is a radial differential operator that does not act on the angular variables whereas the operators Lab and Na are angular operators that do not act on the radial variable r and commute with the operator ∂r ,

θa,

[∂r , Lab ] = [∂r , Na ] = 0.

(1.90)

It is easy to see that the commutator of the operators Na is equal to the operator Lab , that is, Lab = [Na , Nb ].

(1.91)

The nice fact about the operators Lab is that they form a Lie algebra. Proposition 1.3 The operators Lab form the algebra so(n) [Lab , Lcd ] = −4δ [a [c Lb] d] .

(1.92)

16

1 Introduction

Proof This is proved by a direct calculation. As a corollary we get the identities L[a c Lb]c = −

(n − 2) Lab , 2

L[ab Lcd] = 0,

(1.93) (1.94)

where, as usual, the square brackets denote the anti-symmetrization over all included indices. Proposition 1.4 The action of the angular operators on the angular coordinates is as follows Na θ b = δ b a − θ b θa ,

(1.95)

Lab θ c = −T c dab θ d ,

(1.96)

T cd ab = δ c a δ d b − δ c b δ d a .

(1.97)

Lab (θ c1 · · · θ ck ) = −kT (c1 dab θ c2 · · · θ ck ) θ d .

(1.98)

where

More generally,

Proof This is proved by using the property Lab θ c = δ c b θa − δ c a θb .

(1.99)

The Casimir operator L2 =

1 Lab Lab 2

(1.100)

is a second-order operator commuting with the operators Lab [Lab , L2 ] = 0.

(1.101)

N 2 = L2 .

(1.102)

It is easy to see that

1.4 Hypergeometric Operators

17

Proposition 1.5 Let Tc1 ...ck be a symmetric traceless tensor and ϕk be a function on the sphere S n−1 defined by ϕk (θ ) = Tc1 ...ck θ c1 · · · θ ck .

(1.103)

L2 ϕk = −k(k + n − 2)ϕk .

(1.104)

T cd ab T ab pq = T cd pq ,

(1.105)

Then

Proof By using (1.98) and

we obtain L2 (θ c1 · · · θ ck ) = −k(k + n − 2)θ (c1 · · · θ ck ) + k(k − 1)δ (c1 c2 θ c3 · · · θ ck) .

(1.106)

The second term vanishes by multiplying it by a traceless tensor Tc1 ...ck . As we will see below, the operator L2 is nothing but the scalar Laplacian on the sphere S n−1 , L2 = ΔS0

n−1

(1.107)

.

The functions ϕk are the eigenfunctions of the operator L2 on S n−1 with eigenvalues   n−1 = k(k + n − 2), λk −ΔS0

k = 0, 1, 2, . . .

(1.108)

Therefore, the classification of the eigenfunctions is reduced to the classification of symmetric traceless tensors. The number of such tensors, that is, the multiplicity of the eigenvalue λk , is given by (6.112) (with n replaced by (n − 1)).

1.4

Hypergeometric Operators

Let μ, ν be two real parameters and γ =

μ+ν , 2

δ=

μ−ν , 2

(1.109)

so that μ = γ + δ and ν = γ − δ. We suppose that μ, ν > −1 so that γ > −1. Let Hμ,ν be a second-order differential operator of the hypergeometric type

18

1 Introduction

Hμ,ν = (1 − x 2 )∂x2 − [μ − ν + (ν + μ + 2)x] ∂x .

(1.110)

Let ω be a complex parameter. We define the following functions of a real variable x ∈ (−1, ∞). 

1 1−x 1 , = F ω + γ + , −ω + γ + ; μ + 1; (1.111) 2 2 2  

1 1−x 1 − x − iε −μ 1 (2) Φμ,ν (ω; x) = ,(1.112) F ω − δ + , −ω − δ + ; −μ + 1; 2 2 2 2

(1) Φμ,ν (ω; x)

(3) (ω; x) = Φμ,ν

Γ (ω + γ + 12 )Γ (−ω + γ + 12 ) Γ (ν + 1) 

1 1 + x + iε 1 , ×F ω + γ + , −ω + γ + ; ν + 1; 2 2 2

(4) (3) Φμ,ν (ω; x) = e−iπ μ Φμ,ν (ω; x)

−ie−iπ(ω+δ)

(1.113)

(1.114)

Γ (ω + γ + 12 )Γ (−ω + γ + 12 )Γ (ω + δ + 12 ) Γ (μ + 1)Γ (ω − δ + 12 )

(1) Φμ,ν (ω; x),

where F (a, b; c; z) is the hypergeometric function and the function (1 − x − iε)−μ is defined by  −μ

(1 − x − iε)

=

(1 − x)−μ ,

for x ∈ (−1, 1),

eiπ μ (x − 1)−μ , for x ∈ (1, ∞).

(1.115)

By using the property of the hypergeometric function [25, 36] F (a, b; c; 1 − z) =

Γ (c)Γ (c − a − b) F (a, b; a + b − c + 1; z) Γ (c − a)Γ (c − b) +zc−a−b

(1.116)

Γ (c)Γ (a + b − c) F (c − a, c − b; c − a − b + 1; z), Γ (a)Γ (b) (3)

(4)

for a non-integer μ the functions Φμ,ν and Φμ,ν can be written in the form (3) (2) Φμ,ν (ω; x) = Γ (μ)Φμ,ν (ω; x)

+

Γ (−μ) Γ (ω + γ + 12 )Γ (−ω + γ + 12 ) (1)    Φμ,ν (ω; x),  Γ (ν + 1) Γ ω − δ + 1 Γ −ω − δ + 1 2

2

(1.117)

1.4 Hypergeometric Operators

19

(4) (2) Φμ,ν (ω; x) = e−iπ μ Γ (μ)Φμ,ν (ω; x) + Γ (−μ)

Γ (ω + γ + 12 )Γ (ω + δ + 12 ) Γ (ω − γ + 12 )Γ (ω − δ + 12 )

(1) Φμ,ν (ω; x).

(1.118) For an integer μ = m a similar formula can be obtained by the analytic continuation, that is, setting μ = m + ε and taking the limit as ε → 0. We will need the following properties of these functions. Proposition 1.6 (i)

1. All functions Φμ,ν satisfy the equation 

Hμ,ν + ω − 2

ν+μ+1 2

2  (i) = 0. Φμ,ν

(1.119)

2. The function Φμ,ν (ω; x) is finite as x → (−1)+ and has the following asymptotics as x → 1− (3)

  (3) Φμ,ν (ω; x)

x→1−

∼ Γ (μ)

1−x 2

−μ ,

(1.120)

for positive μ > 0 and   (3) Φ0,ν (ω; x)

x→1−

∼ − log

1−x 2

 ,

(1.121)

for μ = 0. (3) 3. For x ∈ (−1, 1) the function Φμ,ν (ω; x) is an even analytic function of ω with simple poles at ±ωk , where ωk = k +

ν+μ+1 , 2

k = 0, 1, 2, . . .

(1.122)

and the residues   (3) Res Φμ,ν (ω; x); ±ωk

(1.123)

 1+x Γ (k + ν + μ + 1) F −k, k + ν + μ + 1; ν + 1; . = ∓(−1)k k!Γ (ν + 1) 2

20

1 Introduction

4. The function Φμ,ν (ω; x) is bounded as x → ∞ and has the asymptotics as x → 1+ (for μ > 0), (4)



 (4) Φμ,ν (ω; x)

x→1+

x−1 ∼ Γ (μ) 2

−μ (1.124)

.

(4) 5. For x > 1 the function Φμ,ν (ω; x) is ananalytic function of ω with simple poles at −ωk , k = 0, 1, 2, . . . and − μ−ν+1 + k , k = 0, 1, 2, . . . ; it is analytic for Re ω > 2

− μ−ν+1 . 2 6. The functions Φ (1) (ω; x), Φ (2) (ω; x) and Φ (3) (ω; x) are even function of ω. The (4) function Φμ,ν (ω; x) has the property (4) (4) Φμ,ν (ω; x) − Φμ,ν (−ω; x) = −

×

  π tan[π(ω + δ)] + tan[π(ω − δ)] Γ (μ + 1)

Γ (ω + γ + 12 )Γ (−ω + γ + 12 ) Γ (ω − δ + 12 )Γ (−ω − δ + 12 )

(1) Φμ,ν (ω; x).

(1.125)

Proof The Eq. (1.119) is the hypergeometric equation with the solutions given by the hypergeometric function. (1) Next, let, first, x ∈ [−1, 1). Obviously, the function Φμ,ν (ω; x) is finite at x = 1 and (3) the function Φμ,ν (ω; x) is finite at x = −1. By using the representation (1.117) we obtain (1) (ω; x) as x → 1− . the asymptotics (1.120) of the function Φμ,ν Now, let x ∈ (1, ∞). As x → 1+ by using (1.120) we obtain (1.124). Further, for (1) (2) Re ω > 0 the functions Φμ,ν (ω; x) and Φμ,ν (ω; x) are unbounded. By using the relation if a − b is not an integer) [25] F (a, b, c; z) =

Γ (c)Γ (b − a) (−z)−a F (a, 1 − c + a; 1 − b + a; z−1 ) Γ (b)Γ (c − a) +

Γ (c)Γ (a − b) (−z)−b F (b, 1 − c + b; 1 − a + b; z−1 ), (1.126) Γ (a)Γ (c − b)

we obtain the following asymptotics as x → ∞ (1) Φμ,ν (ω; x) ∼

 x ω− 1 (ν+μ+1)

Γ (μ + 1)Γ (2ω)

2

Γ (ω + δ + 12 )Γ (ω + γ + 12 ) 2

(2) Φμ,ν (ω; x) ∼ eiπ μ

 x ω− 1 (ν+μ+1)

Γ (−μ + 1)Γ (2ω) Γ (ω − δ +

(3) Φμ,ν (ω; x) ∼ ieiπ (−ω+γ )

1 2 )Γ (ω

+γ −

2

1 2)

2

(1.127)

,

,

Γ (−ω + γ + 12 )Γ (2ω)  x ω− 12 (ν+μ+1) . 2 Γ (ω − δ + 12 )

(1.128)

(1.129)

1.4 Hypergeometric Operators

21 (4)

By using these equations we prove the result for the function Φμ,ν . Proposition 1.7 Let w = w(x) be the function defined by w(x) = (1 − x)μ (1 + x)ν .

(1.130)

The spectrum of the operator Hμ,ν on the Hilbert space L2 ([−1, 1], w) consists of simple eigenvalues

λk −Hμ,ν = k(k + ν + μ + 1) ,

k = 0, 1, 2, . . .

(1.131)

with the eigenfunctions being the Jacobi polynomials (μ,ν)

Pk

(x) =

 1−x Γ (μ + k + 1) F −k, k + μ + ν + 1; μ + 1; . k!Γ (μ + 1) 2 (μ,ν)

Proof This is a well known property of the Jacobi polynomials Pk example, [25].

(1.132) (x); see, for

For the future reference we obtain the following intertwining relation. Proposition 1.8 Let ρ = ρ(x) be a function defined by ρ(x) = (1 − x)α (1 + x)β ,

(1.133)

with real parameters α and β. Then there holds α(α + 2μ) β(β + 2ν) − 2(1 − x) 2(1 + x) 

 α+β α+β 1/2 Hμ+α,ν+β − + μ + ν + 1 ρ −1/2 . =ρ 2 2

Hμ,ν −

(1.134)

Proof This is proved by direct calculation. Proposition 1.9 The operators Hμ,ν satisfy the intertwining relation

Hμ,ν ∂x = ∂x Hμ−1,ν−1 + ν + μ ,

(1.135)

and, therefore, for any complex λ and real t (Hμ,ν + λ)−1 ∂x = ∂x (Hμ−1,ν−1 + λ + μ + ν)−1 ,

(1.136)

22

1 Introduction



 exp(tHμ,ν )∂x = ∂x exp t Hμ−1,ν−1 + ν + μ .

(1.137)

Proof This is easily proved by direct calculation. We will need two particular types of operators Ln = H n2 −1, n2 −1 = (1 − x 2 )∂x2 − nx∂x ,

(1.138)

Mn = H n2 −1, n2 = (1 − x 2 )∂x2 − [(n + 1)x − 1]∂x ,

(1.139)

where n ≥ 1 is a positive integer. Proposition 1.10 The operators Ln , Mn satisfy the intertwining relations Ln ∂x = ∂x (Ln−2 + n − 2) ,

(1.140)

Mn ∂x = ∂x (Mn−2 + n − 1) .

(1.141)

Proof This follows from (1.135). Proposition 1.11 For a positive integer m > 0 there holds Ln ∂xm = ∂xm [Ln−2m + m(n − m − 1)] , Mn ∂xm

=

∂xm [Mn−2m

+ m(n − m)] .

(1.142) (1.143)

Therefore, for odd n = 2α + 1,   L2α+1 ∂xα = ∂xα L1 + α 2 ,

(1.144)

M2α+1 ∂xα = ∂xα [M1 + α(α + 1)] ,

(1.145)

L2β+2 ∂xβ = ∂xβ [L2 + β(β + 1)] ,

(1.146)

M2β+2 ∂xβ = ∂xβ [M2 + β(β + 2)] .

(1.147)

and for even n = 2β + 2,

Proof This is proved by induction. As a corollary we obtain the following formal relations.

1.4 Hypergeometric Operators

23

Proposition 1.12 For any complex λ and a real t there holds (L2α+1 + λ)−1 ∂xα = ∂xα (L1 + λ + α 2 )−1 ,

(1.148)

(M2α+1 + λ)−1 ∂xα = ∂xα [M1 + λ + α(α + 1)]−1 ,

(1.149)

(L2β+2 + λ)−1 ∂xβ = ∂xα [L2 + λ + β(β + 1)]−1 ,

(1.150)

(M2β+2 + λ)−1 ∂xβ = ∂xα [M2 + λ + β(β + 2)]−1 ,

(1.151)

exp(tL2α+1 )∂xα = e

tα 2

∂xα exp(tL1 ),

exp(tM2α+1 )∂xα = etα(α+1) ∂xα exp(tM1 ), exp(tL2β+2 )∂xβ

=

etβ(β+1) ∂xα

(1.152) (1.153)

exp(tL2 ),

(1.154)

exp(tM2β+2 )∂xβ = etβ(β+2) ∂xα exp(tM2 ).

(1.155)

These equations enable one to reduce the calculation of the resolvent and the heat kernel for the scalar and the spinor case to two basic cases: n = 1 and n = 2, that is, one and two dimensions.

2

Geometry of Simple Groups

2.1

Right-invariant and Left-invariant Basis

Let G be a simple Lie group of dimension m with the Lie algebra .g. Let .h be the Cartan subalgebra (that is, the maximal Abelian subalgebra) of the Lie algebra .g. The dimension r of the Cartan subalgebra is the rank of the Lie algebra .g. The exponentiation of the Cartan subalgebra defines the maximal torus, that is, the maximal flat submanifold. Let .C j ik be the structure constants in some fixed basis and .Ci be the matrices of the adjoint representation of .g defined by (Ci )j k = C j ik ;

(2.1)

[Ci , Cj ] = C k ij Ck .

(2.2)

.

they satisfy the algebra .

The Cartan-Killing metric .γij is defined by tr Ci Cj = C k il C l j k = −c2 γij ,

.

(2.3)

where c is a normalization constant. We use this metric to raise indices. This metric is positive definite for compact groups such as .SO(n) and indefinite for non-compact groups such as .SO(1, n). The Casimir operator of the adjoint representation is C 2 = γ ij Ci Cj = −c2 I ,

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_2

(2.4)

25

26

2 Geometry of Simple Groups

where .γ ij is the inverse matrix of .γij and I is the unit matrix. Then one can show that the matrices .γ Cj are anti-symmetric, that is, .(γ Cj )T = −γ Cj , and CjT = −γ Cj γ −1 ,

(2.5)

.

and, therefore, the matrices .Ci are traceless, .tr Ci = C j ij = 0. This means that the structure constants .Cabc = γad C d bc are completely anti-symmetric. Let .x i be the canonical (normal geodesic) coordinates on the group G. Let .F (x, x  ) be the group multiplication map in canonical coordinates. Below we will abuse notation by denoting the group element by the same symbol as the canonical coordinates, that is, we will denote the group element .z = F (x, x  ) simply by .xx  ; this should not cause any confusion. This function is an analytic function of the canonical coordinates with a Taylor series 1 F i (x, x  ) = x i + x i + C i j k x j x k + . . . . 2

(2.6)

.

Let .C = C(x) be the matrix defined by C(x) = Ci x i .

(2.7)

.

The matrix .C(x) is real anti-symmetric; so, it must have pairs of purely imaginary eigenvalues. The eigenvalues of the matrix .C(x) are determined by the positive roots .αj , .j = 1, . . . , p, of the algebra .g; the number p of positive roots is such that .m = r + 2p. Then the eigenvalues of the matrix .C(x) are Spec(C(x)) =

.

⎧ ⎨

⎫ ⎬

0, . . . , 0, iα1 (x), −iα1 (x), . . . , iαp (x), −iαp (x) . ⎩   ⎭

(2.8)

r

This allows one to compute the traces and the determinants of functions of the matrix .C(x) tr f (C(x)) = rf (0) +

.

p

 f (iαj (x)) + f (−iαj (x)) , .

(2.9)

j =1

det f (C(x)) = [f (0)]r

p 

f (iαj (x))f (−iαj (x)).

(2.10)

j =1

Let .Ta be the generators of some representation of the group G satisfying the algebra .g [Ta , Tb ] = C c ab Tc ,

.

(2.11)

2.1 Right-invariant and Left-invariant Basis

27

and T 2 = γ ab Ta Tb

.

(2.12)

be the corresponding Casimir operator. A group element in this representation is represented by the matrix D(x) = expx, T  ,

.

(2.13)

where .x, T  = x a Ta . For the adjoint representation we use the notation Ad(x) = exp C(x).

(2.14)

D(x)D(y) = D (xy) .

(2.15)

.

Then .

Let .adA : g → g be the operator on the Lie algebra defined by adA B = [A, B].

(2.16)

D(x)Tb = Ta Ada b (x)D(x) , .

(2.17)

Tb D(x) = D(x)Ta Ad b (−x).

(2.18)

.

Proposition 2.1 There holds .

a

Proof This is proved by using   D(x)Tb [D(x)]−1 = exp adT (x) Tb

(2.19)

adT (x) Tb = Ta [C(x)]a b .

(2.20)

.

and noticing that .

The second equation is obtained by replacing .x → −x. Proposition 2.2 There holds a a dD(x) = D(x)ω+ (x)Ta = ω− (x)Ta D(x),

.

a are the one-forms where .ω±

(2.21)

28

2 Geometry of Simple Groups a ω+ (x) = Y a i (x)dx i , .

(2.22)

a ω− (x) = Yi a (x)dx i ,

(2.23)

.

and .Y = (Y a i ) is the matrix (in canonical coordinates) Y (x) =

.

I − exp(−C(x)) . C(x)

(2.24)

Proof Let f (t, x) = [D(tx)]−1 dD(tx).

.

(2.25)

Then we show that   ∂t f (t, x) = [D(tx)]−1 T (dx)D(tx) = exp −t adT (x) T (dx).

.

(2.26)

Integrating this equation with the initial condition .f (0, x) = 0 we get f (1, x) =

.

  1 − exp −adT (x) T (dx). adT (x)

(2.27)

It is easy to see that the operator .adT (x) acts by multiplication on the matrix .T (dx), adT (x) T (dx) = Ta [C(x)]a i dx i ,

(2.28)

f (1, x) = Ta Y a i (x)dx i ;

(2.29)

.

and, therefore, .

a (x). The one-forms .ωa (x) can be obtained by noticing that this gives the one-forms .ω+ − −1 .D(−x) = [D(x)] and .Y (−x) = Y T (x) and replacing .x → −x. a (x) and .ωb (x) are right-invariant and left-invariant Proposition 2.3 The one-forms .ω+ −  one-forms, that is, for any .x a a ω+ (x  x) = ω+ (x), .

(2.30)

a ω− (xx  )

(2.31)

.

=

a ω− (x).

a (xx  ) = ωa (x). Similarly, for Proof By differentiating (2.15) with respect to x we get .ω− − a .ω+ .

2.1 Right-invariant and Left-invariant Basis

29

a are related by Proposition 2.4 The one-forms .ω± a b ω− = Ada b ω+

.

(2.32)

and satisfy the equations 1 a b c dω+ = − C a bc ω+ ∧ ω+ ,. 2 1 a b c dω− = C a bc ω− ∧ ω− . 2

.

(2.33) (2.34)

Proof This is proved by computing   a dω+ Ta = d D −1 dD = −D −1 dD ∧ D −1 dD

.

1 b c b c Tb ∧ ω+ Tc = − C a bc Ta ω+ ∧ ω+ . = −ω+ 2

(2.35)

Let .R(x) be the inverse of the matrix .Y (x), so that in canonical coordinates it has the form R(x) =

.

C(x) . I − exp(−C(x))

(2.36)

We define two collections of dual vector fields ∂ ,. ∂x i ∂ = R i a (x) i , ∂x

Ka− = Ra i (x)

(2.37)

Ka+

(2.38)

.

so that .

a ω± , Kb±  = δ a b ,

(2.39)

a ω+ , Kb−  = Adb a , .

(2.40)

a ω− , Kb+  = Ada b .

(2.41)

and .

Proposition 2.5 The vector fields .Ka+ and .Kb− are right-invariant and left-invariant vector fields, that is, for any .x 

30

2 Geometry of Simple Groups

Ka+ (x  x) = Ka+ (x), .

(2.42)

Ka− (xx  ) = Ka− (x).

(2.43)

.

Proof This follows from eqs. (2.31). Proposition 2.6 The vector fields .Ka± are related by Kb− = Ada b Ka+ ,

.

(2.44)

and form two mutually commuting representations of the algebra .g, [Ka+ , Kb+ ] = C c ab Kc+ , .

(2.45)

[Ka− , Kb− ]

(2.46)

.

= −C

c

− ab Kc , .

[Ka+ , Kb− ] = 0.

(2.47)

Proof This can be derived from Eqs. (2.34). It is easy to see that the Casimir operators of these representations are equal to each other, K 2 = γ ab Ka− Kb− = γ ab Ka+ Kb+ .

.

(2.48)

Proposition 2.7 The action of the left-invariant and the right-invariant vector fields on the group element has the form Ka+ D(x) = D(x)Ta , .

(2.49)

Ka− D(x)

(2.50)

.

= Ta D(x), .

Ka+ Ad(x) = Ad(x)Ca , .

(2.51)

Ka− Ad(x) = Ca Ad(x),

(2.52)

K 2 D(x) = D(x)T 2 = T 2 D(x), .

(2.53)

K 2 Ad(x) = D(x)T 2 = T 2 Ad(x).

(2.54)

and, therefore, .

Proof This follows from (2.21). Proposition 2.8 The left-invariant and the right-invariant vector fields act on the group elements by the left and right group multiplication. Let .p, K ±  = pa Ka± . Then for any .p, q,

2.2 Metric and Connection

31

.

expp, K + D(x) = D(xp), .

(2.55)

expq, K − D(x) = D(qx), .

(2.56)

exp[p, K +  + q, K − ]D(x) = D(qxp).

(2.57)

Proof This follows from Proposition 2.7 by using the Taylor series. Proposition 2.9 The action of the right-invariant and the left-invariant vector fields on a smooth scalar function f is given by the right and left group multiplication, that is, for any .p, q, .

expp, K + f (x) = f (xp), .

(2.58)

expq, K − f (x) = f (qx), .

(2.59)

exp[p, K +  + q, K − ]f (x) = f (qxp).

(2.60)

Proof This can be proved by using the flows of the vector fields .p, K ± .

2.2

Metric and Connection

The metric .Gij on the group is defined by a b ds 2 = γab ω± ⊗ ω± = Gij (x)dx i ⊗ dx j ,

(2.61)

Gij = γab Y a i Y b j , .

(2.62)

Gij = γ ab R i a R j b .

(2.63)

.

that is, .

The matrix .G = (Gij ) is given by  G(x) = γ

.

sinh(C(x)/2) C(x)/2

2 .

(2.64)

The Riemannian volume element is then 1 n dvol G (x) = |γ |1/2 ω+ ∧ · · · ∧ ω+ = |G|1/2 (x)dx,

.

(2.65)

32

2 Geometry of Simple Groups

where .|γ | = det γij , .|G| = det Gij = |γ |(det Y )2 and .dx = dx 1 ∧· · ·∧dx n is the standard Lebesgue measure. By using the property .det[exp(−C(x)/2)] = 1 it is easy to see that .

det Y = (det R)−1 = JG2 ,

(2.66)

where 

sinh[C(x)/2] .JG (x) = det C(x)/2

1/2

p  sin(αj (x)/2) = , αj (x)/2

(2.67)

j =1

and, therefore, |G|1/2 = |γ |1/2 JG2 .

(2.68)

.

Proposition 2.10 The volume element of the group is bi-invariant, that is, for any .x  , dvol (x) = dvol (x −1 ) = dvol (xx  ) = dvol (x  x).

.

(2.69)

Proof By using the right invariance of the vector fields .Ka+ , (2.43), we have R i a (x  x) =

.

∂(x  x)i j R a (x). ∂x j

(2.70)

By taking the determinant of this equation we get 

∂(x  x)i . det ∂x j

 =

JG2 (x)

.

(2.71)

= |G|1/2 (x)

(2.72)

JG2 (x  x)

Thus, |G|

.

1/2



∂(x  x)i (x x) det ∂x j 



which gives the result. The proof for the right shift is similar. The metric .Gij induces the Levi-Civita connection .∇ G on the group. Proposition 2.11 The right-invariant and the left-invariant vector fields satisfy 1 c + G + ∇K + Kb = − C ba Kc , . a 2

.

(2.73)

2.2 Metric and Connection

33

− G ∇K − Kb = a

1 c C ba Kc− . 2

(2.74)

Proof This is proved by showing that this connection is symmetric and compatible with the metric. This also means that the coefficients of the Levi-Civita connection in the right-invariant and the left-invariant bases are 1 Γ± a bc = ∓ C a bc . 2

(2.75)

.

Proposition 2.12 The metric .Gij is bi-invariant and the vector fields .Ka± are the Killing vectors of this metric, that is, LKa± G = 0.

(2.76)

.

Proof This is proved by showing that + G + G + (Ka+ , ∇K + Kc ) + (Kb , ∇ + Kc ) = 0, K

.

a

b

(2.77)

which, in turn, follows from (2.73) and (2.5). Proposition 2.13 The Riemann curvature tensor of the Levi-Civita connection is RG a bcd =

.

1 a C bf C f cd . 4

(2.78)

The Ricci curvature tensor and the scalar curvature are RGab =

.

RG =

1 2 c γab , 4 m 2 c . 4

.

(2.79) (2.80)

Proof This is proved by using Eqs. (2.73) and (2.74). The curves x i (t) = tξ i ,

.

(2.81)

where .ξ i is a unit vector, that is, .γij ξ i ξ j = 1, define the one-parameter subgroups. These are circles wrapped around the group infinitely many times. The geodesics of the metric starting at the origin are precisely the one-parameter subgroups defined above. Therefore,

34

2 Geometry of Simple Groups

i i the  geodesic distance of the point .x (t) = tξ from the origin is precisely .d(x, 0) = |x| = γij x i x j = |t|.

Proposition 2.14 The geodesic distance between two points x and .x  in canonical coordinates is d(x, x  ) = d(0, x −1 x  ) = |F (−x, x  )|.

.

(2.82)

The group G is covered by the canonical coordinates if they range over the domain    O = x ∈ Rm  for each root |αj (x)| < 2π .

.

(2.83)

If we let the canonical coordinates .x i range over the whole .Rm , then the group will be covered infinitely many times. It is convenient to complexify the group by extending the coordinates .x j to the complex space .Cm . Then all the functions defined above are analytic functions of .x j with some singularities. That is, if we move away from the real section .Rm in the complex space .Cm , then the volume element is not vanishing. This can be used to regularize some integrals below. Note that the metric and, therefore, the square of the geodesic distance are even analytic functions of the canonical coordinates Gij (x) = γij −

.

1 γim C m kn C n lj x k x l + . . . , . 12

d 2 (x, x  ) = γij (x i − x i )(x j − x j ) + . . . .

(2.84) (2.85)

Therefore, the metric is well defined for the imaginary coordinates, which enables one to define  sin[C(x)/2] 2 ,. C(x)/2   sin[C(x)/2] 1/2 ˜ JG (x) = JG (ix) = det . C(x)/2 ˜ G(x) = G(ix) = γ



.

(2.86) (2.87)

˜ = (G ˜ ij ) is real and defines locally a metric of a non-compact Riemannian The matrix .G manifold dual to the group G with negative curvature; this dual manifold is not a group but rather an m-dimensional submanifold of the 2m-dimensional complexification of the group G. The square of the geodesic distance in this manifold is d˜ 2 (x, x  ) = −d 2 (ix, ix  ) = −|F (ix, −ix  )|2 .

.

(2.88)

2.3 Heat Kernel on Simple Groups

2.3

35

Heat Kernel on Simple Groups

The scalar Laplacian on a Riemannian manifold is determined by its metric −1/2 ΔG 0 = |G|

.

∂ ∂ |G|1/2 Gij j , i ∂x ∂x

(2.89)

which is equal to ab G G ab G G ΔG 0 = γ ∇K + ∇K + = γ ∇K − ∇K − .

.

a

a

b

(2.90)

b

˜ ij defines the scalar Laplacian .Δ˜ G on the dual manifold. The dual metric .G 0 Proposition 2.15 The scalar Laplacian is determined by the Casimir operator (2.48) of the left-invariant and the right-invariant vector fields, 2 ΔG 0 =K .

(2.91)

± ab G G ab G ab ± ± ΔG 0 f = γ ∇K ± ∇K ± f = γ ∇K ± Kb f = γ Ka Kb f.

(2.92)

.

Proof This follows from .

a

a

b

We will need the following fundamental fact about compact simple groups. Notice that the function .JG (x) defined by (2.67) is an analytic function of the canonical coordinates on the whole .Rm . Proposition 2.16 The functions .JG−1 and .J˜G−1 are the eigenfunctions of the scalar Laplacian −1 ΔG 0 JG =

.

m 2 −1 c JG , . 24

˜−1 Δ˜ G 0 JG = −

m 2 ˜−1 c JG . 24

(2.93) (2.94)

−1 Proof This can be proved by showing that the function .JG ΔG 0 JG is a group invariant 2 equal to .c m/24. For a detailed proof for semisimple groups see [14, 21, 33]. Similar equation also holds for the dual Laplacian .Δ˜ G 0. Now, let .ΦG (t; x) be a function on .R+ × Rm defined by

ΦG (t; x) = |γ |1/2 (4π t)−m/2 exp

.

   x, γ x c2 t JG (x) exp − , 24 4t

m

(2.95)

36

2 Geometry of Simple Groups

where .|γ | = det γij and .x, γ x = γij x i x j . For the future use we also define the dual function Φ˜ G (t; x) = i m ΦG (−t + iε, ix)

.

= |γ |

1/2

(4π t)

−m/2

   m  x, γ x 2 ˜ , exp − c t JG (x) exp − 24 4t

(2.96)

where .J˜G (x) is defined by (2.87). Proposition 2.17 The functions .ΦG and .Φ˜ G satisfy the heat equations .

  −2 ∂t − J 2 Δ G ΦG (t; x) = 0 , . 0J

(2.97)

  ˜−2 Φ˜ G (t; x) = 0 , ∂t − J˜2 Δ˜ G J 0

(2.98)

ΦG (0; x) = Φ˜ G (0, x) = δ(x) .

(2.99)

with the initial conditions .

Proof This is proved by using Eq. (2.94); see [6]. Proposition 2.18 The integral of the functions .ΦG and .Φ˜ G over .Rm is equal to 1, 

 .

dx ΦG (t; x) =

Rm

dx Φ˜ G (t; x) = 1 .

(2.100)

Rm

Proof This is proved by noting that this is certainly true at .t = 0 and further showing that the time derivative is equal to zero. Indeed, we have 

 dx ΦG (t, x) =

∂t

.

Rm

−2 dx JG2 ΔG ΦG (t, x). 0J

(2.101)

−1/2 dx |G|1/2 ΔG ΦG (t, x) = 0. 0 |G|

(2.102)

Rm



= Rm

The function .ΦG (t, x) is an analytic function of canonical coordinates on the whole Rm . It is also analytic in t with a cut along the negative real axis. It decreases at infinity exponentially provided .Re x, γ x > 0. This is not exactly the heat kernel on the group G since it is not periodic. Rather, it is a pseudo- heat kernel defined on the whole tangent space .Rm . Notice also that we absorbed the volume form .G1/2 in it to make it a density. In this form the volume form, dx, which is just the Lebesgue measure, is not vanishing.

.

2.3 Heat Kernel on Simple Groups

37

To make it the proper heat kernel we would need to make it periodic by summing over the closed geodesics. Our integral over the whole .Rm does this automatically, there is no need to introduce any extra summation. Proposition 2.19 Let T be a representation of the group G with generators .Ta satisfying the commutation relations (2.11) and .T 2 be the corresponding Casimir operator (2.12). Then the heat semigroup has the following integral representation  .

exp(tT ) = 2

dx ΦG (t, x)D(x), .

(2.103)

dx Φ˜ G (t, x)D(ix),

(2.104)

Rm



exp(−tT 2 ) = Rm

where .D(x) = exp T (x) is defined by (2.13). Proof This is obviously true for .t = 0. We compute the time derivative 

 dx ΦG (t, x)D(x) =

∂t

.

Rm

−2 dx D(x)JG2 ΔG 0 JG ΦG (t, x)

Rm



=

−1/2 dx D(x)|G|1/2 ΔG ΦG (t, x). 0 |G|

(2.105)

Rm

Now, by integrating by parts twice and using (2.91) and (2.53) we obtain 

 dx ΦG (t, x)D(x) =

∂t

.

Rm

dx ΦG (t, x)ΔG 0 D(x) Rm



=

dx ΦG (t, x)K 2 D(x) Rm



= T2

dx ΦG (t, x)D(x).

Rm

By integrating this equation we get the desired formula.

(2.106)

38

2.4

2 Geometry of Simple Groups

Spin-tensor Bundles over Simple Groups

Let .V be a spin-tensor vector bundle realizing the representation G of the spin group with generators .Gab satisfying the algebra (1.35) and 1 T c = − C c ab Gab . 2

.

(2.107)

Recall that the indices are raised with the Cartan metric .γab defined by (2.3). Proposition 2.20 The matrices .Tc form a representation of the group G, [Ta , Tb ] = C c ab Tc .

.

(2.108)

Proof This is proved by using the algebra .so(n), (1.35), and the Jacobi identity C d f [a C f bc] = 0.

.

(2.109)

We choose the basis of the right-invariant vector fields .Ka+ so that the Levi-Civita connection is (2.75) 1 Γ+ a bc = − C a bc , 2

.

(2.110)

and, therefore, the spin connection one form on the bundle .V is A+ =

.

1 c Tc ω+ . 2

(2.111)

The covariant derivative of a section .ϕ of the bundle .V is given by (1.40) so that   1 G + T ∇K ϕ = K + + a ϕ, . a a 2   1 G Ka− + Ada c Tc ϕ. ∇K −ϕ = a 2

.

(2.112) (2.113)

The Laplacian (1.57) takes the form 1 G G G ab ΔG = γ ab ∇K Ta Kb+ + T 2 , + ∇ + = Δ0 + γ K a 4 b

.

where .T 2 = γ ab Ta Tb .

(2.114)

2.5 Geometry of SO(n + 1) and SO(1, n)

39

We want to rewrite the Laplacian in terms of Casimir operators of some representations G do not form a representation of the algebra of the group G. The covariant derivatives .∇K + a .g. The operators that do are the covariant Lie derivatives along the right-invariant and the left-invariant vector fields; we denote them by Ka± = LKa± .

(2.115)

.

Proposition 2.21 The Lie derivatives along the vector fields .Ka± have the form Ka+ = Ka+ + Ta , .

(2.116)

Ka− = Ka−

(2.117)

.

and satisfy the algebra [Ka+ , Kb+ ] = C c ab Kc+ , .

(2.118)

[Ka− , Kb− ] = −C c ab Kc− , .

(2.119)

[Ka+ , Kb− ] = 0.

(2.120)

.

Proof This is proved by using the definition (1.54) and the algebra (2.45). Proposition 2.22 The Laplacian is given by the sum of the Casimir operators ΔG =

.

1 2 1 2 1 K+ + K− − T 2, 2 2 4

(2.121)

2 = γ ab K± K± . where .K± a b

Proof This is proved by direct calculation.

2.5

Geometry of SO(n + 1) and SO(1, n)

We will use heavily the properties of the special orthogonal group .SO(n). The Lie group SO(n) is a simple compact doubly connected group (with the universal cover .Spin(n)) of dimension

.

m = dim SO(n) =

.

n(n − 1) 2

(2.122)

and the rank r = rank SO(n) =

.

n 2

.

(2.123)

40

2 Geometry of Simple Groups

Therefore, the number of positive roots of the Lie algebra .so(n) is .p = (m − r)/2, which is equal to ⎧  ⎪ n−1 2 ⎪ ⎪ , if n is odd, ⎨ 2 .p = ⎪ ⎪ n(n − 2) ⎪ ⎩ , if n is even. 4

(2.124)

The basis of the Lie algebra .so(n) is labeled by two indices .Tab , where .1 ≤ a < b ≤ n. To describe the fundamental representation of the group .SO(n) we introduce the .n × n matrices .Mab by (Mab )c d = δ c a δbd .

.

(2.125)

The matrices .Mab satisfy the equations M a a = I, .

.

M ab Mcd = M a d δ b c ,

(2.126) (2.127)

where, as usual, .I = (δ a b ) is the unit matrix. The generators of the fundamental (vector) representation of the group .SO(n) are given by the anti-symmetric matrices Tab = Mab − Mba .

.

(2.128)

These matrices satisfy the equations Tab Tcd = δbc Mad − δac Mbd − δbd Mac + δad Mbc ,

.

(2.129)

and, therefore, the commutation relations (1.35) [T ab , Tcd ] = −δ a c T b d + δ b c T a d + δ a d T b c − δ b d T a c .

(2.130)

T a b Tad = −(n − 2)Mbd − δbd I,

(2.131)

.

Furthermore, .

which gives the Casimir operator T2 =

.

1 Tab T ab = −(n − 1)I. 2

These matrices define the tensor .Tab cd = (Tab )cd by

(2.132)

2.5 Geometry of SO(n + 1) and SO(1, n)

41

.

T cd ab = δ c a δ d b − δ c b δ d a ;

(2.133)

T cd ab T ab pq = T cd pq , .

(2.134)

it satisfies the equations .

T ac ab = (n − 1)δ c b .

(2.135)

The Weyl-Cartan basis in the complexification of the algebra .so(n) is constructed as follows. Let the Hermitian matrices .H1 , . . . , Hr be defined as follows (recall that .r = [n/2] is the rank of the algebra). For an even n, H1 = iT12 ,

H2 = iT34 ,

.

···

Hr = iTn−1,n ,

(2.136)

Hr = iTn−2,n−1 .

(2.137)

and for odd n, H1 = iT12 ,

H2 = iT34 ,

.

···

These matrices mutually commute [Hi , Hj ] = 0,

.

1 ≤ i, j ≤ r

(2.138)

and form the basis in the Cartan subalgebra. The remaining Weyl generators .Eα are labeled by the roots of the Lie algebra .α = (αi ), which are covectors in .Rr , and satisfy the commutation relations [Hi , Eα ] = αi Eα , .

(2.139)

[Eα , E−α ] = α(H ), .

(2.140)

.

[Eα , Eβ ] = Nα,β Eα+β ,

(2.141)

where .α(H ) = α i Hi and .Nα,β are some constants (for details, see [12, 29]). An N -dimensional representation of the algebra .so(n) is formed by .N × N matrices A .Gab = (G Bab ), .A, B = 1, . . . , N , satisfying the commutation relations (1.35). Here and below we denote the representation by the same symbol as the generators. Notice that this implies a useful relation Gc a Gcb − Gc b Gca = −(n − 2)Gab .

.

(2.142)

The Casimir operator of this representation is G2 =

.

1 Gab Gab . 2

(2.143)

42

2 Geometry of Simple Groups

For an irreducible representation of dimension N there holds tr Gab Gcd = 4

.

N G2 δ c [a δ d b] , n(n − 1)

(2.144)

N 2 G δab . n

(2.145)

in particular, tr Gc a Gcb = 2

.

The Cartan metric .β of the .so(n) is defined as follows: if .X = then we define

1 ab 2 Y Gab

β(X, Y ) =

.

1 ab 2 X Gab

1 1 Xab Y ab = βabcd Xab Y cd , 2 4

and .Y =

(2.146)

where .

βabcd = δac δbd − δad δbc ;

(2.147)

1 n(n − 1) βabcd β abcd = . 4 2

(2.148)

notice that .

We use the structure constants of the group .SO(n), F ef abcd = 8δ [e [a δb][c δ f ] d] ,

.

(2.149)

to define .m × m matrices .Fab by (Fab )ef cd = F ef abcd .

.

(2.150)

Note that the product of anti-symmetric matrices involves a factor . 12 if we sum over all indices, not restricted by .a < b. The matrices .Fab satisfy the commutation relations (1.35) and form the adjoint representation of the algebra .so(n). They satisfy the equation .

1 ef F abpq F pq cdef = −2(n − 2)βabcd , 4

(2.151)

which fixes the normalization constant .c2 = 2(n − 2) in (2.3). We also have .

1 ef F abpq F pq abcd = −4(n − 2)δ e [c δ f d] , 4

(2.152)

2.5 Geometry of SO(n + 1) and SO(1, n)

43

which gives the Casimir operator of the adjoint representation 1 Fab F ab = −2(n − 2)I, 2

F2 =

.

(2.153)

where I is the unit matrix in the space of anti-symmetric matrices, that is, I ab cd = 2δ a [c δ b d] .

.

(2.154)

The scalar curvature of the group .SO(n) is RSO(n) =

.

1 n(n − 1)(n − 2) . 4

(2.155)

We denote the canonical coordinates on the group .SO(n) by .ωab with .1 ≤ a < b ≤ n. We define the norm |ω|2 =

1 ωab ωab 2

(2.156)

F (ω) =

1 Fab ωab , 2

(2.157)

.

and the matrix .

with the elements [F (ω)]ab cd = 4δ [a [c ωb] d] .

.

(2.158)

The pseudo-heat kernel .ΦSO(n) (2.95) has the form ΦSO(n) (t; ω) = (4π t)−n(n−1)/4 exp

.



   n(n − 1)(n − 2) |ω|2 t JSO(n) (ω) exp − , 24 4t (2.159)

where 

sinh[F (ω)/2] .JSO(n) (ω) = det F (ω)/2

1/2

p  sin[αj (ω)/2] , = αj (ω)/2

(2.160)

j =1

with .αj (ω) being the positive roots of .SO(n). The dual representation of this function is     n(n − 1)(n − 2) |ω|2 −n(n−1)/4 ˜ ˜ t JSO(n) (x) exp − , .ΦSO(n) (t; ω) = (4π t) exp − 24 4t (2.161)

44

2 Geometry of Simple Groups

where .J˜SO(n) (x) is defined by (2.87),   p sin[F (ω)/2] 1/2  sinh[αj (ω)/2] J˜SO(n) (ω) = det = . F (ω)/2 αj (ω)/2

.

(2.162)

j =1

Next, we consider the .n(n + 1)/2-dimensional Lie group G with the Lie algebra .g generated by the operators .Pa and .Lab with .1 ≤ a < b ≤ n. [Pa , Pb ] = −

.

1 Lab , . a2

(2.163)

[Lab , P c ] = −2δ c [a Pb] , .

(2.164)

[Lab , Lcd ] = −4δ [a [c Lb] d] ,

(2.165)

where .a 2 is a real parameter. Obviously, the group G has a subgroup .SO(n) formed by the generators .Lab . We will allow the parameter .a 2 to be positive or negative. One can show that for .a 2 > 0 the group G is nothing but the group .SO(n + 1), which is compact, and for .a 2 = −b2 < 0 the group G is the group .SO(1, n), which is non-compact. We introduce capital Latin indices, .A, B, C, . . . , that run from 0 to n and let .X0b = Pb and .Xab = Lab . Then the algebra (2.163) takes the form [XAB , XCD ] =

.

1 EF C ABCD XEF , 2

(2.166)

with the structure constants .C EF ABCD of the form C cd 0a0b = −

.

2 c d δ [a δ b] , . a2

(2.167)

C 0d ab0c = 2δ d [a δb]c , .

(2.168)

C ef abcd = 8δ [e [a δb][c δ f ] d] .

(2.169)

The metric .γABCD of the algebra .g has the form γ0b0d = δbd ,

.

.

γabcd = a 2 (δac δbd − δad δbc ) ,

(2.170) (2.171)

with the inverse metric .γ ABCD given by γ 0b0d = δ bd ,

.

.

(2.172)

2.5 Geometry of SO(n + 1) and SO(1, n)

1 (δac δbd − δad δbc ) , a2

γ abcd =

45

(2.173)

so that, in particular,  .

 1 X, γ −1 Y = γ ABCD XAB YCD 2 1 = X0b Y0b + 2 Xab Yab . 2a

(2.174)

The Casimir operator of this representation is 1 ABCD γ XAB XCD 2 1 = P 2 + 2 L2 , a

X2 =

.

(2.175)

where .P 2 = Pb P b and .L2 = 12 Lab Lab . The determinant of the metric is |γ | = a n(n−1) ,

.

(2.176)

and the scalar curvature of the group G is RG = −

.

1 ABCD EF n(n2 − 1) γ C ABP Q C P Q CDEF = . 64 4a 2

(2.177)

We denote the canonical coordinates on the group G by .(k AB ), with .0 ≤ A < B ≤ n and split them as follows .(k AB ) = (q a , ωbc ), where .q a = k 0a , .ωbc = k bc , and .b < c. Then .

k, γ k =

1 γABCD k AB k CD = |q|2 + a 2 |ω|2 , 4

(2.178)

where .|q|2 = q a qa and .|ω|2 = 12 ωab ωab . Let .C = C(q, ω) = (C AB CD ) be the matrices defined by [C(q, ω)]cd 0b = 2δ [c b q d] , .

(2.179)

[C(q, ω)]0d 0c = ωd c , .

(2.180)

[C(q, ω)]ef cd = 4δ [e [c ωf ] d] ,

(2.181)

.

and .JG (q, ω), J˜G (q, ω) be the functions defined by

46

2 Geometry of Simple Groups

 JG (q, ω) = det

.

sinh[C(q, ω)/2] C(q, ω)/2

1/2 (2.182)

,.

  sinh[C(q, −iω)/2] 1/2 ˜ JG (q, ω) = det . C(q, −iω)/2

(2.183)

The pseudo-heat kernel (2.95) on the group G has the form n(n2 − 1) t 24 a2   q  |q|2 + a 2 |ω|2 , ×JG 2 , ω exp − 4t a

ΦG (t; q, ω) = a n(n−1)/2 (4π t)−n(n+1)/4 exp

.

!

(2.184)

and the dual function .Φ˜ G is defined by n(n2 − 1) t 24 b2   q  |q|2 + b2 |ω2 | . ×J˜G 2 , ω exp − 4t b

Φ˜ G (t; q, ω) = bn(n−1)/2 (4π t)−n(n+1)/4 exp −

.

!

(2.185)

Proposition 2.23 There hold .

!  1 = exp t P 2 + 2 L2 a



 dω

dq ΦG (t; q, ω) exp[q, P + ω, L],

Rn

Rn(n−1)/2

(2.186)

.

!  1 = exp t P 2 − 2 L2 b



 dω

Rn(n−1)/2

dq Φ˜ G (t; q, ω) exp[q, P + i ω, L],

Rn

(2.187) where .dω =

" a 0 for S n and Λ < 0 for H n . Changing the sign of the curvature Λ → −Λ is the duality transformation that allows one to relate formally many results for these manifolds. In the limit Λ → 0 we recover the Euclidean space Rn . For the reference, the Ricci tensor and the scalar curvature are R a b = (n − 1)Λδ a b ,

R = n(n − 1)Λ .

(4.13)

We set Λ=

1 , a2

(4.14)

so that a is real for S n and purely imaginary for H n . That is why we denote |a| = b for H n . Then the duality transformation between S n and H n is a → ib. In symmetric spaces, when the curvature tensor is parallel, one can compute the twopoint functions introduced above explicitly. Let K be a matrix defined by K a b = R a cbd y c y d .

(4.15)

60

4 Maximally Symmetric Spaces

Proposition 4.1 The one-forms 

√  sin K dy b √ K ab

ω = a

(4.16)

provide an orthonormal frame with the corresponding Levi-Civita connection  Γ

a

b

= −R

a

bcd y

d

I − cos K

√ c K

e dy

e

.

(4.17)

The Van Vleck–Morette determinant has the form  √  K Δ(x, x ) = det . √ sin K 

(4.18)

Proof This is proved in our papers [2, 5]. In maximally symmetric spaces these expressions can be computed explicitly. We have K=

r2 Π, a2

(4.19)

where r = |y| and Π a b = δa b −

y a yb r2

(4.20)

is a projection onto a hyperplane orthogonal to y a . Therefore, the matrix K has the eigenvalue r 2 /a 2 with multiplicity (n − 1) and the eigenvalue 0 with multiplicity 1. Proposition 4.2 Let f be a function defined by f (r) = a sin

r  a

(4.21)

,

that is, f (r) = a sin

r 

f (r) = b sinh The vector fields

a r  b

for for

Sn, H n.

(4.22) (4.23)

4.2 Maximally Symmetric Spaces

 ea =

61

  y b ya ∂ r y b ya b + − , δ a 2 2 f (r) ∂y b r r

(4.24)

and the one-forms   ya yb f (r) a y a yb dy b δ b− 2 ω = + r r2 r

a

(4.25)

provide orthonormal frames for the tangent space and the cotangent space. The metric and the volume form are ya yb f 2 (r)  ya yb  δ dy a dy b , + − ab r2 r2 r2   f (r) n−1 = dy. r 

ds 2 = dvolM

(4.26) (4.27)

The commutation coefficients (1.32) are given by γ c ba =

1 − f  (r) c (δ b ya − δ c a yb ), rf (r)

(4.28)

and the Levi-Civita connection has the form 1 − f  (r) c (δ b ya − δ c a yb ), rf (r)

(4.29)

 1 − f  (r) (y a dy b − y b dy a ) . r2

(4.30)

Γabc = so that the connection one-form is  Γab =

The Van Vleck–Morette determinant is Δ(x, x  ) =



r f (r)

n−1 .

(4.31)

Proof This is proved by using (4.19) (see [2, 5]). Note that γ b ba = (n − 1)

1 − f  (r) ya . rf (r)

(4.32)

62

4.3

4 Maximally Symmetric Spaces

Local Coordinates

The hyperbolic space H n is homeomorphic to the Euclidean space Rn and can be covered by a single coordinate chart. However, there is no global coordinate system that covers the whole sphere S n . For the sphere we should have at least two coordinate patches, one containing the South pole, and another, containing the North pole. Let y a be the normal geodesic coordinates with the origin at the North pole. The location of the indices (up or down) on the coordinates does not matter, it will be chosen for convenience. Recall that √ r = |y| = ya y a . We have 0 ≤ r < aπ for S n and 0 ≤ r < ∞ for H n . The South pole patch of S n is covered by the coordinates y b = εb

πa − r b y , r

(4.33)

where εb = −1 for b = 1 and εb = +1 for b = 2, . . . , n, so that r = πa − r .

(4.34)

The Jacobian matrix of this transformation is  ∂y b y b yc b πa − r b , δ c − πa 3 =ε ∂y c r r

(4.35)

so that the Jacobian is 

∂y b det ∂y c



 =

πa − r r

n−1 .

(4.36)

Notice that for 0 < r < π a the Jacobian is finite and not equal to zero, moreover, it is strictly positive. Therefore, this change of variables preserves the orientation. We will also view the manifolds S n and H n as hypersurfaces in Rn+1 . We will use capital Latin indices to run over 0, 1, . . . , n. Let (XA ) = (X0 , Xa ), a = 1, . . . , n, be the Cartesian coordinates in Rn+1 . Then S n and H n are defined by the equations S n : (X0 )2 + |X|2 = a 2 , H n : −(X0 )2 + |X|2 = −b2 ,

(4.37) X0 > 0,

(4.38)

where |X|2 = (X1 )2 + · · · + (Xn )2 .

(4.39)

4.3 Local Coordinates

63

So, the formal duality transformation between these manifolds is a → ib and X0 → iX0 (to change the signs of a 2 and (X0 )2 ). Obviously, these equations are preserved by the action of the orthogonal group SO(n + 1) for S n and the pseudo-orthogonal group SO(1, n) for H n , which are their isometry groups. The stereographic coordinates ua , a = 1, . . . , n, (based at the South pole) are defined by by ua =

1+



Xa 1 − |X|2 /a 2

(4.40)

.

As usual, we have a 2 > 0 for S n and a 2 < 0 for H n . The inverse transformation is Xa =

2 ua , 1 + u2 /a 2

(4.41)

X0 1 − u2 /a 2 , = a 1 + u2 /a 2

(4.42)

√ √ where u = |u| = ua ua . These coordinates are related to the geodesic coordinates introduced earlier by ua = where r =

u(r) a y , r

(4.43)

|y|2 is the geodesic distance and u(r) =

f (r) , f  (r)

(4.44)

where f (r) is defined by (4.22) and (4.23). That is,  r  , 2a r  , u(r) = b tanh 2b

u(r) = a tan

for for

Sn, H n.

(4.45) (4.46)

Therefore, the geodesic coordinates are related to the coordinates Xa by Xa =

f (r) a y , r

X0 = f  (r). a

(4.47) (4.48)

64

4 Maximally Symmetric Spaces

We will also find it useful to use the following coordinates x(r) a y , r

(4.49)

x(r) = f  (r),

(4.50)

xa = where

that is, x(r) = cos

r 

x(r) = cosh

, a r  b

for ,

for

Sn H n.

(4.51) (4.52)

For the lack of a better name we will call these coordinates the hypergeometric coordinates. The origin of this name will become clear later. As we already mentioned above the geodesic coordinates y a vary over the whole Rn for n H and over the ball of radius aπ for S n so that r = |y| < aπ . It should be clear that the stereographic coordinates ua vary over the whole Rn for S n and over the ball of radius a for H n so that |u| < a. The hypergeometric coordinate x(r) varies over the interval [−1, 1] for S n and over the interval [0, ∞) for H n . Also, note that while (Xa ) = (X1 , . . . Xn ) can be used as local coordinates on H n they only cover the Northern hemisphere of S n . Normal coordinates are distinguished by the fact that the geodesics connecting the South pole with the point y have very simple form y i (t) = tθ0i ,

(4.53)

where θ0i is a unit vector, θ0i θ0i = 1, so that y(0) = 0 and y i (r) = y i . That is, the geodesics emanating from the origin are r(t) = |t| , θ i (t) = sign(t)θ0i ,

(4.54) (4.55)

where t ∈ [−π, π ]. That is why, ar is exactly the geodesic distance from the South pole (with coordinates y = 0) to the point y. The geodesics on the hyperbolic space H n look exactly the same r(t) = |t| , θ i (t) = sign(t)θ0i ,

(4.56) (4.57)

4.4 Geodesic Spherical Coordinates

65

where now t ∈ R. The fundamental difference between the sphere S n and the hyperbolic space H n is that the geodesics on S n will wind around the sphere infinitely many times forming closed geodesics, whereas in H n the geodesics do not self-intersect; they go to infinity in both directions without self-intersections; in other words, there are no closed geodesics.

4.4

Geodesic Spherical Coordinates

We consider an embedded (n − 1)-sphere S n−1 of radius r centered at the North pole. A family of such spheres provides a foliation of our maximally symmetric manifold, both of S n and H n , and defines the geodesic spherical coordinates θ a = y a /r, with r = |y| = √ ya y a . The metric (4.26) in spherical coordinates takes the form ds 2 = dr 2 + f 2 (r)dθ 2 ,

(4.58)

where dθ 2 = dθ a dθ a . That is why the sphere S n and the hyperbolic space H n can be described as warped products S n = S 1 ×f S n−1 , H = R ×f S n

n−1

.

(4.59) (4.60)

It is not difficult to obtain the metric in other coordinate systems. In particular, the metric in the hypergeometric coordinates (with x = cos(r/a)) has the form

ds 2 = a 2

 dx 2 2 2 . + (1 − x )dθ 1 − x2

(4.61)

Next, for the geodesic coordinates we have dy = dy 1 ∧ · · · ∧ dy n = r n−1 dr dvol S n−1 (θ ),

(4.62)

where dvol S n−1 (θ ) =

1 εa ...a θ a1 dθ a2 ∧ · · · ∧ dθ an (n − 1)! 1 n

(4.63)

is the Riemannian volume element in S n−1 . Therefore, the Riemannian volume element becomes dvol M (y) = f n−1 (r)rdr dvol S n−1 (θ ).

(4.64)

66

4 Maximally Symmetric Spaces

The basis vector fields in these coordinates have the form ea = θa ∂r +

1 b θ Lba , f (r)

(4.65)

and the orthonormal frame (4.25) of one-forms reads ωa = θ a dr + f (r)dθ a .

(4.66)

We compute [ea , eb ] =

1 − f  (r) Lab , f 2 (r)

(4.67)

and, therefore, γ c ab =

1 − f  (r) c (δ a θb − δ c bθa ), f (r)

(4.68)

and Γabc = γcba =

1 − f  (r) c (δ b θa − δ c a θb ). f (r)

(4.69)

1 − f  (r) θa . f (r)

(4.70)

Notice also that γ b ba = (n − 1)

The corresponding Levi-Civita connection one-form is

 Γab = 1 − f  (r) (θ a dθ b − θ b dθ a ) .

(4.71)

Of course, the curvature of this connection is given by (2.13).

4.5

Isometries

Proposition 4.3 Let h be a function defined by h(r) = that is,

f  (r) f (r)

(4.72)

4.5 Isometries

67

r  1 cot a a r  1 h(r) = coth b b

h(r) =

for for

Sn, H n.

(4.73) (4.74)

The vector fields  Pa =

  ∂ ya yb ya yb ab + h(r)r δ − 2 ∂y b r2 r

= θa ∂r + h(r)θ b Lba , Lab = y a

∂ ∂ − yb a , b ∂y ∂y

(4.75) (4.76)

form a basis in the vector space of Killing vector fields. Proof This is proves by using the results of [2,5,6] where the Killing vectors on symmetric spaces were computed. Furthermore, we have the proposition. Proposition 4.4 The isometry algebra g of the maximally symmetric space has the form [Pa , Pb ] = −

1 Lab , a2

[Lab , P c ] = −2δ c [a Pb] , [Lab , Lcd ] = −4δ [a [c Lb] d] .

(4.77) (4.78) (4.79)

Proof See, for example, [6, 33]. One can actually find the function h(r) by requiring the operators Pa to satisfy this algebra. The isometry algebra g is so(n + 1) for S n and so(1, n) for H n . The vector fields Lab form the subalgebra h = so(n), which is the isotropy algebra of both S n and H n . The dimension of the isotropy algebra is dim h = n(n−1)/2 and the dimension of the isometry algebra is dim g = n(n + 1)/2. Of course, in the limit of infinite radius a → ∞ the Killing vector fields Pa become Pa = ∂a , and the isometry algebra becomes the isometry algebra E(n) = Rn  SO(n) of the Euclidean space Rn . The isometry algebra can be written in a more compact form as follows. To explain this we let capital Latin indices run over 0, 1, . . . , n. We extend the operators Lab , with a, b = 1, . . . , n, to the set of operators LAB , with A, B = 0, 1, . . . , n, by L0b = Pb .

(4.80)

68

4 Maximally Symmetric Spaces

Then the algebra (4.79) takes the form [LAB , LCD ] =

1 EF C ABCD LEF , 2

(4.81)

where the non-zero structure constants C EF ABCD are C cd 0a0b = −

2 c d δ [a δ b] , a2

(4.82)

C 0d ab0c = 2δ d [a δb]c ,

(4.83)

C ef abcd = 8δ [e [a δb][c δ f ] d] .

(4.84)

Note that since the generators are labeled with two indices the structure constants are labeled with six indices. The factor 1/2 appears because we formally sum over all indices, whereas we should only sum over pairs of indices ab with a < b. Such factors will appear elsewhere too. The invariant Cartan metric γABCD of the isometry algebra g is defined by 1 EF 1 C ABP Q C P Q CDEF = −2(n − 1) 2 γABCD , 4 a

(4.85)

where the normalization factor is chosen so that γ0b0d = δbd ,

(4.86)

γabcd = a 2 (δac δbd − δad δbc ) .

(4.87)

This defines the inner product (X, Y ) =

1 a2 γABCD XAB Y CD = X0a Y0a + Xab Yab . 4 2

(4.88)

Note that it is positive definite for S n and indefinite for H n . Also, the determinant of the metric is |γ | = det γ = a n(n−1) .

(4.89)

The dual metric is, of course, γ 0b0d = δ bd ,

(4.90)

4.6 Lie Derivatives

69

γ abcd =

 1  ac bd ad bc δ . δ − δ δ a2

(4.91)

Proposition 4.5 The Killing vectors satisfy the equations ωa , Pc ωb , Pc + ωa , Pc ωb , ∇ep Pc +

1 a ω , Lcd ωb , Lcd = δ ab , a2

1 a ω , Lcd ωb , ∇ep Lcd = 0. a2

(4.92) (4.93)

Proof For the proof see [6]. For the future reference we compute the derivatives of the Killing vectors. Proposition 4.6 There hold Γ ab , Pc = −2[1 − f  (r)]h(r)δ [a c θ b] , Γ ab , Lcd = −4[1 − f  (r)]θ [a δ b] [c θd] , ωa , ∇eb Pc = −

2 f (r)δ [a c θ b] . a2

ωa , ∇eb Lcd = −2δ a [c δ b d] − 4[1 − f  (r)]θ [a δ b] [c θd] ,

(4.94) (4.95) (4.96) (4.97)

where f (r) is given by (4.21) and h(r) is given by (4.73). Proof We use (4.71) and (4.75), (4.76), to compute (4.94), (4.95). Next, we compute the derivatives 

1 f (r) θb ec − δbc θ a ea , 2 a   = 2δ b [c ed] − 2[1 − f  (r)] θ b θ[c ed] − θ[c δ b d] θ a ea ,

∇eb Pc = − ∇eb Lcd

(4.98) (4.99)

to get (4.96),(4.97).

4.6

Lie Derivatives

Let V be a spin-tensor vector bundle realizing a representation G of the spin group Spin(n) with the generators Gab satisfying the algebra (2.130). The covariant derivative of a section of the vector bundle V along a Killing vector ξ is given by (1.40) and the Lie derivative by (1.52). To simplify notation we denote the Lie derivatives along the Killing vectors Pa and Lab by

70

4 Maximally Symmetric Spaces

P a = L Pa ,

(4.100)

Lab = LLab .

(4.101)

Proposition 4.7 The Lie derivatives along the Killing vectors Pa and Lab are Pa = Pa +

1 − f  (r) Wa (θ ), f (r)

Lab = Lab + Gcd ,

(4.102) (4.103)

where f (r) is given by (4.21) and Wa (θ ) = Gab θ b .

(4.104)

Proof We use Eqs. (4.94), (4.95), to get ∇Pa = Pa − [1 − f  (r)]h(r)Wa ,

(4.105)

∇Lcd = Lcd − 2[1 − f  (r)]θ[c Wd] .

(4.106)

Further, by using eqs. (4.96), (4.97), and (1.53) we compute SPa =

1 f (r)Wa , a2

SLcd = Gcd + 2[1 − f  (r)]θ[c Wd] .

(4.107) (4.108)

By using these equations we finally obtain the Lie derivatives (4.102), (4.103), from (1.52). Proposition 4.8 The Lie derivatives Pa and Lab form a representation of the isometry algebra g [Pa , Pb ] = −

1 Lab , a2

(4.109)

[Lab , P c ] = −2δ c [a Pb] ,

(4.110)

[Lab , Lcd ] = −4δ [a [c Lb] d] .

(4.111)

Proof This is proved by using the equations (for details, see [6]) SPa Pa +

1 SL Lab = 0 , 2a 2 ab

(4.112)

4.7 Laplacian

71

∇Pa SPa + SPa SPa +

1 ∇L SL = 0 , 2a 2 ab ab

(4.113)

1 1 SLab SLab = 2 G2 , 2 2a a

(4.114)

where G2 = 12 Gab Gab .

4.7

Laplacian

First, we express the scalar Laplacian in terms of the Killing vectors. Proposition 4.9 The scalar Laplacian is equal to the Casimir operator of the isometry algebra 1 ABCD LAB LCD γ 4 1 = P 2 + 2 L2 , a

Δ0 =

(4.115)

where P 2 = Pa P a and L2 = 12 Lab Lab . Proof This is proved by using the definition Δ0 = ∇ea ∇ea and Eqs. (4.92), (4.93). Proposition 4.10 The scalar Laplacian has the form Δ0 = ∂r2 + (n − 1)h(r)∂r +

1 f 2 (r)

L2 .

(4.116)

Proof By using the explicit form of the Killing vectors Pa we get P 2 = ∂r2 + (n − 1)h(r)∂r + h2 (r)L2 .

(4.117)

The result follows. The scalar Laplacian can also be written in the form  Δ0 = f −(n−1)/2

1 ∂r2 + 2 f



(n − 1)(n − 3) L2 + 4



(n − 1)2 + 4a 2

 f (n−1)/2 ,

(4.118)

Notice that for n = 1 and n = 3 the potential term vanishes and we have 1 1 ΔS0 = ∂r2 + 2 L2 , f

(4.119)

72

4 Maximally Symmetric Spaces  3 1 1 ΔS0 = f −1 ∂r2 + 2 L2 + 2 f . f a

(4.120)

There is a deep reason for that, namely it is because the sphere S 3 is the group manifold of the group SU (2). This leads to the fact that in n = 3 the function 1/f is the eigenfunction of the Laplacian, that is, ΔS0

3

1 1 1 = 2 . f a f

(4.121)

This will be discussed later in more detail. Next, we do the same for the Laplacian on the vector bundle. We extend the set of Lie derivatives Lab to LAB by adding L0a = Pa .

(4.122)

Proposition 4.11 The Laplacian on the vector bundle V is equal to the sum of the Casimir operators 1 ABCD 1 LAB LCD − 2 G2 γ 4 a   1 = P 2 + 2 L2 − G2 a

Δ=

(4.123)

where L2 = 12 Lab Lab and P 2 = Pa P a . Proof We use Eqs. (4.92), (4.93), to get Δ = ∇ea ∇ea = ∇Pa ∇Pa +

1 ∇L ∇L . 2a 2 ab ab

(4.124)

Further, we have P2 +

1 2 1 L = (∇Pa + SPa )(∇Pa + SPa ) + 2 (∇Lab + SLab )(∇Lab + SLab ). 2 a 2a

(4.125)

Now, by using the properties (4.112)–(4.114) of the quantities SPa , SLab , we get the result. By using the form of the Lie derivatives (4.102)–(4.103) we can compute the Laplacian explicitly. Proposition 4.12 The Laplacian on the vector bundle V has the form Δ = ∂r2 + (n − 1)h(r)∂r +

1 ˆ Δ, f 2 (r)

(4.126)

4.7 Laplacian

73

where Δˆ = L2 + (1 − f  )Gab Lab + (1 − f  )2 W 2

(4.127)

W 2 = W a Wa = Ga b Gac θ b θ c .

(4.128)

with

Proof The Laplacian is given by (4.123). First, we have L2 = L2 + Gab Lab + G2 .

(4.129)

To compute P 2 we first notice that θ a Wa = 0; also, by using (1.99) we show that Lba Wa = Wb .

(4.130)

Therefore, by using (4.75) we have Pa

1 − f Wa = 0. f

(4.131)

Next, by using (1.89) we get 1 f  ab G Lab . 2f

(4.132)

(1 − f  )f  (1 − f  )2 2 G L + W , ab ab f2 f2

(4.133)

Wa Pa = − Therefore, we obtain from (4.102) P2 = P 2 −

Thus, by using the scalar Laplacian (4.115) we obtain Δ = Δ0 +

 1    2 2 (1 − f . )G L + (1 − f ) W ab ab f2

(4.134)

Finally, by using the explicit form of the scalar Laplacian (4.116) we get the result. The Laplacian can also be written in the form 

 (n − 1)(n − 3) (n − 1)2 1 f (n−1)/2 , (4.135) + Δ = f −(n−1)/2 ∂r2 + 2 Δˆ + 4 f 4a 2

74

4 Maximally Symmetric Spaces

where Δˆ is given by (4.127). It is worth noting that the operator Δˆ can be written in the form Δˆ = ∇ˆ a ∇ˆ a ,

(4.136)

  ∇ˆ a = θ b Lba + (1 − f  )Gba .

(4.137)

where

Notice that the coefficients of the Laplacian (4.127) depend on the angular coordinates θ only through the quantity W 2 , (4.128). There are three important cases when this quantity does not depend on θ and is constant: 1. Scalar representation. Obviously, in this case W 2 = 0. 2. Spinor representation. In this case the quantity W 2 is given by (1.50), W2 = −

n−1 I. 4

(4.138)

3. Two dimensions, n = 2. In this case there is only one generator G12 = G of the group SO(2) and, therefore, W 2 = G2 .

(4.139)

In all these cases all coefficients of the Laplacian depend only on the radial coordinate r. This leads to the fact that the heat kernel U (t; x, x  ) depends only on the geodesic distance d(x, x  ) between the points x and x  .

5

Three-dimensional Maximally Symmetric Spaces

5.1

Geometry of S 3

A remarkable property of the 3-sphere .S 3 is that there exists an orthonormal basis consisting of the right-invariant and the left-invariant vector fields and 1-forms. This is not so on other maximally symmetric spaces of constant curvature, not even on .S 2 . The reason for that is that .S 3 is in fact a Lie group, namely .SU (2). The sphere .S 3 can be described as the homogeneous space .S 3 = SO(4)/SO(3) = [SU (2) × SU (2)] /SU (2). It is well known that .SO(4) = SO(3) × SO(3) . This means that the 3-sphere .S 3 is the double covering of the group .SO(3). On another hand, the universal covering of the group .SO(3) is the spin group .Spin(3) = SU (2), which is isomorphic to the group .SU (2). Therefore, .S 3 = SU (2) . We consider the sphere .S 3 of radius a. Let .y i be the normal geodesic coordinates  on 3 .S with the origin at the North pole. We introduce the radial coordinate .r = |y| = yi y i , i i 3 and the angular coordinates .θ = y /r . The geodesic coordinates on .S are related to the canonical coordinates .x i on the group .SU (2) by yi ,. a r ρ=2 , a

xi = 2

.

(5.1) (5.2)

 with .ρ = |x| = xi x i ; of course, the angular coordinates .θ i are the same. Therefore, the group .SU (2) in the standard normalization is the sphere .S 3 of radius .a = 2; in particular, it is easy to see that

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_5

75

76

5 Three-dimensional Maximally Symmetric Spaces

vol (S 3 (a)) =

.

 a 3 2

vol (SU (2)) = 2π 2 a 3 .

(5.3)

The metric on .S 3 has the form (4.58) ds 2 = dr 2 + a 2 sin2

.

r  a

dθ 2 .

(5.4)

The Killing vectors .Pa and .Lab are defined by (4.75) and (4.76). We relabel the Killing vectors .Lab by La =

.

∂ 1 εabc Lbc = εabc y b c . 2 ∂y

(5.5)

Notice that .θ a La = 0. Then the Killing vectors .Pa are Pa = θa ∂r +

.

r  1 Qac Lc , cot a a

(5.6)

where .Qac = εacb θ b is defined by (3.28). The isometry algebra of the sphere .S 3 is then given by [Pa , Pb ] = −

.

1 c ε ab Lc , . a2

(5.7)

[La , Pb ] = −εc ab Pc , .

(5.8)

[La , Lb ] = −εc ab Lc .

(5.9)

This is nothing but the Lie algebra .so(4), with the vector fields .La forming the isotropy subalgebra .so(3). This algebra can be factorized as follows. Let .Ka± be the vector fields defined by 1 (aPc ± Lc ) 2    r  1 aθc ∂r + cot Qcb ± δc b Lb . = 2 a

Kc± =

.

(5.10)

Proposition 5.1 The vector fields .Kb± are the right-invariant and the left-invariant vector fields (3.40) on .SU (2) forming two different mutually commuting representations of the algebra .so(3), [Ka+ , Kb+ ] = −εc ab Kc+ , .

(5.11)

[Ka− , Kb− ] = εc ab Kc− , .

(5.12)

.

5.1 Geometry of S 3

77

[Ka+ , Kb− ] = 0.

(5.13)

Proof This follows from the algebra (5.7)–(5.9) as well as from the explicit form of the vector fields in canonical coordinates.

Proposition 5.2 The scalar Laplacian on .S 3 is equal to the Casimir operator of .SO(4) 3

4 2 4 2 K = 2 K− a2 + a r  2 1 2 ∂r + = ∂r2 + cot ΔS0 , a a a 2 sin2 (r/a)

ΔS0 =

.

(5.14)

2 = K ± K ± and .ΔS is the scalar Laplacian on .S 2 . where .K± a a 0 2

Proof This follows from Eqs. (4.116) and (4.115). 2 2 is equal to the scalar Laplacian .ΔSU (2) on the Recall that .ΔS0 = L2 = Li Li and .K± 0 group .SU (2), given by (3.58), therefore, 3

ΔS0 =

.

4 SU (2) Δ . a2 0

(5.15)

The standard orthonormal bases (4.66) and (4.65) have the form ωa = θ a dr + a sin

.

ea = θa ∂r +

r  a

dθ a , .

1 Qac Lc . a sin(r/a)

(5.16) (5.17)

The Levi-Civita connection one-form (4.71) in this basis reads   r  (θ a dθ b − θ b dθ a ) , Γab = 1 − cos a

.

(5.18)

and the curvature two-form is equal to Rab =

.

1 a ω ∧ ωb , a2

(5.19)

so that R ab cd =

.

 1  a b a b δ . δ − δ δ c d d c a2

(5.20)

However, on the three-dimensional sphere .S 3 we can choose another orthonormal basis b with the right-invariant basis .ωa on .SU (2) defined by (3.39) (of course, .ω ˜ b = (a/2)ω+ +

78

5 Three-dimensional Maximally Symmetric Spaces

we could have chosen also the left-invariant basis) ω˜ a = θ a dr + a sin

.

r   r  r  cos δaj − sin Qaj dθ j , a a a

(5.21)

and the dual basis of the vector fields .e˜b = (2/a)Kb+ , with .Kb+ defined by (3.40),  1  r  cot Qac + δac Lc . a a

e˜a = θa ∂r +

.

(5.22)

Proposition 5.3 The orthonormal bases .ωa and .ω˜ a are related by an orthogonal transformation ωa = Λa b ω˜ b , .

(5.23)

ea = Λa b e˜b ,

(5.24)

.

where .Λ = (Λab ) is the orthogonal matrix, Λ = exp

.

r a

 Q ,

(5.25)

equal to r 

Λab = θa θb + cos

.

a

(δab − θa θb ) + sin

r  a

Qab .

(5.26)

Proof This is proved by using the algebra of the matrices Q, (3.31), and Eq. (3.32). The right-invariant basis .ω˜ a satisfies d ω˜ a =

.

1 a ε bc ω˜ b ∧ ω˜ c , . a

2 [e˜a , e˜b ] = − εc ab e˜c , . a ∇e˜a e˜b =

1 c ε ba e˜c . a

(5.27) (5.28) (5.29)

Therefore, in this basis the Levi-Civita connection is simply Γ˜ a bc =

.

1 a ε bc , a

(5.30)

5.1 Geometry of S 3

79

and, hence, the connection one-form is 1 εabc ω˜ c . a

Γ˜ab =

.

(5.31)

The curvature of this connection is .

˜ ab = 1 ω˜ a ∧ ω˜ b , R a2

(5.32)

˜ cd . Rab = Λa c Λb d R

(5.33)

so that .

Now, we consider a spin-tensor vector bundle .V realizing a representation G of the group .SO(3) with generators .Gab forming the algebra (1.35). We relabel them by Ta =

.

1 εabc Gbc 2

(5.34)

so that they satisfy the algebra (3.9). The spin connection one-form (1.38) is   r  Qbc Tc dθ b , A = 1 − cos a

.

(5.35)

with the curvature 1 εabc Tc ωa ∧ ωb . 2a 2

(5.36)

1 A˜ = Td ω˜ d , a

(5.37)

1 F˜ = 2 εabc Tc ω˜ a ∧ ω˜ b . 2a

(5.38)

F=

.

The right-invariant spin connection is .

with the curvature .

The Lie derivatives .LPa and .LPa along the Killing vector fields .Pa and .La of sections of the bundle .V in the standard basis are given by (4.102), (4.103), Pa = Pa −

.

1 − cos(r/a) Qac Tc , . a sin(r/a)

L a = La + Ta .

(5.39) (5.40)

80

5 Three-dimensional Maximally Symmetric Spaces

We denote the Lie derivatives along the Killing vectors .Ka± in the standard basis by ± .Ka = L ± . Ka Proposition 5.4 The Lie derivatives .Kb± along the Killing vector fields .Ka± in the standard basis are

1 − cos(r/a) 1 ± ± − .K (5.41) Qbc ± δbc Tc . b = Kb + 2 sin(r/a) Proof Eq. (5.41) follows from the definition .Kb± =

1 2

(aPb ± Lb ) and (5.39), (5.40).

Proposition 5.5 The Lie derivatives in the standard basis, .Ka± , form two mutually commuting representations of the algebra .so(3), [Ka+ , Kb+ ] = −εc ab Kc+ , .

(5.42)

[Ka− , Kb− ] = εc ab Kc− , .

(5.43)

[Ka+ , Kb− ]

(5.44)

.

= 0.

Proof This follows from the algebra formed by the operators .Pa and .Lb and the operators Ka± .

.

Proposition 5.6 The Laplacian in the standard basis is equal to the sum of the Casimir operators 3

ΔS =

.

 1  2 2 2K+ + 2K− −T2 , 2 a

(5.45)

2 = K± K± and .T 2 = T T . where .K± a a a a

Proof This follows from the equation (4.123). Proposition 5.7 The Laplacian has the form 3

ΔS = ∂r2 +

.

r  2 1 2 Δˆ S , cot ∂r + a a a 2 sin2 (r/a)

(5.46)

where   r    r 2 2 Ta La − 1 − cos Δˆ S = L2 + 2 1 − cos (δab − θa θb ) Ta Tb . a a

.

(5.47)

5.1 Geometry of S 3

81

Proof This follows from Eqs. (4.126) and (4.127). It can also be proved by direct calculation using the explicit form of the Lie derivatives (5.39), (5.40). 2 It is worth noting that the operator .Δˆ S can be written in the form 2 Δˆ S = ∇ˆ a ∇ˆ a ,

(5.48)

   r   Tc . ∇ˆ a = −Qac Lc + 1 − cos a

(5.49)

.

where .

We denote the Lie derivatives along the Killing vectors .Ka± in the right-invariant basis by .K˜ a± = L˜ Ka± . Proposition 5.8 The Lie derivatives in the right-invariant basis have the form K˜ a+ = Ka+ + Ta , .

(5.50)

K˜ a−

(5.51)

.

=

Ka− .

Proof Equations (5.50), (5.51) follow from the definition of the Lie derivative (1.52) and Eqs. (5.27)–(5.29). Proposition 5.9 The Lie derivatives in the right-invariant basis form two mutually commuting representations of the algebra .so(3), [K˜ a+ , K˜ b+ ] = −εc ab K˜ c+ , .

(5.52)

[K˜ a− , K˜ b− ] = εc ab K˜ c− , .

(5.53)

[K˜ a+ , K˜ b− ] = 0.

(5.54)

.

Proof This follows from (5.50), (5.51), and the algebra formed by the operators .Ka± . Proposition 5.10 The Laplacian in the right-invariant basis has the form  1  ˜2 2 2K+ + 2K˜ − −T2 2 a  1  3 = ΔS0 + 2 4Ta Ka+ + T 2 . a

3 Δ˜ S =

.

Proof This follows from Eqs. (5.50) and (5.51).

(5.55)

82

5 Three-dimensional Maximally Symmetric Spaces

Proposition 5.11 Let .Λ = (Λab ) be an orthogonal matrix of the form Λ = exp

.

r a

 Q

(5.56)

with some anti-symmetric matrix Q. Let .Gab be the generators of the group .SO(n) satisfying the algebra (1.35). Then Λc a Λd b Gcd = OGab O −1 ,

.

(5.57)

where O = exp

.

  r Qab Gab . 2a

(5.58)

Proof It is easy to see that the matrices ˜ ab = Λc a Λd b Gcd G

.

(5.59)

satisfy the same commutation relations (2.130). Therefore, they are related by a similarity transformation (5.57). The explicit form of the matrix O is found by considering the curve in the orthogonal group .Λ(t) = exp (t rQ/a) and solving the corresponding differential equation obtained from (5.57). For the group .SU (2) (or .SO(3)) this means the following. Let .Ta be the generators of .SU (2) defined by .Ta = (1/2)εabc Gbc , (3.9), and satisfying the algebra (3.9). Let M be the matrix defined by Mba =

.

1 bij ε εakm Λi k Λj m . 2

(5.60)

Let .Qab = εabc θ c and .T (θ ) = Ti θ i . Then M b a Tb = OTa O −1 ,

(5.61)

 T (θ )) .

(5.62)

.

where O = exp

.

r a

The standard connection and the right-invariant connection and their curvatures are related by the gauge transformation .

˜ −1 − (dO)O −1 , . A = O AO

(5.63)

˜ −1 . F = O FO

(5.64)

5.2 Geometry of H 3

83

The covariant derivatives of a section of the spin-tensor vector bundle .V are related by the gauge transformation as well ∇ξ ϕ = O ∇˜ ξ O −1 ϕ.

(5.65)

.

Therefore, the Laplacian s are also related by the gauge transformation 3 3 ΔS = O Δ˜ S O −1 .

(5.66)

.

5.2

Geometry of H 3

The hyperbolic space .H 3 is dual to the sphere .S 3 and, therefore, to the group .SU (2) in the sense described in Sect. 2. It can be described as the homogeneous space H 3 = SO(1, 3)/SO(3) = Spin(1, 3)/Spin(3).

.

(5.67)

The group .Spin(1, 3) is isomorphic to the group .SL(2, C) and the group .Spin(3) is just SU (2), therefore, the hyperbolic space .H 3 is, in fact, the homogeneous space .H 3 = SL(2, C)/SU (2) . It is dual to the sphere .S 3 by the transformation .a → ib, that is, it is a pseudo-sphere with an imaginary radius. The group .SL(2, C) is the six-dimensional manifold which is the complexification of the group .SU (2). Both .S 3 and .H 3 are threedimensional submanifolds of .SL(2, C); the .S 3 is the real section and the .H 3 is the imaginary section. The difference is that .S 3 = SU (2) is a group and .H 3 is not; also, 3 3 .S is compact and .H is non-compact. Almost all of the local formulas for the sphere have the dual analog on the hyperbolic space. The proofs are also the same. That is why, we  will just list some of the results. Let i 3 .y be the normal geodesic coordinates on .H , .r = yi y i , and .θ i = y i /r . The metric on 3 .H has the form (4.58) .

ds 2 = dr 2 + b2 sinh2

.

r  b

dθ 2 .

(5.68)

The standard orthonormal basis (4.66) has the form ωa = θ a dr + b sinh

.

ea = θa ∂r +

r  b

dθ a , .

1 b2 sinh2 (r/b)

Qac Lc ,

(5.69) (5.70)

where .La are defined by (5.5). The Levi-Civita connection one-form (4.71) in this basis reads

84

5 Three-dimensional Maximally Symmetric Spaces

  r  (θ a dθ b − θ b dθ a ) , Γab = 1 − cosh b

.

(5.71)

and the curvature two-form is equal to Rab = −

.

1 a ω ∧ ωb , b2

(5.72)

so that R ab cd = −

.

 1  a b δ c δ d − δa d δb c . 2 b

(5.73)

The Killing vectors are .La and Pa = θa ∂r +

.

r  1 coth Qac Lc . b b

(5.74)

The isometry algebra of .H 3 is then given by 1 c ε ab Lc , . b2 [La , Pb ] = −εc ab Pc , . [Pa , Pb ] =

.

[La , Lb ] = −εc ab Lc .

(5.75) (5.76) (5.77)

This is nothing but the algebra .so(1, 3) with the vector fields .La forming the isotropy subalgebra .so(3). The scalar Laplacian has the form (4.116) 3

1 2 L b2 r  2 1 2 ∂r + = ∂r2 + coth ΔS0 . b b b2 sinh2 (r/b)

2 ΔH 0 =P −

.

(5.78)

The spin connection one-form (1.38) is   r  Qbc Tc dθ b A = 1 − cosh b

.

(5.79)

with the curvature F =−

.

1 εabc Tc ωa ∧ ωb . 2b2

(5.80)

5.2 Geometry of H 3

85

The Lie derivatives .LPa and .LPa along the Killing vector fields .Pa and .La of sections of the bundle .V in the standard basis are given by (4.102), (4.103), Pa = Pa −

.

1 − cosh(r/b) Qac Tc , . b sin(r/b)

La = L a + Ta .

(5.81) (5.82)

The Laplacian in the standard basis has the form 3

 1  2 L −T2 2 b r  2 1 2 Δˆ S , ∂r + = ∂r2 + coth b b b2 sinh2 (r/b)

ΔH = P 2 −

.

(5.83)

where   r    r 2 2 Δˆ S = L2 + 2 1 − cosh Ta La − 1 − cosh [δab − θa θb ]Ta Tb . b b

.

(5.84)

The crucial difference between .S 3 and .H 3 is that the hyperbolic space .H 3 is not a group manifold. The isometry algebra .so(1, 3) cannot be factorized as the algebra .so(4) since the vector fields Kb± =

.

1 (ibPb ± Lb ) 2

(5.85)

are complex even though, formally, they satisfy the same algebra .so(3) × so(3) [Ka+ , Kb+ ] = −εc ab Kc+ , .

(5.86)

[Ka− , Kb− ] = εc ab Kc− , .

(5.87)

[Ka+ , Kb− ] = 0.

(5.88)

.

Formally, we still have for the scalar Laplacian 3

ΔH 0 =−

.

4 2 4 2 K = − 2 K− . b2 + b

(5.89)

The Lie derivatives .LK ± along the vector fields .Ka± in the standard basis are b

± .K b

=

Kb±

1 − cosh(r/b) 1 −i Qbc ± δbc Tc . + 2 sinh(r/b)

(5.90)

86

5 Three-dimensional Maximally Symmetric Spaces

They satisfy the algebra [Ka+ , Kb+ ] = −εc ab Kc+ , .

(5.91)

[Ka− , Kb− ] = εc ab Kc− , .

(5.92)

[Ka+ , Kb− ] = 0.

(5.93)

.

The Laplacian has the form 3

ΔH = −

.

 1  2 2 2 2K . + 2K − T + − b2

(5.94)

Since .H 3 is not a group there is no right-invariant orthonormal basis. Formally we can define a complex twisted basis ω˜ a = θ a dr + b sinh

.

r   r  r  cosh δja + i sinh Qaj dθ j , b b b

(5.95)

r  1 1 coth Qac Lc − i La . b b b

(5.96)

and the dual basis e˜b = θb ∂r +

.

In this basis the Levi-Civita connection is imaginary 1 Γ˜ a bc = −i εa bc , . b 1 A˜ = −i Td ω˜ d . b

.

(5.97) (5.98)

The Lie derivatives .K˜ a± = L˜ Ka+ in the twisted basis have the form K˜ a+ = Ka+ + Ta , .

.

K˜ a− = Ka− ,

(5.99) (5.100)

and satisfy the algebra [K˜ a+ , K˜ b+ ] = −εc ab K˜ c+ , .

(5.101)

[K˜ a− , K˜ b− ] = εc ab K˜ c− , .

(5.102)

[K˜ a+ , K˜ b− ] = 0.

(5.103)

.

The Laplacian in the twisted basis has the form

5.2 Geometry of H 3

87 3 Δ˜ H = −

.

 1  ˜2 2 2 ˜− 2 K , + 2 K − T + b2

(5.104)

which can be expressed in terms of the scalar Laplacian 3 Δ˜ H = −

.

 1  H3 + 2 Δ . + 4T K + T a a 0 b2

(5.105)

The Laplacian s are related by the gauge transformation 3 3 ΔH = O Δ˜ H O −1 ,

(5.106)

   r  r O = exp −i Qab Gab = exp −i T (θ ) , 2b b

(5.107)

.

where .

with .T (θ ) = Ti θ i .

Part II Heat Kernel

6

Scalar Heat Kernel

6.1

Reduction Formulas

In the scalar case, the Laplacian has the form (4.116). We use the hypergeometric coordinates defined by (4.49)–(4.52). Notice that for the sphere S n the variable x = cos(r/a) varies in the interval [−1, 1] and ∂x = −

a ∂r . sin(r/a)

(6.1)

For the hyperbolic space H n , when a = ib is imaginary, the variable x = cosh(r/b) is ranging in the interval [1, ∞) and b ∂r . sinh(r/b)

(6.2)

  1 1 2 2 2 L )∂ − nx∂ + (1 − x . x x a2 (1 − x 2 )

(6.3)

∂x = The scalar Laplacian takes the form Δ0 =

Therefore, the scalar heat kernel U0 (t, y) on a maximally symmetric space (S n or H n ) depends only on the geodesic distance, that is, the radial coordinate r, and does not depend on the angular coordinates θ i , so, abusing the notation slightly, we write U0 (t, y) = U0 (t, r),

(6.4)

G0 (λ, y) = G0 (λ, r).

(6.5)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_6

91

92

6 Scalar Heat Kernel

The equations and the initial condition for the heat kernel and the resolvent on H n have the same form as for S n . On the hyperbolic space H n , they have to be supplemented by the boundedness condition at infinity, as r and x go to infinity. Then the heat equation becomes   a 2 ∂t − Ln U0 (t, r) = 0,

(6.6)

with the asymptotic condition as t → 0 −n/2

U0 (t, r) ∼ (4π t)



r2 exp − 4t

 (6.7)

.

The corresponding resolvent G0 (λ, r) satisfies the equation for r = 0,   Ln + a 2 λ G0 (λ, r) = 0,

(6.8)

with the asymptotic condition as r → 0, n

−1 1 G0 (λ, r) ∼ 4π n/2 r n−2    n Γ n2 − 1 1 − x − 2 +1 = , 2 (4π )n/2 a n−2 Γ

2

1 log r 2π 1 log(1 − x), =− 4π

n ≥ 3,

(6.9)

G0 (λ, r) ∼ −

1 1 G0 (λ, r) ∼ √ − r, 2 −λ 2

n = 2,

(6.10)

n = 1.

(6.11)

Proposition 6.1 The scalar heat kernel and the resolvent are given by the derivatives of the basic cases n = 1 and n = 2. For odd n = 2α + 1 ≥ 1, they have the form n U0S (t, r)

n



α2 t = exp a2

GS0 (λ, r) =





1 ∂x 2π a 2

1 ∂x 2π a 2

α GS0

1

α

1

U0S (t, r),

  α2 λ + 2,r , a

(6.12)

(6.13)

6.2 Scalar Heat Kernel on S 1 and R

93

α   α2 1 n U0H (t, r) = exp − 2 t − ∂ U0R (t, r), x b 2π b2 n GH 0 (λ, r)

α    α2 1 R = − ∂x G0 λ − 2 , r . 2π b2 b

(6.14)

(6.15)

For even n = 2β + 2 ≥ 2, we have 

n

U0S (t, r) = exp

n



GS0 (λ, r) =

n U0H (t, r)



β(β + 1) t a2

1 ∂x 2π a 2

β GS0

2

1 ∂x 2π a 2

 λ+

β

2

U0S (t, r),

 β(β + 1) , r , a2

β   β(β + 1) 1 2 = exp − t − ∂x U0H (t, r), 2 2 b 2π b

β    β(β + 1) 1 n H2 λ − GH (λ, r) = − ∂ G , r . x 0 0 2π b2 b2

(6.16)

(6.17)

(6.18)

(6.19)

Proof This is proved by using the intertwining relations (1.148)–(1.155). The normalization factor is chosen to satisfy the asymptotic conditions as t → 0 for the heat kernel and as r → 0 for the resolvent.

6.2

Scalar Heat Kernel on S 1 and R

Notice that the operator L1 = a 2 ∂r2

(6.20)

is just the one-dimensional Laplacian. The hyperbolic space in one dimension is just the line H 1 = R. Proposition 6.2 The heat kernel and the resolvent on the line R are U0R (t, r)

= (4π t)

GR 0 (λ, r) = where μ =

√

−λ.

−1/2



r2 exp − 4t

1 exp(−μr), 2μ

 ,

(6.21) (6.22)

94

6 Scalar Heat Kernel

Proof The resolvent and the heat kernel on the real line are well known. They are just given by the solutions of the differential equations with the correct asymptotic conditions. The resolvent has a cut in the complex plane λ along the real axis from 0 to ∞ with the jump R GR 0 (λ + iε, r) − G0 (λ − iε, r) = i

cos

√  λr , √ λ

λ > 0.

(6.23)

It is easy to see that Eq. (1.16) obviously holds. Proposition 6.3 The heat kernel and the resolvent on the circle S 1 are 1 U0S (t; r)

   r 1

t 2 , = exp − 2 k cos k 2π a a a k∈Z

= (4π t)

−1/2

k∈Z

1

GS0 (λ; r) =

  (r + 2π ak)2 , exp − 4t

1 cosh [μ(r − π a)] . 2μ sinh(μπ a)

(6.24)

(6.25)

(6.26)

Proof The heat kernel and the resolvent on the circle of radius a are obtained by the Fourier expansion 1

U0S (t; r) =

  1

t r , exp − 2 k 2 + ik 2π a a a

(6.27)

k∈Z

 r 1 a

. exp ik G0 (λ; r) = 2π a k2 − a2λ S1

(6.28)

k∈Z

The heat kernel can be expressed directly in terms of the Jacobi theta function [25] 1

U0S (t; r) =

1 θ3 2π a



 r  it .  2π a π a 2

(6.29)

Equation (6.25) is proved by using the Poisson summation formula [37],

k∈Z

f (k) =



dp exp (i2π kp) f (p)

(6.30)

k∈Z R

 for the function f (p) = exp −tp 2 + ipξ or, equivalently, by the duality transformation for the Jacobi theta function. Notice that the heat kernel on S 1 is nothing but the heat kernel

6.2 Scalar Heat Kernel on S 1 and R

95

on R with the points r and r + 2π a identified, that is,

1

U0S (t, r) =

U0R (t; r + 2π ak).

(6.31)

k∈Z

The corresponding expression for the resolvent 1

GS0 (λ; r) =

 1

exp − μ|r + 2π ak| 2μ

(6.32)

k∈Z

can be summed up to get (6.26). We introduce a useful function Ψ1 (t, r) =

1 4π i



dω cot C− −C+

ω−r 2



  ω2 exp − , 4t

(6.33)

where C+ is the contour just above the real axis that goes from −∞ + iε to +∞ + iε and C− is a similar contour that goes just below the real axis from −∞ − iε to +∞ − iε. For real positive t > 0, the integrand has poles on the real line at r + 2π k,

k ∈ Z,

(6.34)

and, therefore, by computing the residues, we get Ψ1 (t, r) =

k∈Z



(r + 2kπ )2 exp − 4t

 .

(6.35)

This function satisfies the equation Ψ1 (t, π ) = Ψ1 (4t, 0) − Ψ1 (t, 0).

(6.36)

By rewriting Eq. (6.33) in the form √ t Ψ1 (t, r) = 4π i

C− −C+

    √ ω2 ω t −r exp − , dω cot 2 4

(6.37)

we can define the function Ψ1 for negative t and imaginary r: there is only one pole on the real line and the integral is  2 r , Ψ1 (−t, ir) = exp − 4t

t > 0.

(6.38)

96

6 Scalar Heat Kernel

Obviously, the heat kernel is 1 U0S (t, r)

−1/2

= (4π t)

 Ψ1

t r , a2 a

 .

(6.39)

This formula also works for R by replacing a → ib. The resolvent on the circle S 1 is a meromorphic function of λ with simple poles on the real line at λk =

k2 , a2

k = 0, 1, 2, . . .

(6.40)

with the residues 

1 1 , Res GS0 (λ; r), λ0 = − 2π a

(6.41)

 r 

1 1 cos k . Res GS0 (λ; r), λk = − πa a

(6.42)

and for k ≥ 1

Equation (1.10) takes the form 1 U0S (t; r)

  ∞  r 1

1 t 2 + . = exp − 2 k cos k 2π a πa a a

(6.43)

k=1

We will find it useful to introduce the following functions: Ω+ (t) =

k∈Z

  π2 exp − k 2 t

 1/2

t exp(−tk 2 ), = π

(6.44)

k∈Z

  π2 2 Ω− (t) = (−1) exp − k t k∈Z   1/2

2   t 1 = +k exp −t . π 2

k

(6.45)

k∈Z

These functions can be expressed in terms of the Jacobi theta functions. They can also be written as the integrals

6.3 Scalar Heat Kernel on S 2

97

Ω+ (t) = −

1 2π i

dω cot C+

Ω− (t) = −

1 2π i



dω

  ω2 exp − , 2 4t

ω

  ω2 1 exp − , sin (ω/2) 4t

(6.46)

(6.47)

C+

where C+ is the contour just above the real axis. Proposition 6.4 The heat kernel diagonal on the circle S 1 has the form S1 U0,diag (t)

  ∞ 1

1 t 2 + = exp − 2 k 2π a πa a k=1   t ,0 . = (4π t)−1/2 Ω+ a2

(6.48)

The heat trace is   ∞  

t 2 S1 Tr exp tΔ0 = 1 + 2 exp − 2 k . a

(6.49)

k=1

Proof This is obtained by setting r = 0 in the above equations and noticing that Ψ1 (t, 0) = Ω+ (t). The heat trace is obtained by multiplying by the volume vol (S 1 ) = 2π a.

6.3

Scalar Heat Kernel on S 2

Proposition 6.5 The scalar resolvent on the sphere S 2 has the form 2

GS0 (λ, r) =  where ω = i −λa 2 −

1 4

1 P 1 (−x), 4 cos(ωπ ) ω− 2

(6.50)

(with Im ω > 0) and Pν (x) is the Legendre function [25].

Proof The solution of the differential equation for the resolvent (6.8), which is finite as (3) (x) x → −1+ and has the correct asymptotics at x → 1− , is given by the function Φ0,0 (1.113). It takes the form (6.50) by using the expression of the Legendre function in terms of the hypergeometric function (see, e.g., [25]),   1 1+x 1 . Pω− 1 (−x) = F ω + , −ω + ; 1; 2 2 2 2

(6.51)

98

6 Scalar Heat Kernel

By using the relation [25] 2 cos(ωπ )Qω− 1 (x) + sin(ωπ )Pω− 1 (x), 2 2 π

Pω− 1 (−x) = 2

(6.52)

this can also be written in the form 2

GS0 (λ, r) =

1 1 Q 1 (x) + tan(ωπ )Pω− 1 (x), 2 2π ω− 2 4

(6.53)

where Qν (x) is the Legendre function of the second kind [25]. Proposition 6.6 The scalar resolvent on S 2 is a meromorphic function of λ with simple poles at the eigenvalues of the Laplacian 1 k(k + 1), a2

λk (−Δ0 ) =

k = 0, 1, 2, . . . ,

(6.54)

with the residues 2

Res {GS0 (λ; r), λk } = −

2k + 1 Pk (x), 4π a 2

(6.55)

where Pk (x) are the Legendre polynomials [25]. Proof Note that the resolvent (6.53) is an even meromorphic function of ω with simple poles at ±ωk , where ωk =

1 + k, 2

k = 0, 1, 2, . . . .

(6.56)

This means that, as a function of λ, the resolvent has simple poles at λk =

1 k(k + 1), a2

k = 0, 1, 2, . . . .

(6.57)

By using the relations [25] F (−k, b; c; 1 − z) =

Γ (c)Γ (c − b + k) F (−k, b; b − c + 1 − k; z), Γ (c + k)Γ (c − b)

(6.58)

and 

1−x Pk (x) = F −k, k + 1; 1; 2

 ,

(6.59)

6.3 Scalar Heat Kernel on S 2

99

and the residues of the gamma function, we compute the residues of the resolvent. Proposition 6.7 The scalar heat kernel on the sphere S 2 has the form 2 U0S (t, r)

 

        ∞  1 1 2 t r 1 t k+ exp − 2 k + = exp Pk cos 2 2 2 2 a 2π a 4a a k=0



=

t 1 exp 4π t 4a 2



 Ψ2

t r , a2 a

(6.60)

 ,

(6.61)

where

  dω ω tan (ωπ ) Pω− 1 (cos r) exp −tω2

Ψ2 (t, r) = −it

2

(6.62)

C+

     ∞ 

1 1 2 k+ exp −t k + = 2t Pk (cos r) 2 2

(6.63)

k=0

−1/2

= (4π t)

π   √ (ω + 2π k) (ω + 2π k)2 , (−1) 2 dω √ exp − 4t cos r − cos ω k∈Z



k

r

(6.64) and C+ is a contour just above the real axis. Proof heat kernel is obtained by the inverse Laplace transform (1.8). By using λ =  The  1 2 − 1 , we obtain (6.61) with ω 4 a2 

Ψ2 (t; r) = −it

dω ω

   2 Qω− 1 (x) + tan(ωπ )Pω− 1 (x) exp −tω2 . 2 2 π

(6.65)

C+

The first term in (6.65) does not contribute to the integral and we obtain (6.62). By computing the residues, this gives (6.63) and the result (6.63) follows. To compute the dual integral representation (6.64) of the function Ψ2 , we rewrite it as the integral −1/2



Ψ2 (t, r) = −i(4π t)

C+

  ω2 , dω f (r, ω)ω exp − 4t

(6.66)

100

6 Scalar Heat Kernel

where f (r, ω) =

∞

k=0

    1 ω Pk (cos r). exp i k + 2

(6.67)

This sum can be evaluated by using the generating function of the Legendre polynomials; for Im ω > 0, we obtain f (r, ω) = √

exp(iω/2) 1 − 2eiω cos r + e2iω

.

(6.68)

This function f (r, ω) is analytic in the upper half-plane of ω satisfying the relations f (r, ω + π ) = if (r + π, ω),

f (r + 2π, ω) = f (r, ω),

(6.69)

and f (r, ω + 2π ) = −f (r, ω),

f (r, −ω) = f (r, ω).

(6.70)

The function f (r, ω) has branch point singularities on the real axis at ωk± = ±r + 2π k,

k∈Z

(6.71)

with some cuts along the real axis; it is equal to (here k ∈ Z) ⎧√ ⎪ (−i) 2 ⎪ ⎪ , for − π + 2π k < ω < −r + 2π k, √ ⎪ ⎪ 2 cos r − cos ω ⎪ ⎪ ⎪ ⎪ ⎨√ 1 2 f (r, ω) = , for − r + 2π k < ω < r + 2π k, √ ⎪ 2 cos ω − cos r ⎪ ⎪ ⎪ ⎪ ⎪ √2 ⎪ i ⎪ ⎪ , for r + 2π k < ω < π + 2π k. √ ⎩ 2 cos r − cos ω

(6.72)

This enables one to compute −1/2

Ψ2 (t, r) = (4π t)

k∈Z

π (−1)

k −π

  (ω + 2π k)2 . dω (−i)f (r, ω + iε)(ω + 2π k) exp − 4t

Finally, by using (6.72) and the symmetries of the integrand, we get (6.64). Proposition 6.8 The scalar heat kernel diagonal on the sphere S 2 has the form

(6.73)

6.3 Scalar Heat Kernel on S 2

101

  ∞ 1

t (2k + 1) exp − k(k + 1) 4π a 2 a2 k=0     t t 1 Ψ exp = , 0 , 2 4π t 4a 2 a2

2

S U0,diag (t) =

(6.74)

(6.75)

where Ψ2 (t, 0) = (4π t)−1/2

 dω

R

  ω2 ω/2 exp − sin(ω/2) 4t

(6.76)

 

(ω + 2π k) (ω + 2π k)2 . (6.77) (−1)k dω exp − sin(ω/2) 4t π

−1/2

= (4π t)

k∈Z

0

The scalar heat trace is   ∞ 

 t 2 Tr exp tΔS0 = (2k + 1) exp − 2 k(k + 1) a

(6.78)

k=0

    t t a2 Ψ2 exp = ,0 . t 4a 2 a2

(6.79)

Proof We compute the value of the function Ψ2 (t, r) at r = 0. First of all, we get the spectral series Ψ2 (t, 0) = 2t

∞ 

k=0

     1 1 2 k+ exp −t k + ; 2 2

(6.80)

this gives (6.74). By multiplying this by the volume of the sphere, we get the heat trace (6.78). Second, Eqs. (6.75) and (6.77) follow from (6.61) and (6.64). The function Ψ2 (t, 0) is not an analytic function of t at t = 0. However, it has the asymptotic expansion in positive powers of t as t → 0+ . Proposition 6.9 There is the asymptotic expansion as t → 0+ , Ψ2 (t, 0) ∼

∞

k=0

ak

tk , k!

where   ak = (−1)k+1 1 − 21−2k B2k

(6.81)

102

6 Scalar Heat Kernel

∞ =π

dω 0

ω2k cosh2 (π ω)

,

(6.82)

and Bn are the Bernoulli numbers [25]. Proof This is proved by using the expansion ∞

z B2k 2k = z , (−1)k+1 2(22k−1 − 1) sin z (2k)!

(6.83)

k=0

(valid for |z| < 2π , see [25]), in (6.76) and computing the Gaussian integrals.

6.4

Scalar Heat Kernel on H 2

Next, we consider the hyperbolic plane H 2 , so that the radius a = ib is imaginary with b real; recall also that now x = cosh(r/b). Proposition 6.10 The scalar resolvent on the hyperbolic plane H 2 has the form 2

1 Q 1 (x) 2π −iν− 2 √ ∞ 2 exp(isν) = ds √ , 4π cosh s − cosh(r/b)

GH 0 (λ, r) =

(6.84)

(6.85)

r/b

 where ν = i −λb2 +

1 4

(with Im ν > 0).

Proof The resolvent, which is a solution of the differential equation (6.7), is now given by (4) the function Φ0,0 (x) (1.114) 2

GH (λ, r) =

   1 1 1 + x + iε 1 F iν + , −iν + ; 1, 4 cosh(π ν) 2 2 2   1 1−x 1 , −ie−π ν F iν + , −iν + ; 1, 2 2 2

(6.86)

which can be written in the form (6.84). Now, by using the integral representation of the Legendre function [25]

6.4 Scalar Heat Kernel on H 2

103

√ ∞ 2 exp(isν) Q−iν− 1 (cosh ρ) = ds √ , 2 2 cosh s − cosh ρ

(6.87)

ρ

we get (6.85). By using the relation [25] Q−iν− 1 (z) − Qiν− 1 (z) = iπ tanh(νπ )Piν− 1 (z), 2

2

2

(6.88)

this can also be written in the form (compare with (6.53)) 2

GH 0 (λ, r) =

1 i Qiν− 1 (x) + tanh(νπ )Piν− 1 (x). 2 2 2π 2

(6.89)

Let us define now the dual function Ψ˜ 2 (t, r) by (here t, r > 0) Ψ˜ 2 (t, r) = Ψ2 (−t, ir) = t



  dν ν tanh (νπ ) Piν− 1 (cosh r) exp −tν 2 . 2

(6.90)

C+

By using the integral representation of the Legendre function √

2 coth(νπ ) Piν− 1 (cosh r) = 2 π

∞ ds √ r

sin(νs) cosh s − cosh r

,

(6.91)

we can compute the integral over ν to get (after renaming s → ν) Ψ˜ 2 (t, r) = (4π t)

−1/2

∞ √ 2 dν √ r

 2 ν . exp − 4t cosh ν − cosh r ν

(6.92)

Proposition 6.11 The scalar heat kernel on the hyperbolic plane H 2 has the form 2

U0H (t; r) =

    t r t 1 exp − 2 Ψ˜ 2 . , 4π t 4b b2 b

(6.93)

Proof The Legendre function Q−iν− 1 (x) is an analytic function of ν. Therefore, the 2 resolvent is an analytic function of λ with a branch cut along the positive real axis from c = 1/(4b2 ) to ∞. By using the relation (6.88), we obtain 2

2

2

H H Jump GH 0 (λ; r) = G0 (λ + iε; r) − G0 (λ − iε; r)

=

i tanh(νπ )Piν− 1 (x). 2 2

(6.94)

104

6 Scalar Heat Kernel

Now, by using (1.16), this gives the heat kernel 2 U0H (t; r)

 ∞    1 t t 2 , (6.95) = exp − dν ν tanh(νπ )P ν 1 (x) exp − iν− 2 2π b2 4b2 b2 0

which is exactly (6.93). Proposition 6.12 The scalar heat kernel diagonal on the hyperbolic plane H 2 can be presented either in the spectral form H2 U0,diag (t)

 ∞    1 t t 2 = exp − dν ν tanh(νπ ) exp − ν 2π b2 4b2 b2

(6.96)

0

or in the dual form 2

H U0,diag (t) =

    t t 1 exp − 2 Ψ˜ 2 , 0 , 4π t 4b b2

(6.97)

where Ψ˜ 2 (t, 0) = (4π t)−1/2

dω R

  ω/2 ω2 exp − . sinh(ω/2) 4t

(6.98)

Proof This is obtained by taking the limit r → 0, that is, x → 1, in (6.95) and (6.93). Note the similarity between the heat kernels on S 2 (6.60) and H 2 (6.93), as well as the heat kernel diagonal s, Eqs. (6.75) and (6.98). It is easy to see that the asymptotic expansion of the function Ψ˜ 2 (t, 0) as t → 0 has the form Ψ˜ 2 (t, 0) ∼

∞ k

t k=0

where the ak are defined by (6.82).

k!

(−1)k ak ,

(6.99)

6.5 Scalar Heat Kernel on S n , n ≥ 3

105

Scalar Heat Kernel on S n , n ≥ 3

6.5

Proposition 6.13 The scalar resolvent on the sphere S n has the form n GS0 (λ, r)

=

Γ (ω +

n−1 n−1 2 )Γ (−ω + 2 ) F (4π )n/2 a n−2 Γ n2

 where ω = i −a 2 λ −

(n−1)2 4

  n−1 n 1+x n−1 , −ω + ; ; , ω+ 2 2 2 2 (6.100)

(with Im ω > 0).

Proof This is the solution of the differential equation for the resolvent (6.8) that is finite as (3) x → −1+ and has the correct asymptotics at x → 1− . It is given by the function Φμ,ν (x) (1.113) with μ = ν = n2 − 1. Let us show how this generates the reduction formula in odd dimensions, when α = (n − 1)/2 is a positive integer. By using the derivatives of the hypergeometric function [25] ∂zk F (a, b; c; z) =

Γ (a + k)Γ (b + k)Γ (c) F (a + k, b + k; c + k; z), Γ (a)Γ (b)Γ (c + k)

(6.101)

we can rewrite it as the multiple derivative n GS0 (λ, r)

  Γ (ω)Γ (−ω) 1 1+x α . = (2∂x ) F ω, −ω; ; √ 2 2 (4π )n/2 a n−2 π 1

(6.102)

Finally, by using [25]   r   r  1 1 1 + cos = cos ω −π , F ω, −ω; ; 2 2 a a

(6.103)

we get n GS0 (λ, r)

where μ =



−λ −

(n−1)2 . 4a 2

 =

1 ∂x 2π a 2

α

cosh[μ(r − aπ )] , 2μ sinh(μaπ )

(6.104)

This gives the previous result (6.13).

Proposition 6.14 The scalar resolvent on the sphere S n has simple poles at the eigenvalues of the scalar Laplacian λk =

1 k(k + n − 1), a2

k = 0, 1, 2, . . . ,

(6.105)

106

6 Scalar Heat Kernel

with the residues n

Res {GS0 (λ; r), λk } = −

  (2k + n − 1)Γ (k + n − 1) n 1−x n . F −k, k + n − 1; , 2 2 (4π a 2 )n/2 Γ 2 k! (6.106)

Proof The resolvent is an even function of ω. If n > 1, then the resolvent is a meromorphic function of ω with simple poles at ±ωk , where ωk =

n−1 + k, 2

k = 0, 1, 2, . . . .

(6.107)

This means that as a function of λ, the resolvent has simple poles at λk =

1 k(k + n − 1), a2

k = 0, 1, 2, . . . .

(6.108)

By using the relation [25] F (−k, b; c; 1 − z) =

Γ (c)Γ (c − b + k) F (−k, b; b − c + 1 − k; z) Γ (c + k)Γ (c − b)

(6.109)

and the residues of the gamma function, we obtain the result (6.106). Notice the residues can be expressed in terms of the Gegenbauer polynomials [25] (with α = (n − 1)/2 > 0) Ck(α) (x) =

  1 1−x Γ (2α + k) F −k, k + 2α; + α; . k!Γ (2α) 2 2

(6.110)

Equation (6.100) also gives the resolvent in the one-dimensional case (6.26) by substituting n = 1 and the previous result in two dimensions (6.50) by setting n = 2 and using Eq. (6.51). Note that the one-dimensional case is special. In this case the resolvent, as a function of ω, has a double pole at ω0 = 0 and simple poles at ±ωk = k, k = 1, 2, . . . . Proposition 6.15 The scalar heat kernel on the sphere S n has the form 

Sn

U0 (t, r) =



n+1  ∞

2 t d (n) exp − 2 k(k k (n+1)/2 n a 2π a k=0

Γ

   n 1−x + n − 1) F −k, k + n − 1; , , 2 2

(6.111) where the multiplicities are d0 (n) = 1 and dk (n) = (2k + n − 1)

(k + n − 2)! , (n − 1)!k!

(6.112)

6.6 Scalar Heat Kernel on H n , n ≥ 3

 =

k+n−1 n−1

107



 +

k+n−2 n−1

 for k ≥ 1.

(6.113)

Proof We use Eq. (1.10) (and some properties of the gamma function) to compute the heat kernel by using the eigenvalues and the residues of the resolvent to get (6.111). In the special case of n = 1, to obtain the correct form of the heat kernel, one has to divide the first term for k = 0 in (6.111) by 2, and this gives the correct result (6.43); this was explained above. In odd dimensions, with integer α = n−1 2 , this formula can also be written in the reduction form (6.12). Proposition 6.16 The scalar heat kernel diagonal and the heat trace on the sphere S n have the form 

Sn U0,diag (t)



n+1  ∞

2 t dk (n) exp − 2 k(k (n+1)/2 n a 2π a k=0

Γ

=

  ∞ 

 t n dk (n) exp − 2 k(k + n − 1) . Tr exp tΔS0 = a

 + n − 1) ,

(6.114)

(6.115)

k=0

Proof The heat kernel diagonal is obtained by setting r = 0, i.e., x = 1, and the heat trace is obtained by multiplying the heat kernel diagonal by the volume of the sphere.

Scalar Heat Kernel on H n , n ≥ 3

6.6

Let us consider now the hyperbolic space H n so that the radius a = ib is imaginary with b real so that x = cosh(r/b), that is, x varies in the interval [1, ∞). Proposition 6.17 The scalar resolvent for the hyperbolic space H n with n ≥ 2 has the form n

GH 0 (λ, r) =

Γ (iν +

n−1 n−1 2 )Γ (−iν+ 2 ) n (4π )n/2 bn−2 Γ 2

(6.116)

  n−1 n − 1 n 1 + x + iε n F iν + , −iν + ; ; × − exp −iπ 2 2 2 2 2    n−1 n 1−x n−1 , −iν + ; ; , −ie−π ν F iν + 2 2 2 2 

 where ν = i −b2 λ +



(n−1)2 4

with Im ν > 0.

108

6 Scalar Heat Kernel

Proof For large real negative λ, that is, imaginary ν, the resolvent should be bounded at (4) (x) (1.114), infinity, as r → ∞, or x → ∞. The resolvent is given by the function Φμ,ν n with μ = ν = 2 − 1. By using the correct asymptotics as x → 1, we get the resolvent (6.116). In odd dimensions n ≥ 3, this formula can also be written in an alternative form n GH 0 (λ, r)

   n   Γ n2 − 1 x − 1 − 2 +1 1 n 1−x 1 , iν + ; − + 2; = F −iν + 2 2 2 2 2 (4π )n/2 bn−2

   Γ − n2 + 1 Γ (−iν + n−1 n−1 n 1−x n−1 2 ) , iν + ; ; . (6.117) F −iν + + n 2 2 2 2 Γ 2 − 1 Γ (−iν − n−3 2 ) Let us show how this generates the reduction formula in odd dimensions when α = ≥ 1 is a positive integer. By using the derivatives of the hypergeometric function [25]

n−1 2

  1 1 3 1 (6.118) (−z) 2 −α F −iν + , iν + ; − α; z 2 2 2    Γ ( 32 − α) 1 3 1 α α 1/2   (−1) ∂z (−z) F −iν + , iν + ; ; z , = 2 2 2 Γ 32 and       Γ 12 + α Γ (−iν)Γ (iν) 1 1 ∂zα F −iν, iν; ; z , F −iν + α, iν + α; + α; z =   2 2 Γ 12 Γ (−iν + α)Γ (iν + α) (6.119) we obtain the reduction formula α       x − 1 1/2 1 3 1−x ib 1 1 n 2iν − iν, + iν; ; GH (λ, r) = − ∂ F x 0 2ν 2 2 2 2 2 2π b2   1 1−x . +F −iν, iν; ; (6.120) 2 2 Now, by using the values of the hypergeometric function [25]

2ν sinh

  r  r  1 2 = cos ν , F iν, −iν; ; − sinh 2 2b b

(6.121)

 r  r  1  r  1 3 F = sin ν , + iν, − iν; ; − sinh2 2b 2 2 2 2b b

(6.122)

6.6 Scalar Heat Kernel on H n , n ≥ 3

109

we obtain α  1 1 n exp(−μr), GH (λ, r) = − ∂ x 0 2 2μ 2π b

(6.123)

 2 where μ = −λ + (n−1) , which coincides with the previous result (6.15). 4b2 We will find it useful to introduce the following functions: T (ν, α) =

Γ (iν + α)Γ (−iν + α) , Γ (iν)Γ (−iν)

    Γ iν + α + 12 Γ −iν + α + 12     T˜ (ν, α) = . Γ iν + 12 Γ −iν + 12

(6.124)

(6.125)

It is easy to see that they satisfy the equations   T (ν, α) = ν 2 + (α − 1)2 T (ν, α − 1), 



1 T˜ (ν, α) = ν + α − 2 2

2 

T˜ (ν, α − 1),

(6.126)

(6.127)

and T (ν, 0) = T˜ (ν, 0) = 1,   1 = ν tanh(π ν), T ν, 2   1 ˜ = ν coth(π ν). T ν, 2

(6.128) (6.129) (6.130)

Therefore, for an integer α, these functions are polynomials of degree 2α, T (ν, α) = ν 2 [ν 2 + 12 ] · · · [ν 2 + (α − 1)2 ],      2    2   1 3 1 2 2 2 2 ˜ T (ν, α) = ν + ν + ··· ν + α − . 2 2 2

(6.131)

(6.132)

Also, by using the properties of the gamma function, it is easy to see that they are related by

110

6 Scalar Heat Kernel

  1 = ν tanh(π ν)T˜ (ν, α) , T ν, α + 2   1 1 ˜ = coth(π ν)T (ν, α + 1) , T ν, α + 2 ν

(6.133) (6.134)

which gives their form also for a half-integer argument. Therefore, for α, the   half-integer 1 function T (ν, α) has simple poles at half-integer imaginary ν = ±i k + 2 , k ∈ Z, and the function T˜ (ν, α) has simple poles at integer imaginary ν = ±ik, k ∈ Z. Proposition 6.18 The scalar resolvent on the hyperbolic space H n is an analytic function 2 to ∞. The jump of the of λ with a branch cut along the positive real axis from c = (n−1) 4b2 resolvent across the cut is (with positive real ν) n

n

n

H H Jump GH (6.135) 0 (λ; r) = G0 (λ + iε; r) − G0 (λ − iε; r)     1 n−1 n−1 n−1 n 1−x 2π i  T ν, F −iν + , iν + ; ; . = 2 2 2 2 2 (4π )n/2 bn−2 Γ n2 ν

Proof The upper side of the cut, λ+iε, corresponds to positive real ν and the lower side of the cut, λ − iε, corresponds to the real negative ν. By using (1.125), we obtain Eq. (6.135) with       ν sinh(π ν) n−1 n−1 n−1 = Γ iν + Γ −iν + . (6.136) T ν, 2 π 2 2 This can be transformed finally to (6.124) by using the properties of the gamma function. When n ≥ 3 is odd, the function T (ν, (n − 1)/2) is a polynomial of degree n − 1. When n ≥ 2 is even, the function T (ν, (n − 1)/2) has the form T

    n−2 n−1 = ν tanh(π ν)T˜ ν, , ν, 2 2

(6.137)

  is a polynomial of degree (n − 2). where T˜ ν, n−2 2 Proposition 6.19 The scalar heat kernel on the hyperbolic space H n has the form n U0H (t; r)

  (n − 1)2  = t exp − 4b2 (4π b2 )n/2 Γ n2 2

∞ ×

dν T 0

(6.138)

      n−1 n−1 n 1−x t n−1 F −iν + , iν + ; ; exp − 2 ν 2 . ν, 2 2 2 2 2 b

6.6 Scalar Heat Kernel on H n , n ≥ 3

111

Proof This is obtained by using Eq. (1.16). For odd dimension, when α = n−1 2 is an integer, we can use the derivatives of the hypergeometric function (6.119) to obtain       Γ (n) 1 n−1 1 F −iν + α, iν + α; + α; z = √ 2 ∂zα F iν, −iν; ; z . ν, 2 2 2 π (6.139) Now, by using (6.121) and computing the integral over ν, we get the heat kernel in the reduction form T

α    2 (n − 1)2 1 r n −1/2 , U0H (t; r) = exp − t − ∂ (4π t) exp − x 4t 4b2 2π b2

(6.140)

which coincides with the previous result (6.14). Proposition 6.20 The scalar heat kernel diagonal on the hyperbolic space H n is     t (n − 1)2 t Hn B0 , U0,diag (t) = (4π t)−n/2 exp − 4 b2 b2

(6.141)

where 2t n/2 B0 (t) =  n Γ 2

∞

    n−1 exp −tν 2 . dν T ν, 2

(6.142)

0

Proof This is obtained by substituting x = 1 in Eq. (6.138). It is easy to see that in odd dimension n ≥ 3 the function B0 (t) is a polynomial of degree n−3 2 , which can be easily computed. Notice that for n = 3 it simplifies to B(t) = 1. In even dimension, the function B0 (t) is not analytic at t = 0 but has an asymptotic expansion in positive powers of t as t → 0+ .

7

Spinor Heat Kernel

7.1

Spinor Heat Trace

We use the notation of Sect. 2.5. Let .γa be the Dirac matrices and .Σab = 12 γab be the generators of the spinor representation of the spin group given by (1.46). Recall that the n dimension of the spinor representation is .N = 2[ 2 ] . Let Z be a first-order differential operators defined by Z = Σ ab Lab ,

.

(7.1)

where .Lab are the operators defined by (4.76). Recall that the operator .L2 = 12 Lab Lab is equal to the scalar Laplacian .L2 = ΔS0

n−1

on the unit sphere .S n−1 .

Proposition 7.1 There holds Z 2 − (n − 2)Z = −I L2 ,

.

(7.2)

where I is the unit matrix. Also, the function .ϕ(θ ) = γc θ c is the eigenfunction of the operators Z and .L2 , Zϕ = (n − 1)I, .

.

L2 ϕ = (n − 1)(n − 2)I.

(7.3) (7.4)

Proof This is proved by using the algebra of the Dirac matrices (1.45), Eqs. (1.93) and (1.94), and the derivatives .Lab θ c given by (1.99). Equation (7.3) is obtained by using

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_7

113

114

7 Spinor Heat Kernel

γac γ c = (n − 1)γa .

.

(7.5)

Proposition 7.2 Let B be the operator defined by  B=

−L2 +

.

(n − 2)2 . 4

(7.6)

Then the operators     n−2 1 I B− + Z ,. .P+ = 2B 2 P− =

    n−2 1 I B+ −Z 2B 2

(7.7)

(7.8)

are orthogonal projections satisfying P+2 = P+ , .

(7.9)

P−2 = P− , .

(7.10)

.

P− P+ = P+ P− = 0, . P+ + P− = I.

(7.11) (7.12)

The operator Z has the form  Z=

.

   n−2 n−2 + B P+ + − B P− . 2 2

(7.13)

Proof This follows from the equation .

  n−2 2 Z− I = I B 2. 2

(7.14)

The spectrum of the operator .L2 is given by the spectrum of the scalar Laplacian on n−1 (6.105) .S   λj −L2 = j (j + n − 2),

.

j = 0, 1, 2, . . . .

(7.15)

Therefore, the spectrum of the operator B is λj (B) = j +

.

n−2 , 2

j = 0, 1, 2, . . .

(7.16)

7.1 Spinor Heat Trace

115

with the same eigenfunctions and multiplicities given by (6.112)   (j + n − 3)! dj (B) = dj −L2 = (2j + n − 2) . (n − 2)!j !

.

(7.17)

Proposition 7.3 The operator Z does not have eigenvalue equal to .(n − 2). Proof Suppose there is a spinor .ϕ such that .Zϕ = (n − 2)ϕ. Then we have L2 ϕ = −Z[Z − (n − 2)I ]ϕ = 0.

.

(7.18)

Therefore, .ϕ is constant and .Zϕ = ϕ = 0. Proposition 7.4 For any analytic function f    n−2 n−2 + B P+ + f − B P− , 2 2

 f (Z) = f

.

(7.19)

and, therefore,        

n−2 n−2 n−2 n−2 1 B− f +B + B + f −B .Tr f (Z) = NTr 2B 2 2 2 2  ∞   j +n−2 {f (j + n − 1) + f (−j )} . (7.20) =N n−2 j =0

Thus, the operator Z has two series of eigenvalues: all nonpositive integers, λ− j (Z) = −j,

j = 0, 1, 2, . . . ,

.

(7.21)

with the multiplicities dj− (Z) = N



.

 j +n−2 , n−2

(7.22)

and the positive integers greater than .(n − 2), λ+ j (Z) = j + n − 1,

.

j = 0, 1, 2, . . . ,

(7.23)

with the same multiplicities + .d (Z) j

 =N

 j +n−2 . n−2

(7.24)

116

7 Spinor Heat Kernel

Proof This follows from (7.13). Proposition 7.5 The spinor Laplacian on .S n has the form n

ΔS1/2 =

.

  n−1 (1 − x 2 )∂x2 − nx∂x + I (7.25) 4

1 1 −Z 2 + (n − 2)Z + −Z 2 + nZ − (n − 1)I . + 2(1 − x) 2(1 + x) 1 a2

Proof This follows from (4.126) by using (7.2) and the algebra (1.47) of the generators Σab .

.

Proposition 7.6 Let .α, β > −1 be real numbers and ρ = (1 − x)α (1 + x)β .

(7.26)

.

Then n a 2 ρ −1/2 ΔS1/2 ρ 1/2 = (1 − x 2 )∂x2 + [β − α − (β + α + n)x]∂x

.

(7.27)

1 n(n − 1) I − (α + β + n − 1)2 + 4 4

1 + α(α + n − 2)I − Z 2 + (n − 2)Z 2(1 − x)

1 β(β + n − 2)I − Z 2 + nZ − (n − 1)I . + 2(1 + x) Proof This is proved by direct calculation using the intertwining relation of the hypergeometric operators (1.134). The idea is now to kill the terms .(1 − x)−1 and .(1 + x)−1 by choosing .α and .β. To simplify notation, we will omit below the unit matrix I ; this should not cause any confusion. Proposition 7.7 Let .μ and .ν be the operators defined by μ = B.

(7.28)

ν = (−1 + B) P+ + (1 + B) P− ,

(7.29)

.

and .ρ be the operator n

n

ρ = (1 − x)μ+1− 2 (1 + x)ν+1− 2 .

.

(7.30)

7.1 Spinor Heat Trace

117

The spinor Laplacian has the form Sn .Δ1/2



1 1/2 1 2 Hμ, ν + n(n − 1) − (ν + μ + 1) ρ −1/2 I, = 2ρ 4 a

(7.31)

where .Hμ,ν is the hypergeometric differential operator defined by (1.110). Proof Let .α = μ + 1 − that

n 2

and .β = ν + 1 − n2 . By using (7.13), it is not difficult to show

  n−2 2 μ2 = Z − , 2

.

 n 2 ν2 = Z − , 2

(7.32)

and, therefore, .

α(α + n − 2) = Z 2 − (n − 2)Z, .

(7.33)

β(β + n − 2) = Z 2 − nZ + (n − 1)I.

(7.34)

Now, the result follows from Eq. (7.27). Proposition 7.8 The eigenvalues of the spinor Laplacian (7.31) are   1 Sn .λk −Δ1/2 = a2



ν+μ+1 k+ 2

2

 n(n − 1) − , 4

(7.35)

with .k = 0, 1, 2, . . . . Proof This follows from Proposition 1.7 and Eq. (7.31). Proposition 7.9 The spinor heat trace has the form  

  ∞    n 2 n(n − 1) k+n−1 t  n Tr exp tΔS1/2 = N 2 exp − 2 k + − (. 7.36) 2 4 n−1 a

.

k=0

Proof The heat trace is    2 ∞    1 t n(n − 1) Sn l + (ν + μ + 1) − .Tr exp tΔ1/2 = Tr exp − 2 .(7.37) 2 4 a l=0

We have .

1 (ν + μ + 1) = BP+ + (B + 1) P− . 2

(7.38)

118

7 Spinor Heat Kernel

Therefore, by using the projections .P± , (7.8), we get



∞     1  n−2 n(n − 1) t n 2 Tr B − exp − Tr exp tΔS1/2 = N exp t + B) (l 2B 2 4a 2 a2

.

+

∞   l=0

l=0





t n−2 2 exp − 2 (l + B + 1) B+ . 2 a

(7.39)

Next, by using the spectrum of the operator B, we compute .



   2  ∞    j + n − 3 t n − 2 n(n − 1) n exp − 2 j + Tr exp tΔS1/2 = N exp t n−2 2 4a 2 a j =1

+

∞  ∞   l=1 j =0

j +n−3 n−2



 +

j +n−2 n−2





t exp − 2 a

  

n−2 2 l+j + 2



∞

  k  n(n − 1)  j +n−2 n 2 t  = N exp t 2 . exp − 2 k + n−2 2 4a 2 a

(7.40)

j =0

k=0

Finally, by using the identity [37]

.

k    j j =0

m

 =

 k+1 , m+1

(7.41)

we compute    k   j +n−2 k+n−1 = . . n−2 n−1

(7.42)

j =0

The result follows.

7.2

Spinor Heat Kernel

As usual we use the geodesic coordinates .y i centered at a fixed point .x  of the manifold and introduce the radial coordinate r and the angular coordinates .θ i . Proposition 7.10 The spinor heat kernel .U1/2 (t; r) and the spinor resolvent .G1/2 (λ; r) in the standard basis do not depend on the angular coordinates .θ i , that is, they depend only on t, .λ, and the radial coordinate r.

7.2 Spinor Heat Kernel

119

Proof This follows from the fact that the spinor Laplacian (7.25) does not depend on the angular coordinates. Alternatively, one can show this directly as follows. It is well known that the antisymmetrized Dirac matrices .γa1 ...ak , defined by (1.44), together with the unit matrix I form a basis in the space of complex .N × N matrices, that is, every such matrix has the form [42] F (r, θ ) = F(0) (r, θ )I +

.

n  1 a1 ...ak F (r, θ )γa1 ...ak , k! (k)

(7.43)

k=1

where the coefficients .F a1 ...ak are some anti-symmetric tensors. Since .F0 is a scalar, it cannot depend on the angular coordinates .θ i , that is, .F(0) (r, θ ) = F˜(0) (r). Since the polynomials .θ a1 · · · θ ak are symmetric tensors, the only non-zero anti-symmetric tensor a (r, θ ) = F˜ (r)θ a . Therefore, the general form of the spinor heat kernel reads is .F(1) (1) U1/2 (t; y) = f1 (t; r)I + f2 (t; r)γa θ a ,

.

(7.44)

where .f1 and .f2 are functions of t and r only. The initial conditions for the heat kernel mean that the functions .f1 and .f2 have the following asymptotics as .t → 0: −n/2

f1 (t; r) ∼ (4π t)

.



r2 exp − 4t

 ,.

f2 (0; r) = 0.

(7.45) (7.46)

By using the explicit form of the Laplacian (7.25) and Eq. (7.3), we obtain two separate heat equations for the functions .f1 and .f2 . The solution of the heat equation for .f2 with the zero initial condition is zero, .f2 (t, r) = 0. The statement follows. Therefore, the spinor resolvent on .S n satisfies the equation .

  n An + a 2 λ GS1/2 (λ; r) = 0,

(7.47)

where An = (1 − x 2 )∂x2 − nx∂x −

.

n−1 n−1 + . 2(1 + x) 4

(7.48)

Proposition 7.11 The operator .An can be intertwined as follows:  n (1 + x)−1/2 , An = (1 + x)1/2 Mn − 4

.

where the operator .Mn = H n2 −1,

n 2

is defined by (1.110),

(7.49)

120

7 Spinor Heat Kernel

Mn = (1 − x 2 )∂x2 + [1 − (n + 1)x)]∂x .

(7.50)

.

Proof This follows from (1.134). Let .β be a positive real number. We intertwine the operator .An as follows: .

(1 + x)−β/2 An (1 + x)β/2 = (1 − x 2 )∂x2 + [β − (β + n)x]∂x

(7.51)

1 n(n − 1) 1 − (β + n − 1)2 + + [β(β + n − 2) − (n − 1)] . 4 4 2(1 + x) Now, by choosing .β = 1, we get (1 + x)−1/2 An (1 + x)1/2 = (1 − x 2 )∂x2 + [1 − (n + 1)x]∂x −

.

n . 4

(7.52)

The statement follows. Thus, the spinor resolvent on the sphere .S n has the form 

n

GS1/2 (λ; r) =

.

1+x 2

1/2

˜ S n (λ; r), G 1/2

(7.53)

˜ S satisfies the equation where .G 1/2 n

.

  n n ˜ S (λ; r) = 0. Mn − + a 2 λ G 1/2 4

(7.54)

Proposition 7.12 The spinor resolvent on .S n has the form        Γ ω + n2 Γ −ω + n2 1 + x 1/2 n n 1+x n n  , −ω + ; + 1; , F ω + 2 2 2 2 2 a n−2 (4π )n/2 Γ 2 + 1 (7.55)  n(n−1) 2 where .ω = i −a λ − 4 , with .Im ω > 0.

Sn .G1/2 (λ; r) =

(3)

Proof The solution of Eq. (7.54) is given by the function .Φμ,ν (x) with .μ = n2 − 1 and n .ν = 2 . Then the solution of Eq. (7.47) that is finite at .x = −1 (that is, .r = aπ ) and has the correct asymptotics as .x → 1− (that is, as .r → 0) is given by (7.55). In the case of odd dimension n, by using the transformation property of the hypergeometric function [25] F (a, b; c; 1 − z) = zc−a−b

.

+

Γ (c)Γ (a + b − c) F (c − a, c − b; c − a − b + 1; z) Γ (a)Γ (b)

Γ (c)Γ (c − a − b) F (a, b; a + b − c + 1; z), Γ (c − a)Γ (c − b)

(7.56)

7.2 Spinor Heat Kernel

121

it can also be written in the form Sn .G1/2 (λ; r)

=

Γ ( n2 − 1)



n

a n−2 (4π) 2 +

1+x 2

1/2 

1−x 2

1− n 2



n 1−x F ω + 1, −ω + 1; 2 − ; 2 2



 

Γ (1 − n2 ) Γ (ω + n2 )Γ (−ω + n2 ) n n n 1−x F ω + , −ω + ; ; . (7.57) Γ ( n2 − 1) Γ (ω + 1)Γ (−ω + 1) 2 2 2 2

In the even dimension n, a similar result can be obtained by the analytic continuation in n. The operator .Mn = H n2 −1, n2 has a well known spectrum (see Proposition 1.7). Its ( n −1, n ) eigenfunctions are the Jacobi polynomials .Pk 2 2 (x), where .k = 0, 1, 2, . . . , with the simple eigenvalues λk (−Mn ) = k(k + n) .

(7.58)

.

( n −1, n ) Therefore, the eigenfunctions of the operator .An are .(1 + x)1/2 Pl 2 2 (x) with the eigenvalues λk (−An ) = k(k + n) +

.

n . 4

(7.59)

Proposition 7.13 The spinor resolvent on .S n has simple poles at the eigenvalues of the spinor Laplacian λk =

.

n 1  , k + n) + (k 4 a2

k = 0, 1, 2, . . .

(7.60)

with the residues 2Γ (k + n)   (4π a 2 )n/2 Γ n2 k!

n

Res {GS1/2 (λ; r), λk } = −

.



1+x 2

1/2

  n 1−x . F −k, k + n; ; 2 2 (7.61)

Proof The resolvent is an even meromorphic function of .ω with simple poles at .±ωk , where ωk = k +

.

n , 2

k = 0, 1, 2, . . . .

(7.62)

Therefore, as a function of .λ, the resolvent has simple poles at the eigenvalues (7.60). The residues are computed from (7.55) by using the residues of the gamma function to get

122

7 Spinor Heat Kernel

  1 + x 1/2 (2k + n)Γ (k + n) n  2 (4π a 2 )n/2 Γ 2 + 1 k!   1+x n . ×F −k, k + n; + 1; 2 2

n

Res {GS1/2 (λ; r), λk } = −(−1)k

.

(7.63)

Now, by using (6.109), we get the result. Notice that the residues can be expressed in terms ( n −1, n2 )

of the Jacobi polynomials .Pk 2

(x).

Proposition 7.14 The spinor heat kernel on .S n has the form Sn .U1/2 (t; r)

  ∞  Γ ( n+1 n k+n−1 t 2 ) = 2 exp − 2 k(k + n) + 4 n−1 a 2π (n+1)/2 a n k=0



1+x × 2

1/2

  n 1−x . F −k, k + n; ; 2 2

(7.64)

Proof The heat kernel is computed by using Eq. (1.10). We use the property of the gamma function 22z−1 Γ (2z) = √ Γ (z)Γ π

.

  1 z+ 2

(7.65)

to get the result. Proposition 7.15 The spinor heat kernel diagonal and the spinor heat trace on .S n have the form Sn .U1/2,diag (t; r)

  ∞  Γ ( n+1 n k+n−1 t 2 ) ,. = 2 exp − 2 k(k + n) + 4 n−1 a 2π (n+1)/2 a n k=0

(7.66)

  ∞    n k+n−1 t Sn . Tr exp tΔ1/2 = N 2 exp − 2 k(k + n) + 4 n−1 a

(7.67)

k=0

Proof The heat kernel diagonal is obtained by setting .r = 0, that is, .x = 1, and the heat trace is obtained by multiplying by the volume of .S n and taking the spinor trace. Let us consider now the hyperbolic space .H n so that the radius .a = ib is imaginary with b real so that .x = cosh(r/b); that is, x varies in the interval .[1, ∞). The spinor resolvent on .H n has the form Hn .G1/2 (λ; r)

 =

1+x 2

1/2

n ˜H G 1/2 (λ; r),

(7.68)

7.2 Spinor Heat Kernel

123

˜ H n satisfies the equation where .G 1/2 .

  n n ˜H Mn − − b 2 λ G 1/2 (λ; r) = 0. 4

(7.69)

Proposition 7.16 The spinor resolvent for the hyperbolic space .H n has the form Hn .G1/2 (λ, r)

     Γ iν + n2 Γ −iν + n2 1 + x 1/2 n = 2 (4π )n/2 bn−2 Γ 2    n n n n 1 + x + iε 2 F iν + , −iν + ; + 1; × − exp −iπ n 2 2 2 2 2  

e−π ν n n n 1−x +i (7.70) F iν + , −iν + ; ; , ν 2 2 2 2

 where .ν = i −b2 λ +

n(n−1) 4 ,

with .Im ν > 0. (4)

Proof The resolvent is given by the function .Φμ,ν (x) (1.114) with .μ = n2 − 1 and .ν = n2 . For large real negative .λ, that is, imaginary .ν with .Im ν > 0, the resolvent is bounded at infinity, as .r → ∞, or .x → ∞, and has the correct asymptotics (6.9) as .r → 0, that is, + .x → 1 . For odd dimension .n ≥ 3, this can be rewritten in the form n

GH 1/2 (λ, r) =

.

  Γ ( n2 − 1) 1 + x 1/2 (7.71) 2 (4π )n/2 bn−2    n  x − 1 − 2 +1 n 1−x 1 F iν + , −iν + 1; − + 2; × 2 2 2 2  

n n n n n 1−x i Γ (− 2 + 1) Γ (−iν + 2 ) F iν + , −iν + ; ; . + ν Γ ( n2 − 1) Γ (−iν − n2 + 1) 2 2 2 2

Proposition 7.17 The spinor resolvent is an analytic function of .λ with a branch cut along to .∞. The jump of the resolvent across the cut is the positive real axis from .c = n(n−1) 4b2 (with positive real .ν) .

n

n

n

H H Jump GH (7.72) 1/2 (λ; r) = G1/2 (λ + iε; r) − G1/2 (λ − iε; r)   1/2   1+x n 1˜ n−1 n n 1−x 2πi   T ν, F iν + , −iν + ; ; , = 2 2 2 2 2 2 (4π)n/2 bn−2 Γ n2 ν

where .T˜ (ν, α) is a function defined by (6.125).

124

7 Spinor Heat Kernel

Proof The upper side of the cut .λ + iε corresponds to positive real .ν and the lower side of the cut .λ − iε corresponds to the real negative .ν. By using (1.125), we obtain Eq. (7.72). In odd dimension .n ≥ 3, the function .T˜ (ν, n−1 2 ) is a polynomial (6.132) of degree .(n − 1), and for even dimension .n ≥ 2, it has the form     1 n−2 n−1 = coth(π ν)T ν, , T˜ ν, 2 ν 2

.

(7.73)

where .T (ν, n−2 2 ) is a polynomial (6.131) of degree .n − 2. Proposition 7.18 The spinor heat kernel on the hyperbolic space .H n has the form Hn .U1/2 (t; r)

   1 + x 1/2 n(n − 1) t   exp − = 4 2 b2 (4π b2 )n/2 Γ n2 2

∞ × 0

(7.74)

      n n n 1−x t 2 n−1 ˜ F −iν + , iν + ; ; exp − 2 ν . dν T ν, 2 2 2 2 2 b

Proof This is obtained by using Eq. (1.16). Proposition 7.19 The spinor heat kernel diagonal on the hyperbolic space .H n is     t n(n − 1) t Hn B , U1/2,diag (t) = (4π t)−n/2 exp − 1/2 2 4 b b2

.

(7.75)

where 2t n/2   .B1/2 (t) = Γ n2

∞

    n−1 exp −tν 2 . dν T˜ ν, 2

(7.76)

0

Proof This is obtained by substituting .x = 1 in Eq. (6.138). Notice that in odd dimension  n ≥ 3 the function .T˜ ν, n−1 is a polynomial of degree .(n − 1), and, therefore, the 2

.

function .B1/2 (t) is a polynomial of degree . n−1 2 .

8

Heat Kernel in Two Dimensions

8.1

Heat Kernel on S 2

The sphere S 2 is the homogeneous space S 2 = SO(3)/SO(2). We denote the  normal 2 1 2 geodesic coordinates on S by y and y and define the radial coordinate r = yi y i and the angular coordinate ϕ. The coordinates range over 0 ≤ r ≤ aπ and 0 ≤ ϕ ≤ 2π. That is, θ 1 = cos ϕ,

θ 2 = sin ϕ.

(8.1)

The isotropy algebra so(2) has only one generator G12 = G and there is only one operator L12 = L = ∂ϕ . Furthermore, the quantity W 2 = Wa Wa (4.128) is equal to W 2 = G2 . Notice that the eigenvalues of the matrix iG are either integer or half-integer so that the matrix  β = 2 −G2

(8.2)

has non-negative integer eigenvalues. For an irreducible representation, it is just proportional to the unit matrix. Therefore, the Laplacian (4.126) has the form 2

ΔS = ∂r2 +

   r  2 r  1 1 L + 1 − cos cot ∂r + G , a a a a 2 sin2 (r/a)

(8.3)

or, by using the variable x = cos(r/a), 2

ΔS =

  1 1 1 2 2 2 2 2 )∂ − 2x∂ − G + + L + 2G) (1 − x .(8.4) (L x x 2(1 − x) 2(1 + x) a2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_8

125

126

8 Heat Kernel in Two Dimensions

Since the coefficients of the Laplacian do not depend on ϕ, then the heat kernel and the resolvent do not depend on the angular coordinate ϕ; they satisfy the equations 

 2 a 2 ∂t − X U S (t, y) = 0, 2

(X + a 2 λ)GS (λ; y) = 0,

(8.5) (8.6)

where X = (1 − x 2 )∂x2 − 2x∂x +

1−x 2 G . 1+x

(8.7)

Proposition 8.1 The operator X can be intertwined as follows:

β (1 + x)−β/2 , X = (1 + x)β/2 H0,β − 2

(8.8)

where Hμ,ν is the hypergeometric operator defined by (1.110). Proof This follows from (1.134). Notice that the operator X can be expressed in terms of a hypergeometric operator X = H0,0 −

β2 β2 + . 2(1 + x) 4

(8.9)

By using the intertwining relation (1.134), we have (1 + x)

−β/2

H0,0 −



β2 β β β/2 (1 + x) +1 . = H0,β − 2(1 + x) 2 2

(8.10)

This gives the result. Recall (Proposition 1.7) that the eigenfunctions of the operator H0,β are the Jacobi (0,β) (x) with the eigenvalues polynomials Pk λk (−H0,β ) = k(k + β + 1),

k = 0, 1, 2, . . . .

(8.11)

The resolvent has the form S2

G (λ, r) =



1+x 2

β/2

˜ S 2 (λ; r), G

(8.12)

8.1 Heat Kernel on S 2

127

˜ S 2 (λ, r) satisfies the equation where G

β ˜ S2 H0,β + a 2 λ − G (λ, r) = 0. 2

(8.13)

Proposition 8.2 The resolvent on S 2 reads S2

G (λ, r) =

 Γ ω+

β+1 2



 Γ −ω +

β+1 2



4π Γ (β + 1)



β +1 1+x 1 + x β/2 β +1 , −ω + ; β + 1; , × F ω+ 2 2 2 2

 where ω = i −a 2 λ −

1 4



(8.14)

β 2 + 1 with Im ω > 0.

Proof This follows from Proposition 1.6. Notice that this coincides with (6.50) for β = 0 and with (7.55) for β = 1 and n = 2. Proposition 8.3 The resolvent is a meromorphic function of λ with simple poles at the eigenvalues of the Laplacian 2

λk (−ΔS ) =

  β 1 k(k + β + 1) + , 2 a2

k = 0, 1, 2, . . .

(8.15)

with the residues 2

Res {GS0 (λ; r), λk } = −



1 + x β/2 (0,β) 1 Pk (x). + β + 1) (2k 2 4π a 2

(8.16)

Proof Note that the resolvent is an even meromorphic function of ω with simple poles at ±ωk , where ωk =

β +1 + k, 2

k = 0, 1, 2, . . . .

(8.17)

This means that, as a function of λ, the resolvent has simple poles at λk =

  β 1 k(k + β + 1) + , 2 a2

k = 0, 1, 2, . . .

(8.18)

128

8 Heat Kernel in Two Dimensions

with the residues (−1)k Γ (k + β + 1) (2k + β + 1) Γ (β + 1) 4π a 2 k!

β/2

1+x 1+x . (8.19) × F −k, k + β + 1; β + 1; 2 2

2

Res {GS0 (λ; r), λk } = −

Now, by using the relation (6.58) and the properties of the gamma function, we obtain F (−k, k + c; c; 1 − z) = (−1)k

k!Γ (c) F (−k, k + c; 1; z), Γ (c + k)

(8.20)

which gives (8.16) by using (1.132). Proposition 8.4 The heat kernel on the sphere S 2 has the form 2

U S (t, r) =

 

 ∞ 1 + x β/2 (0,β) t 1  β + β + 1) exp − Pk (x). k(k + β + 1) + (2k 2 2 4πa 2 a2 k=0

(8.21) Proof The heat kernel is obtained by the inverse Laplace transform (1.8) and (1.10) with the eigenvalues λk (8.15) and the residues (8.16) computed above. Proposition 8.5 The heat kernel diagonal and the heat trace on the sphere S 2 have the form 2

S Udiag (t) =

   ∞ β 1  t k(k + β + 1) + ,(8.22) + β + 1) exp − (2k 2 4π a 2 a2 k=0

   ∞    β t S2 , = tr (2k + β + 1) exp − 2 k(k + β + 1) + Tr exp tΔ 2 a

(8.23)

k=0

where tr is the fiber trace. Proof The heat kernel diagonal is obtained by setting r = 0, i.e., x = 1, and the heat trace is obtained by taking the fiber trace and by multiplying by the volume of the sphere S2. This can be written in the form 2

U S (t) =





 t r t 1 1 + x β/2 2 exp , (β + 1) Ψ , 2,β 4π t 2 4a 2 a2 a

(8.24)

8.1 Heat Kernel on S 2

129

where  

 ∞  β +1 β +1 2 (0,β) Ψ2,β (t, r) = 2t (x). Pk k+ exp −t k+ 2 2

(8.25)

k=0

Proposition 8.6 The heat kernel diagonal on S 2 has the dual representation in the form  

t t 1 2 exp (β + 1) Ψ , 0 , 2,β 4π t 4a 2 a2

2

S Udiag (t) =

(8.26)

where  Ψ2,β (t; 0) = R



 ω ω/2 ω2 dω cos β exp − . √ 2 4t 4π t sin (ω/2)

(8.27)

Proof The function Ψ2,β has the form 

 ∞  β +1 2 β +1 exp −t k + Ψ2,β (t; 0) = t 2 k+ . 2 2

(8.28)

k=0

We rewrite this function in the form Ψ2,β (t, 0) = −i



  β +1 dω ω2 +i k+ ω , ω exp − √ 4t 2 4π t

∞   k=0C +

(8.29)

where the contour C+ goes just above the real axis so that Im ω > 0. We sum the series ∞  k=0

  i exp(iβω/2) β +1 ω = exp i k + 2 2 sin (ω/2)

(8.30)

for Im ω > 0. Therefore, we obtain  Ψ2,β (t, 0) = C+



ω/2 ω2 ω dω exp − + iβ . √ 4t 2 4π t sin (ω/2)

(8.31)

By deforming the contour of integration, one can show that the value of the integral for the contour just below the real axis is the same, and, therefore, it is determined by the Cauchy principal value of the integral. This integral can be further transformed to the sum

130

8 Heat Kernel in Two Dimensions

Ψ2,β (t, 0) =

2π

∞ 

(−1)

k=−∞

k(β+1) 0

 ω dω (ω + 2π k)/2 cos β √ 2 4π t sin (ω/2)

  1 × exp − (ω + 2π k)2 . 4t

(8.32)

Note that for β = 0 this function coincides with the function Ψ2 (t, 0) (6.80) Ψ2,0 (t, 0) = Ψ2 (t, 0).

(8.33)

This function can also be written as a contour integral  Ψ2,β (t, 0) = t



   β dν iν tan π ν − exp −tν 2 , 2

(8.34)

Γβ

where Γβ is a contour that goes counterclockwise from iε + ∞ to β+1 the point β+1 2 to the point 2 − iε, and finally to −iε + ∞.

β+1 2

+ iε, then around

Proposition 8.7 The function Ψ2,β (t, 0) has the asymptotic expansion as t → 0 Ψ2,β (t, 0) ∼

∞ 

bk (β)

k=0

tk , k!

(8.35)

where bk (β) = (−1)k B2k

β +1 , 2

(8.36)

and Bk (x) are Bernoulli polynomials [25]. √ Proof By rescaling the integration variable ω → 2 tω, we obtain 1 Ψ2,β (t; 0) = √ π



 √t ω exp iβ √t ω √ dω exp −ω . sin t ω 

2

(8.37)

C+

By using the generating function of the Bernoulli polynomials [25], ∞

 zk zexz = Bk (x), z e −1 k! k=0

(8.38)

8.2 Heat Kernel on H 2

131

and computing the Gaussian integral, we obtain (8.36). Notice that the coefficients bk (β) coincide with the coefficients ak (6.82) for β = 0, that is, bk (0) = ak , by using [25], B2k

8.2

β +1 2

=2

1−2k



β B2k (β) − B2k . 2

(8.39)

Heat Kernel on H 2

Next, we consider the hyperbolic plane H 2 , so that the radius a = ib is imaginary with b real; recall also that now x = cosh(r/b). The resolvent satisfies the equation

β ˜ H2 2 H0,β − b λ − G (λ, r) = 0. 2

(8.40)

Proposition 8.8 The resolvent on the hyperbolic plane H 2 has the form H2

G

×

1 (λ, r) = 4π

  Γ iν +

β+1 2





1+x 2

β/2 (8.41)

  Γ −iν + β+1 2

Γ (β + 1)



β +1 β +1 1 + x + iε F iν + , −iν + ; β + 1; 2 2 2

  

 exp iπ iν + β2 β +1 1−x β +1   F iν + −iπ , −iν + ; 1; , 2 2 2 cos π(iν + β2 )  where ν = i −b2 λ +

1 4



β 2 + 1 with Im ν > 0.

Proof The solution of the differential equation for the resolvent (8.40), which is bounded as x → ∞ and has the correct singularity as x → 1+ , is given by Proposition 1.6 and has the form (8.41). For β = 0, this formula coincides with the scalar case on H 2 (6.84), and for β = 1, it coincides with the spinor case (7.70) for n = 2. 2

Proposition 8.9 The resolvent GH (λ, r) is an analytic function of λ with a cut along the real axis from c = 14 (β 2 + 1) to ∞. The jump of the resolvent across the cut is 2

2

2

Jump GH (λ; x) = GH (λ + iε; x) − GH (λ − iε; x) (8.42) 



β/2

1+x β +1 i β β +1 1−x F iν + = tanh π ν + i , −iν + ; 1; . 2 2 2 2 2 2

132

8 Heat Kernel in Two Dimensions

Proof The resolvent is an analytic function of λ with a cut along the real axis from c = 1 2 4 (β + 1) to ∞. By using the property (1.125), we compute the jump of the resolvent. It also easily follows from (8.41). Proposition 8.10 The heat kernel on the hyperbolic plane H 2 has the form 2

U H (t; r) =





 t r t 1 1 + x β/2 exp − 2 (β 2 + 1) , , Ψ˜ 2,β 4π t 2 4b b2 b

(8.43)

where Ψ˜ 2,β (t, r) = t

  β dν ν tanh π ν + i 2

 R

(8.44)



  β +1 β +1 1−x ×F iν + , −iν + ; 1; exp −tν 2 . 2 2 2 Proof The heat kernel is computed by using Eq. (1.16). Proposition 8.11 The heat kernel diagonal on the hyperbolic plane H 2 has the form H2 Udiag (t)

 

t t 1 2 ˜ exp − 2 (β + 1) Ψ2,β = ,0 , 4π t 4b b2

(8.45)

where Ψ˜ 2,β (t, 0) = t

 R

    β exp −tν 2 , dν ν tanh π ν + i 2

= (4π t)−1/2

 dω R



 ω ω2 ω/2 cosh β exp − . sinh(ω/2) 2 4t

(8.46)

Proof Equation (8.46) follows from (8.44) by setting x = 1. Notice that Ψ˜ β (t, 0) is an analytic function of β. We compute it for imaginary β and then analytically continue to the real values of β. Furthermore, by using the integral representation (valid for real z) 1 tanh(π z) = 2π

 dω R

sin(zω) , sinh(ω/2)

(8.47)

8.2 Heat Kernel on H 2

133

we compute (for imaginary β) Ψ˜ 2,β (t, 0) = t

1 2π

 R

1 dω sinh(ω/2)

 R



   β dν ν sin ν + i ω exp −tν 2 . (8.48) 2

By computing the integral over ν, we obtain (8.46). For β = 0, this result coincides with (6.98) and (6.96), and for β = 1 it coincides with (7.75) for n = 2. This result is also dual to the result for the sphere S 2 given by (8.27) and (8.34) in the sense that Ψ˜ β (t, 0) = Ψ2 (−t, 0).

(8.49)

This function is not an analytic function of t, but it has the asymptotic expansion as t → 0+ Ψ˜ 2 β(t) ∼

∞  k=0

where the bk (β) are defined by (8.36).

(−1)k bk (β)

tk , k!

(8.50)

Heat Kernel on S 3 and H 3

9.1

9

Scalar Heat Kernel on S 3 and H 3

Since the case of three dimensions is rather special, we quote the result for n = 3 explicitly. Proposition 9.1 The scalar heat kernel on the hyperbolic space H 3 has the form 3 U0H (t; r)

−3/2

= (4π t)

  r r2 t exp − 2 − , b sinh(r/b) 4t b

(9.1)

with the diagonal value   t H3 U0,diag (t; r) = (4π t)−3/2 exp − 2 . b

(9.2)

Proof For n = 3, we have from (6.138) 3 U0H (t; r)

     ∞  1 t 3 t 2 2 = exp − 2 dν ν F −iν + 1, iν + 1; ; z exp − 2 ν , 2 2π 2 b3 b b 0

(9.3) where z = (1 − x)/2 = − sinh2 (r/2b. By using the relation [25] 

 r  3 sin(νr/b) 2 F −iν + 1, iν + 1; ; − sinh = 2 2b ν sinh(r/b)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_9

(9.4)

135

9 Heat Kernel on S 3 and H 3

136

and the integral ∞

 2   √π r dν ν sin(νr) exp −tν 2 = t −3/2 r exp − , 4 4t

(9.5)

0

we obtain (9.1). Proposition 9.2 The scalar heat kernel on the sphere S 3 has the form 3

U0S (t, r) =

 ∞ 1  sin[(k + 1)r/a] t (k + 1) k(k + 2) . exp − sin(r/a) 2π 2 a 3 a2

(9.6)

k=0

The scalar heat kernel diagonal and the heat trace are S3 U0,diag (t)

 ∞ 1  t 2 = (k + 1) exp − 2 k(k + 2) , 2π 2 a 3 a

(9.7)

k=0

 ∞    t 3 (k + 1)2 exp − 2 k(k + 2) . Tr exp tΔS0 = a

(9.8)

k=0

Proof This is obtained by using the relation of the hypergeometric function to Chebyshev polynomials 

3 1 − cos(r/a) (k + 1)F −k, k + 2; , 2 2

 =

sin[(k + 1)r/a] . sin(r/a)

(9.9)

Proposition 9.3 The scalar heat kernel on S 3 is 3 U0S (t; r)

−3/2

= (4π t)



t exp a2



 Ψ3,0

t r , a2 a

 ,

(9.10)

where

 ∞  (r + 2π k) (r + 2π k)2 Ψ3,0 (t; r) = exp − sin r 4t k=−∞

=

1 2π i

 dω

C− −C+

(9.11)

    ω/2 ω−r ω2 cot exp − . sin ω 2 4t

Proof It is easy to check that the heat kernel on H 3 (with the substitution a = ib) locally satisfies both the heat equation for S 3 and the initial condition. Thus, this function is a good

9.1 Scalar Heat Kernel on S 3 and H 3

137

candidate for the heat kernel on S 3 . However, this function is not well defined globally since it has a singularity at r → π a. A scalar function that is well defined on S 3 must be periodic in r with period 2π a. Therefore, the heat kernel on S 3 is obtained by replacing r → r + 2π ak and summing over k from −∞ to +∞. 3

U0S (t, r) =



3 U0H (t, r + 2π ak)

k∈Z

b→ia

.

(9.12)

Notice also that the scalar heat kernel on the sphere S 3 of radius a is related to the scalar heat kernel on the group SU (2) by 3

U0S (t, r) =

  8 SU (2) r t U , 2 4 . a3 0 a2 a

(9.13)

Proposition 9.4 The scalar heat kernel diagonal on S 3 has the form S3 U0,diag (t)

= (4π t)

−3/2



t exp a2



 Ψ3,0

 t ,0 , a2

(9.14)

where    ∞   2π 2 2 π2 2 1− k exp − k Ψ3,0 (t, 0) = t t

(9.15)

k=−∞



=−

   ω  ω2 ω2 dω 1− exp − , cot 2π i 2 2t 4t

C+  = Ω+ (t) − 2π tΩ+ (t)

(9.16)

with the function Ω+ (t) defined in (6.44). Proof We rewrite Eq. (9.10) in the form   2   ∞ r π2 2 Ψ3,0 (t; r) = exp − exp − k 4t t k=−∞    

π kr 2π k π kr r cosh − sinh × sin r t sin r t and take the limit r → 0; this gives (9.15). We also have

(9.17)

9 Heat Kernel on S 3 and H 3

138

Ψ3,0 (t; 0) = −

1 2π i

 dω C+

  ω2 . exp − 4t sin2 (ω/2) ω/2

(9.18)

Now, by integrating by parts, we get (9.15). Notice that Ψ3,0 (t, 0) = (1 − 2t∂t )Ψ1 (t, 0).

(9.19)

It is easy to see that it is also related to the function Ω+ (t) as noted.

Spinor Heat Kernel on S 3 and H 3

9.2

Recall that the dimension of the spinor representation in three dimensions is N = 2. Proposition 9.5 The spinor heat kernel on H 3 has the form H U1/2 (t; r) = (4π t)−3/2 3



  3 t r2 t r exp − − + . b sinh(r/b) 2b2 cosh2 (r/2b) 2 b2 4t (9.20)

Proof The spinor heat kernel on the hyperbolic space H 3 has the form given by (6.138) and (6.124) H3 U1/2 (t; r)

   1 3 t 1 + x 1/2 = exp − 2 2b 2 2π 2 b3 ∞ ×

(9.21)

      1 3 3 3 t 2 2 dν ν + F −iν + , iν + ; ; z exp − 2 ν , 4 2 2 2 b

0

where z = (1 − x)/2 = − sinh2 (r/2b). We use [25] F (a + 1, b + 1; c + 1; z) =

c ∂z F (a, b; c; z) ab

(9.22)

to get       1 1 3 3 3 1 1 1 2 F −iν + , iν + ; ; z = ∂z F −iν + , iν + ; ; z . ν + 4 2 2 2 2 2 2 2

(9.23)

9.2 Spinor Heat Kernel on S 3 and H 3

139

Next, we use [25]   r  1 1 1 cos(νr/b) F iν + , −iν − ; ; − sinh2 = , 2 2 2 2b cosh(r/2b)

(9.24)

to compute ∞

      3 1 3 3 t 2 2 F −iν + , iν + ; ; z exp − 2 ν dν ν + 4 2 2 2 b

0 2 −1/2

= −b t

√  2 1 r π 1 ∂r exp − . 2 sinh(r/b) cosh(r/2b) 4t

(9.25)

(9.26)

Now, computing the derivative gives the result. Proposition 9.6 The spinor heat kernel diagonal on the hyperbolic space H 3 is    t 3 t H3 1+ 2 . U1/2,diag (t) = (4π t)−3/2 exp − 2 2b 2b

(9.27)

Proof This follows from Eq. (7.75) or directly from (9.20). Proposition 9.7 The spinor heat kernel on the sphere S 3 has the form    ⎫   ⎧ 3  sin k + 3 r ∞ ⎨ ⎬ cos k +  2 2 r 3 1 1 S3 k + − U1/2 (t; r) = ⎩ 2 sin r 4 2π 2 a 3 cos2 (r/2) ⎭ k=0

  3 t . × exp − 2 k(k + 3) + 4 a

(9.28)

The spinor heat kernel diagonal and the spinor heat trace are S3 U1/2,diag (t)

  ∞ 3 1  t , (9.29) = (k + 2)(k + 1) exp − 2 k(k + 3) + 4 2π 2 a 3 a k=0

  3 Tr exp tΔS1/2 = 2

∞  k=0

  3 t . (k + 2)(k + 1) exp − 2 k(k + 3) + 4 a

(9.30)

Proof The spinor heat kernel on the sphere is given by Eqs. (7.64), (7.66), and (7.67), 3

S U1/2 (t; r) =

  ∞ 3 t 1  r  k(k + 3) + exp − , ϕ k a 4 2π 2 a 3 a2 k=0

(9.31)

9 Heat Kernel on S 3 and H 3

140

where ϕk (r) = (k + 1)(k + 2) cos

  r  3 F −k, k + 3; ; sin2 . 2 2 2

r 

(9.32)

It is well known that [25]     r  cos k + 12 r 1 F −k, k + 1, ; sin2 = . 2 2 cos(r/2) 

(9.33)

Furthermore, we use (9.22) to get     r  cos k + 12 r 3 cos(r/2) k(k + 1) cos F −k + 1, k + 2; ; sin2 =− ∂r . 2 2 2 sin r cos(r/2) (9.34) Therefore, 

r 



3 ϕk (r) = k + 2

 sin



k+

3 2

sin r

  r

   3 1 cos k + 2 r , − 4 cos2 (r/2)

(9.35)

which proves (9.28). Also, by noticing that ϕk (0) = (k + 1)(k + 2), we get (9.29). Proposition 9.8 Let       1 1 2 f (t, r) = cos k + r exp −t k + , 2 2 k=0         ∞   1 1 2 1 g(t, r) = sin k + r exp −t k + k+ . 2 2 2 ∞ 



(9.36)

(9.37)

k=0

Then the following duality relations hold: f (t, r) = (4π t)−1/2 π

   (r + 2π k)2 (−1)k exp − , 4t

(9.38)

k∈Z

g(t, r) = (4π t)−1/2

  π  (r + 2π k)2 (−1)k (r + 2π k) exp − . 2t 4t

(9.39)

k∈Z

Proof Notice that g(t, r) = −∂r f (t, r). Also, these functions are anti-periodic, they change sign under the transformation r → r + 2π.

9.2 Spinor Heat Kernel on S 3 and H 3

141

We rewrite the function f as an integral f (t, r) = (4π t)−1/2

 dω

∞ 

cos

k=0

C+

        1 1 ω2 k+ r exp iω k + exp − . 2 2 4t (9.40)

This series can be summed up for Im ω > 0, 

      1 i 1 1 1 cos k + r = + , exp iω k + 2 2 4 sin[(ω + r)/2] sin[(ω − r)/2] k=0 (9.41) to get ∞ 

f (t, r) =

i (4π t)−1/2 4



 dω

C+

  1 ω2 1 + exp − . sin[(ω + r)/2] sin[(ω − r)/2] 4t (9.42)

We compute the integrals by the residues  dω C+

     ω2 1 (r + 2π k)2 exp − = −2π i (−1)k exp − sin[(ω − r)/2] 4t 4t k∈Z

 +

R

dω

  ω2 1 exp − , sin[(ω − r)/2] 4t

(9.43)

to obtain f (t, r) = (4π t)−1/2 π

   (r + 2π k)2 (−1)k exp − 4t k∈Z

i + (4π t)−1/2 4

 R

dω

(9.44)

   1 ω2 1 + exp − . sin[(ω − r)/2] sin[(ω + r)/2] 4t

The integrand is an odd function and, hence, the integral vanishes. Thus, finally, we obtain f (t, r) = (4π t)

−1/2

   (r + 2π k)2 k , π (−1) exp − 4t

(9.45)

k∈Z

which also enables one to compute the function g g(t, r) = (4π t)−1/2

  π  (r + 2π k)2 . (−1)k (r + 2π k) exp − 2t 4t k∈Z

(9.46)

9 Heat Kernel on S 3 and H 3

142

Proposition 9.9 The spinor heat kernel on the sphere S 3 can be written in the dual form S3 U1/2 (t; r)

−3/2

= (4π t)



3 t exp 4 a2



 Ψ3,1/2

t r , a2 a

 ,

(9.47)

where     (r + 2π k)2 t 3  k r + 2π k exp − Ψ3,1/2 (t, r) = exp (−1) t − . 4 sin r 4t 2 cos2 (r/2) k∈Z (9.48) 

Proof We have 

3 Ψ3,1/2 (t, r) = exp t 4



 4 t 3/2 1 g(t, r) − tan(r/2)f (t, r) . √ 2 π sin r

(9.49)

By using (9.38) and (9.39), we obtain  Ψ3,1/2 (t, r) = exp

    (r + 2π k)2 t (r + 2π k) 3  exp − (−1)k t − . 4 sin r 4t 2 cos2 (r/2) k∈Z (9.50)

We notice, once again, that the spinor heat kernel on the sphere S 3 of radius a is obtained from the spinor heat kernel on H 3 by replacing the pseudo-radius b → ia to an imaginary one and making it anti-periodic by replacing r → r + 2π ak and summing over k with the weight (−1)k , that is, 3

S U1/2 (t, r) =

 H3 (−1)k U1/2 (t, r + 2π ak)

b→ia

k∈Z

.

(9.51)

Notice that the value of the function f (t, r) at r = 0 is related to the function Ω− (t) defined by (6.45), f (t, 0) = (4π t)−1/2 π Ω− (t).

(9.52)

Proposition 9.10 The spinor heat kernel diagonal on the sphere S 3 has the form S U1/2,diag (t) = (4π t)−3/2 exp 3



3 t 4 a2



 Ψ3,1/2

 t , 0 , a2

(9.53)

9.3 Heat Trace

143

where      t π 2k2 3  π 2k2 exp − (−1)k 1 − − 2 t 4 2 t t k∈Z     t 3  t 1− Ω− (t) − 2tΩ− = exp (t) . 4 2 

Ψ3,1/2 (t, 0) = exp

(9.54)

Proof We rewrite the function Ψ3,1/2 in the form      r t π kr 3  k t − cosh Ψ3,1/2 (t, r) = exp (−1) 4 sin r t 2 cos2 (r/2) 

k∈Z

   2  π kr r + 4π 2 k 2 2π k sinh exp − − sin r t 4t

(9.55)

and compute the limit r → 0 to get the result.

9.3

Heat Trace

Let T be an irreducible representation of the group SU (2) labeled by an integer (or halfinteger) j with generators Ta forming the algebra su(2) (3.9) with the Casimir operator T 2 = Ta Ta = −j (j + 1)I. The Laplacian is given by (5.45) 3

ΔSj =

 1  2 2 2K , + 2K + j (j + 1)I j + − a2

(9.56)

2 = K± K± and K± are the Lie derivatives (5.41) and I is the unit matrix. where K± j a a a On the sphere S 3 , there exists the right-invariant basis (5.21) and (5.22). The Laplacian in this basis is given by (5.55) 3 Δ˜ Sj =

 1  ˜2 ˜2 + j (j + 1)I , 2 K + 2 K j + − a2

(9.57)

where K˜± a are the Lie derivatives in this basis given by (5.50) and (5.51). The Laplacians in these bases are related by (5.66), and, therefore, the heat semigroups are related by     3 3 exp tΔSj = O exp t Δ˜ Sj O −1 ,

(9.58)

9 Heat Kernel on S 3 and H 3

144

where O is the matrix defined by (5.62)  1 O(y) = D = exp T (y) , a a y 

(9.59)

with T (y) = Ti y i . Therefore, the heat semigroup in the twisted basis is determined by the heat semigroup of the Casimir operators,



   t t  2 3 2 exp t Δ˜ Sj = exp j (j + 1) 2 exp 2 2 K˜+ + K˜− . a a

(9.60)

In particular, this means that the heat kernels are related by 3 3 UjS (t; y) = O(y)U˜ jS (t; y)O −1 (y),

(9.61)

and the heat traces for both operators are equal tr UjS (t; y) = tr U˜ jS (t; y),     3 3 Tr exp tΔSj = Tr exp t Δ˜ Sj . 3

3

(9.62) (9.63)

Recall that K˜a+ = Ka+ + Ta and K˜a− = Ka− , with Ka+ being the Killing vector fields (5.10), and that they commute with each other (5.11). The Laplacian can also be expressed in terms of the scalar Laplacian 3 3 Δ˜ Sj = ΔS0 +

 1  F − j (j + 1)Ij , a2

(9.64)

3

2 /a 2 = 4K 2 /a 2 is the scalar Laplacian and F is the operator defined by where ΔS0 = 4K− +

F = 4Ta Ka+ .

(9.65)

Since the Killing vector fields Ka+ commute with the scalar Laplacian, we get 

      t t 3 3 exp t Δ˜ Sj = exp −j (j + 1) 2 exp F exp tΔS0 . 2 a a We will use the following general statement.

(9.66)

9.3 Heat Trace

145

Proposition 9.11 Let Ta and Ka be the generators of two different irreducible representations of a simple compact Lie group of rank 1 and X = Ta ⊗ Ka . Then tr T exp X = f (T 2 , K 2 )IK ,

(9.67)

where the trace is taken only over the representation T and IK is the identity operator in the representation K. Proof We have tr T Xn = t a1 ···an Ka1 · · · Kan ,

(9.68)

  t a1 ...an = tr T T a1 . . . T an

(9.69)

where

are the invariant tensors of the group. This quantity is a group invariant; since the representations are irreducible and the group has rank 1, then the only such invariant could be the Casimir operator K 2 . 3

Proposition 9.12 The fiber trace of the off-diagonal heat kernel, tr UjS (t; y), does only depend on the radial coordinate r. Proof By using the heat semigroup property (9.66), we can express the fiber trace of the general heat kernel in terms of the scalar one. By using (9.66), we have

   t t 3 3 F U0S (t, r), U˜ jS (t, y) = exp −j (j + 1) 2 exp a a2

(9.70)

with the fiber trace 

t 3 3 tr U˜ jS (t, y) = exp −j (j + 1) 2 Bj (t)U0S (t, r), a

(9.71)

where 

 t Bj (t) = tr exp F . a2

(9.72)

By using the property (9.67), we see that the operator Bj (t) is a function of the scalar 3 Laplacian K 2 only. Therefore, when acting on the scalar heat kernel U0S (t, r), which depends only on the radial coordinate r, it gives a function which only depends on r.

9 Heat Kernel on S 3 and H 3

146

Proposition 9.13 2 ) = det (F − λ) (where det is the determinant 1. The characteristic polynomial χ (λ, K+ T T 2 = with respect to the representation T ) depends only on the Casimir operator K+ + + Ka Ka . 2. The operators μk , k = 1, . . . , N, with N = 2j + 1, defined by

detT (F − λ) = (μ1 − λ) · · · (μN − λ),

(9.73)

2 and commute with the operator F . depend only on K+ 3. The operators Πk defined by

Πk =

(μ1 − F ) · · · (μ k − F ) · · · (μN − F ) (μ1 − μk ) · · · (μk−1 − μk )(μk+1 − μk ) · · · (μN − μk )

(9.74)

are projection operators satisfying Πk2 = Πk ,

N 

Pk = I,

Πj Πj = 0,

i = j.

(9.75)

k=1

4. There is the spectral representation of the operator F F =

N 

μk Πk .

(9.76)

k=1

5. The heat trace has the form  N      t 3 3 Tr exp tΔSj = Tr exp [μk − j (j + 1)] 2 tr Πk exp tΔS0 . a

(9.77)

k=1

Proof The operators μk are the eigenvalues of the operator F , viewed as a matrix with operator-valued elements, and the operators Πk are the corresponding projections. Then, by using (9.66), we get  N    t 3 3 exp t Δ˜ Sj = exp [μk − j (j + 1)] 2 Πk exp(tΔS0 ), a

(9.78)

k=1

and the statement follows. Notice that tr Πk does only depend on the scalar Laplacian as well.

9.4 Heat Kernel on S 3

147

Heat Kernel on S 3

9.4

As usual, we use the geodesic coordinates y i on the sphere S 3 of radius a. We identify the sphere S 3 with the group SU (2) and use also the dimensionless coordinates x i = 2y i /a so that ρ = |x| = 2r/a. Proposition 9.14 There holds 

t 2 exp 2 2 K˜+ a





 =

dq ΦSU (2)

 2t ; q exp q, K˜+ , a2

(9.79)

R3

where q, K˜+  = q i K˜i+ and ΦSU (2) (t, q) is defined by (3.68). Proof This is a particular case of Eq. (3.72) for the representation K˜+ . Proposition 9.15 There is the integral representation of the heat semigroup

       t S3 2t t 3 exp t Δ˜ Sj = exp j (j + 1) 2 dq D(q)ΦSU (2) ; q exp q, K  exp , Δ + 2 0 a a2 R3

(9.80)

where D(q) = exp q, T . Proof This follows from Eqs. (9.60) and (9.79) and the form of the operators K˜a− = Ka− 3 and K˜+ = K + + T by recalling that ΔS = 4K 2 /a 2 . a

a

a

−

0

Proposition 9.16 The twisted heat kernel on S 3 has the form   

t y t 3 , ; U˜ jS (t, y) = (4π t)−3/2 exp j (j + 1) 2 Ψ3,j a a2 a

(9.81)

where Ψ3,j (t, y) =

 2    s + (d + 4π k)2 sin(s/2) (d + 4π k) exp − , (4π t)−3/2 et dq D(q) s sin(d/2) 8t k∈Z

R3

(9.82) with s = |q| and d = d(x, −q) being the geodesic distance in SU (2) between the points x = 2y/a and (−q) defined by (3.62).  Proof We use Eq. (9.80). The action of the heat semigroup exp S3

t S3 2 Δ0

 on the delta

function δ(y) gives the scalar heat kernel U0 (t/2, y). Furthermore, the action of the group

9 Heat Kernel on S 3 and H 3

148

element exp q, K+  right-shifts the argument x → xq, and, therefore, since y = ax/2, we get y → a2 F (x, q). The norm of the vector xq = F (x, q) is the geodesic distance |F (x, q)| = d(x, −q). Finally, by using the scalar heat kernel on S 3 (9.10), we get the result. The geodesic distance d(x, −q) depends on the norms of the vectors x and (−q) as well as the angle ϕ between them. Also, the function ΦSU (2) (2t, q) depends only on the norm s = |q| of the vector q. Let x i = ρθ i so that ρ = |x| and cos ϕ = x, −q /(ρs). We choose the coordinates q such that x = (0, 0, −ρ), that is, the z-axis is along the vector (−x) so that ϕ is the azimuthal angle and ψ is the polar angle, q = s(sin ϕ cos ψ, sin ϕ sin ψ, cos ϕ).

(9.83)

dq = ds dψ dϕ s 2 sin ϕ,

(9.84)

Then

with 0 ≤ s < ∞, 0 ≤ ψ ≤ 2π, and 0 ≤ ϕ ≤ π. Then the function Ψ3,j takes the form  ∞ π   2π t t d 2 Ψ3,j (t, y) = 2 2 exp ds dϕ dψ s sin ϕ D(q)ΦSU (2) (2t; q) Ψ3,0 , . 2 2 2 √

0

0

0

(9.85) The trace of this function is simplified by using the character   ∞ π 2π  √ t tr Ψ3,j (t, y) = 2 2 exp ds dϕ dψ s 2 sin ϕ 2 |μ|≤j

0

0

× cos(μs)ΦSU (2) (2t; q) Ψ3,0



0

 t d , . 2 2

(9.86)

Now, by using the explicit form of the functions ΦSU (2) and Ψ3,0 and integrating over ψ, we obtain tr Ψ3,j (t, y) =

  |μ|≤j k∈Z

1 √ t −3/2 et 4 π

∞

π dϕ sin ϕ

ds 0

0

 2 sin(s/2) s + (d + 4π k)2 ×s cos(μs)(d + 4π k) exp − . sin(d/2) 8t

(9.87)

9.4 Heat Kernel on S 3

149

This integral can be computed exactly for the heat kernel diagonal by setting y = x = 0, which means that the point x is the identity element of the group SU (2) and, therefore, d = |F (0, −q)| = s. Proposition 9.17 The heat kernel diagonal and the heat trace on S 3 for an irreducible representation of SU (2) have the form  

 t t S3 −3/2 ˜ exp j (j + 1) 2 Ψ3,j ;0 , (9.88) Uj,diag (t) = (4π t) a a2 √  −3/2  

   t π t t 3 (2j + 1)Ψ Tr exp tΔSj = exp j (j + 1) ; 0 ,(9.89) 3,j 4 a2 a2 a2 where Ψ3,j (t; 0) =

    π 2k2 1 (−1)2j k 1 − 2μ2 t − 2 2j + 1 t |μ|≤j k∈Z



   π 2k2 exp (1 − μ2 )t t      (1 − 2μ2 t)Ωj (t) − 2tΩj (t) exp (1 − μ2 )t ; (9.90)

× exp − =

1 2j + 1

|μ|≤j

here Ωj (t) = Ω+ (t) for integer j and Ωj (t) = Ω− (t) for half-integer j and the functions Ω± (t) are defined in (6.44) and (6.45). Proof The heat kernel diagonal is given by √ Ψ3,j (t, 0) = 2 2 exp

    t s t , . dq D(q)ΦSU (2) (2t; q) Ψ3,0 2 2 2

(9.91)

R3

The integrand depends only on the radial coordinate s = |q|. Therefore, it is reduced to the average over the sphere S 2 of the matrix D(q) which is determined by its character. Thus one can replace here the matrix D(q) by the rescaled character. Therefore, we have      √ t s 1 t , . Ψ3,j (t, 0) = 2 2 exp dq cos(μs)ΦSU (2) (2t; q) Ψ3,0 2j + 1 2 2 2 |μ|≤j

R3

(9.92) By using the explicit form of the functions ΦSU (2) , (3.68), and Ψ3,0 , (3.70), we obtain

9 Heat Kernel on S 3 and H 3

150

Ψ3,j (t, 0) =

  1 1 √ t −3/2 et 2j + 1 2 π

(9.93)

|μ|≤j k∈Z

∞ ×

 2 s + 4π ks + 8π 2 k 2 ds cos(μs)s(s + 4π k) exp − . 4t

0

This integral can be transformed to the Gaussian form Ψ3,j (t, 0) =

    1 1 π 2k2 (−1)2j k √ t −3/2 et exp − 2j + 1 t 4 π  ×

|μ|≤j k∈Z



s2 ds cos(μs)(s − 4π k ) exp − 4t 2

R



2 2

,

(9.94)

which is easily computed to give the result. The heat trace is obtained by taking the fiber trace and multiplying by the volume of the sphere vol (S 3 ) = 2π 2 a 3 .

9.5

Heat Kernel on H 3

The heat kernel on H 3 of the pseudo-radius b is obtained similarly. Most equations remain the same as for S 3 of radius a with the substitution a = ib. We use the geodesic coordinates y i = rθ i and the dimensionless coordinates x k = 2y k /b and, therefore, ρ = 2r/b. The right-invariant and the left-invariant vector fields are given by (3.40) Ka± = iθ a ∂ρ +

ρ   1 i coth Qac ± δac Lc . 2 2

(9.95)

Contrary to the case of S 3 , the operators Ka± on H 3 are complex. The action of the isometries on H 3 is as follows:   ib exp[K+ (iq)]f (y) = f (9.96) F (−ix, iq) , 2 where the function F (x, q) is the group multiplication in canonical coordinates on SU (2) given by (3.16). 3 2 2 2 2 The scalar Laplacian is equal to the Casimir operator ΔH 0 = −4K− /b = −4K+ /b . The Laplacian in the representation j is given by (5.94) 3

ΔH j =−

 1  2 2 + j (j + 1)Ij , 2K+ + 2K− 2 b

(9.97)

9.5 Heat Kernel on H 3

151

2 = K± K± and K± are the Lie derivatives (5.90). Although on H 3 there is no where K± a a a right-invariant basis, we introduce the complex operators K˜a+ = Ka+ + T+ and K˜a− = Ka− ,Eqs. (5.99) and (5.100). The Laplacian in this basis is given by (5.104) 3 Δ˜ H j =−

 1  ˜2 ˜2 + j (j + 1)I . 2 K + 2 K j + − b2

(9.98)

The Laplacians in these bases are related by (5.66), and, therefore, the heat semigroups are related by     3 3 = O exp t Δ˜ H O −1 , exp tΔH j j

(9.99)

where O is the matrix defined by (5.62)  1 = exp T (y) , O(y) = D ib ib y

(9.100)

and 

   t  ˜2 t 3 ˜2 exp −2 = exp −j (j + 1) K . exp t Δ˜ H + K + − j b2 b2

(9.101)

In particular, this means that the heat kernels are related by 3 3 UjH (t; y) = O(y)U˜ jH (t; y)O −1 (y),

(9.102)

and the fiber trace of the heat kernel for both operators are equal 3 3 tr UjH (t; y) = tr U˜ jH (t; y).

(9.103)

2 is exactly the scalar Laplacian, Since K˜a− = Ka− , then the Casimir operator K˜− 2 2 = − b ΔH 3 , and since the left-invariant K˜ and the right-invariant Lie derivatives K˜− − 4 0 K˜+ commute, we obtain

    

  t ˜2 t H3 t 3 exp −2 exp . Δ = exp −j (j + 1) exp t Δ˜ H K j 2 0 b2 b2 +

(9.104)

Proposition 9.18 There holds      2t t 2 = dq Φ˜ SU (2) exp −2 2 K˜+ ; q exp iq, K˜+ , b b2 R3

(9.105)

9 Heat Kernel on S 3 and H 3

152

k and Φ ˜ SU (2) (t, q) is defined by (3.69). where iq, K˜+  = iqk K˜+

Proof This is a particular case of Eq. (3.73) for the representation K˜+ . Proposition 9.19 There is the integral representation of the heat semigroup 

   t 2t H3 ˜ ˜ exp t Δj dq D(iq)ΦSU (2) ;q = exp −j (j + 1) 2 b b2 

R3

 × exp iq, K+  exp

 t H3 Δ0 , 2

(9.106)

where D(iq) = exp iq, T . Proof This follows from Eqs. (9.104) and (9.105) and the form of the operators K˜a+ = Ka+ + Ta . Proposition 9.20 The twisted heat kernel on H 3 has the form

   t t y 3 U˜ jH (t, y) = (4π t)−3/2 exp −j (j + 1) 2 Ψ˜ 3,j ; , b b2 b

(9.107)

where 

Ψ˜ 3,j (t; y) = (4π t)−3/2 e−t

R3

  s 2 + d˜ 2 sinh(s/2)d˜ exp − dq D(iq) , (9.108) ˜ 8t s sinh(d/2)

˜ q) is the analytic continuation of the geodesic distance between with s = |q| and d˜ = d(x, x = 2y/b and q in SU (2) defined by (2.88) and (3.63). 3

Proof We use Eq. (9.106). The action of the heat semigroup exp(tΔH 0 /2) on the delta 3 function δ(y) gives the scalar heat kernel U0H (t/2, y), given by (9.1), and the action of the group element exp iq, K+  right-shifts the argument y → ibF (−ix, iq)/2. The norm ˜ q)/2. Therefore, we get of this vector is equal to the geodesic distance bd(x, Ψ˜ 3,j

    ˜ d/2 d˜ 2 t exp − dq D(iq)Φ˜ SU (2) (2t; q) . (t; y) = 2 2 exp − ˜ 2 8t sinh(d/2) √

R3

By using the form of the function Φ˜ SU (2) , we obtain (9.108).

(9.109)

9.5 Heat Kernel on H 3

153

A more explicit formula can be obtained by taking the fiber trace. By using the same coordinate system as for the case of S 3 , we obtain tr Ψ˜ 3,j (t, y) =

 |μ|≤j

1 √ t −3/2 e−t 4 π

∞

π dϕ sin ϕ

ds 0

0

  2 + d˜ 2 s sinh(s/2) exp − ×s d˜ cosh(μs) , ˜ 8t sinh(d/2)

(9.110)

where s = |q| and ϕ is the angle between the vectors x and q. This integral can be computed exactly for the heat kernel diagonal, that is, for y = x = 0. Proposition 9.21 The heat kernel diagonal on H 3 for an irreducible representation j of SU (2) has the form  

 t t H3 (t) = (4π t)−3/2 exp −j (j + 1) 2 Ψ˜ 3,j ; 0 , U˜ j,diag b b2

(9.111)

where Ψ˜ 3,j (t; 0) =

     1 1 + 2μ2 t exp (μ2 − 1)t . 2j + 1

(9.112)

|μ|≤j

Proof The heat kernel diagonal is determined by its trace (9.110) for y = 0. In this case, the geodesic distance d˜ between x = 0 and q is equal to d˜ = s = |q|. Therefore, the integral over ϕ just gives a factor 2; thus, we obtain from (9.110)  2   1 1 s . √ t −3/2 e−t ds s 2 cosh(μs) exp − 2j + 1 4t 2 π ∞

Ψ˜ 3,j (t, 0) =

|μ|≤j

0

This integral is Gaussian and can be easily computed to give the result.

(9.113)

Algebraic Method for the Heat Kernel

10.1

10

Algebraic Method for Heat Kernel

We fix one point x  to be the origin (the North pole) and denote the heat kernel U (t; x, x  ) and the resolvent G(λ; x, x  ) in the geodesic coordinates by U (t; y) and G(λ; y). For maximally symmetric spaces, the Ruse–Synge function σ (x, x  ) and the Van Vleck– Morette determinant Δ(x, x  ) are given by (4.7) and (4.31) and the operator of parallel transport P(x, x  ) is simply equal to the unit matrix. Therefore, as t → 0, the asymptotic form (1.17) becomes U (t; y) ∼ (4π t)−n/2



r f (r)

(n−1)/2

 2 r exp − . 4t

(10.1)

By using the semigroup property of the heat kernel, we have U (t; y) = lim exp(tΔ)U (s; y).

(10.2)

s→0

By using the Laplacian in the form (4.135), we get 

(n − 1)2 U (t; y) = exp t 4a 2



r f (r)

(n−1)/2

Uˆ (t, y),

(10.3)

where Uˆ (t; y) is the heat kernel of the operator   1 (n − 1) (n − 1)(n − 3) 1 1 ˆ Hˆ = ∂r2 + + 2 Δ, + ∂r + r 4 r2 f 2 (r) f (r)

(10.4)

with Δˆ defined by (4.127). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0_10

155

156

10 Algebraic Method for the Heat Kernel

Although this formula could be used to compute the heat kernel, we will employ another purely algebraic method following [6]. Let M be a maximally symmetric n-dimensional manifold and V be a homogeneous spin-tensor vector bundle over M = G/H . As we described earlier, the isometry group G of a maximally symmetric n-dimensional manifold M is a simple Lie group G (either SO(n + 1) or SO(1, n)) of dimension m = n(n+1) 2 with the structure constants (4.82). The Lie derivatives LAB (4.101)–(4.103) form a representation of the isometry group. We label the canonical coordinates on the group G by k AB , with 1 ≤ A < B ≤ m. Since the Laplacian is expressed in terms of the Casimir operators as in (4.123), then by using (2.186) one can express the heat semigroup in terms of the action of isometries via     t n exp tΔS = exp − 2 G2 dk ΦSO(n+1) (t, k) expk, L, (10.5) a Rm

AB , k, L = 1 k AB L where dk = AB , and ΦSO(n+1) (t, k) is defined 1≤A 0. We study a Killing vector field ξ = q a Pa + ωL. Let ψτ : M → M be the one-parameter diffeomorphism ˆ ; q, ω, y) depends also generated by the vector field ξ and yˆ = ψτ (y). Of course, yˆ = y(τ on q, ω, and y as parameters. We will omit all parameters where it does not cause any confusion. By using the explicit form of the Killing vectors (10.8) and (10.9), we show that these functions satisfy the equations d yˆ a 1 − h(ˆr )ˆr = qa h(ˆr )ˆr + qb yˆ b yˆ a − ωεab yˆ b , dτ rˆ 2

(10.17)

where rˆ 2 = yˆa yˆa , with the initial condition yˆ a (0) = y a .

(10.18)

158

10 Algebraic Method for the Heat Kernel

We introduce the polar coordinates in the q-plane q 1 = s cos α, where s = |q| = coordinates of yˆ a )

q 2 = s sin α,

(10.19)

√ qa qa . Then the equations take the form (here rˆ and ϕˆ are the polar d rˆ = s cos(ϕˆ − α), dτ   d ϕˆ s rˆ sin(ϕˆ − α). = ω − cot dτ a a

(10.20) (10.21)

with the initial conditions rˆ (0) = r,

ϕ(0) ˆ = ϕ.

(10.22)

For q = s = 0, the equations are linear, so the solution is a circle y(τ ˆ ) = exp(−τ ωε)y = cos(τ ω)y − sin(τ ω)εy,

(10.23)

where ε = (εab ) is the anti-symmetric matrix with ε12 = −ε21 = 1, or, in polar coordinates, rˆ (τ ) = r, ϕ(τ ˆ ) = ϕ + τ ω.

(10.24)

Now, assume that q = 0, that is, s = 0. Proposition 10.1 The solution of Eqs. (10.20), (10.21) with the initial condition (10.22) is determined implicitly by cos

  ρ   ρ  aω rˆ = A cos τ + B sin τ + 2 C, a a a ρ

sin(ϕˆ − α) = where ρ =

√

s 2 + a 2 ω2 and

C − aω cos(ˆr /a) , s sin(ˆr /a)

(10.25)

(10.26)

10.2 Algebraic Method for S 2 and H 2

159

 r  r  s  − aω sin(ϕ − α) sin , s cos a a ρ2 r  s cos(ϕ − α), B = − sin ρ a r  r  + s sin(ϕ − α) sin . C = aω cos a a A=

(10.27) (10.28) (10.29)

Proof Let u = sin(ϕˆ − α).

(10.30)

Then Eqs. (10.20), (10.21) leads to the linear equation   rˆ ω du 1 + cot u= , d rˆ a a s

(10.31)

which can be easily solved; the solution has the form (10.26), where C = C(q, ω, y) is an integration constant (which is a function of the parameters q, ω, and y) determined from the initial condition (10.22); this gives (10.29). Furthermore, let v = cos with ρ =

√

  rˆ aω − 2 C, a ρ

(10.32)

s 2 + a 2 ω2 . Then, by using (10.20) and (10.26), we have  a

2

dv dτ

2 = −ρ 2 v 2 +

s2β 2 , ρ2

(10.33)

where β 2 = ρ2 − C2.

(10.34)

Notice that if β 2 < 0, then there are no solutions. For β 2 > 0, this equation can be easily integrated; the solution can be written in the form v=

 ρ βs τ + γ , cos a ρ2

(10.35)

where γ = γ (q, ω, y) is an integration constant. Therefore, the solution is determined by the equation

160

10 Algebraic Method for the Heat Kernel

   ρ 1  rˆ = 2 βs cos τ + γ + aωC . cos a a ρ

(10.36)

The integration constant γ is determined from the initial condition (10.22) and has the form cos γ =

r   r  1 s cos − aω sin(ϕ − α) sin . β a a

(10.37)

By taking the derivative of (10.36) at τ = 0 and using Eq. (10.20), we also get sin γ =

r  ρ sin cos(ϕ − α). β a

(10.38)

Therefore, the solution can be written in the form (10.25). The coordinate ϕˆ is then determined from Eq. (10.26). Notice that for 0 ≤ rˆ ≤ aπ this uniquely defines rˆ and, ˆ therefore, ϕ. To simplify the calculation, we study the diagonal case, r = 0, in more detail. Proposition 10.2 The solution of Eq. (10.20) with the initial condition with rˆ (0) = 0 has the form −1

rˆ (τ ; q, ω, 0) = 2a sin

 ρ  s sin τ . ρ 2a

(10.39)

There holds rˆ (τ ; 0, ω, 0) = 0 and  ∂ sin(τ ω/2)  rˆ (τ ; q, ω, 0) . = s=0 ∂s ω/2

(10.40)

Proof Substituting r = 0 in the above solution (10.25)–(10.29), we obtain A=

s2 , ρ2

B = 0,

C = aω.

(10.41)

  ρ  s2 rˆ = 1 − 2 2 sin2 τ . a 2a ρ

(10.42)

Therefore, the solution has the form cos This gives  sin

rˆ 2a

 =

ρ  s sin τ , ρ 2a

(10.43)

10.2 Algebraic Method for S 2 and H 2

161

and the result (10.39) follows. Obviously, for s = 0, the solution vanishes, rˆ = 0. By computing the linear term in the Taylor series in s, we get (10.40). We define the Jacobian W = W (τ ; q, ω, y) by  W = det

∂ yˆ a ∂q b

 (10.44)

.

Proposition 10.3 For q = y = 0 and τ = 1, there holds  W (1; 0, ω, 0) =

sin(ω/2) ω/2

2 (10.45)

.

Proof By using a useful relation 2 c ∂ yˆ c ∂ yˆ c 1 ∂ 2 rˆ 2 c ∂ yˆ = − y ˆ , ∂q a ∂q b 2 ∂q a ∂q b ∂q a ∂q b

(10.46)

we obtain 

2 c 1 ∂ 2 rˆ 2 c ∂ yˆ W = det − y ˆ 2 ∂q a ∂q b ∂q a ∂q b 2

 (10.47)

.

Since for q = y = 0 the solution vanishes, y(τ ˆ ; 0, ω, 0) = 0, by substituting q = y = 0 in (10.47) we get  W (1; 0, ω, 0) = det

∂ 2 f (q, ω) ∂q a ∂q b

1/2   

q=y=0

,

(10.48)

where f (q, ω) =

1 2 rˆ (1; q, ω, 0). 2

(10.49)

Since the function f depends only on the radial coordinate s = |q| of the vector q, we obtain ∂f qa ∂s f, = ∂q a s  ∂ 2f qa qb 2 qa qb  1 ∂s f. = ∂ f + δ − ab s ∂q a ∂q b s s2 s2

(10.50) (10.51)

162

10 Algebraic Method for the Heat Kernel

This matrix has two simple eigenvalues, ∂s2 f and (∂s f )/s; therefore, the determinant of this matrix is easily computed 

∂ 2f det ∂q a ∂q b

 =

1 2 (∂ f )(∂s f ). s s

(10.52)

By using (10.39), we compute the derivatives of the function f with respect to s and take the limit s → 0 to obtain the Jacobian (10.45). Proposition 10.4 The heat semigroup on S 2 can be represented in form of an integral  

   t 1 2 − G2 exp tΔS = a(4π t)−3/2 exp (10.53) 4 a2     ρ2 sin (ρ/2a) exp − exp [q, P + ωL] , dq × dω ρ/2a 4t R2

C+

where C+ is the horizontal contour just above the real axis and q, P = q a Pa . Proof This follows from Eq. (3.72) by using the isometry algebra and the representation of the Laplacian in terms of the twisted Lie derivatives. We define the matrix Oab =

∂ yˆ a . ∂y b

(10.54)

This Jacobian is an orthogonal matrix and, therefore, it defines a function θ = θ (q, ω, y) by O = exp (θ ε) = I cos θ + ε sin θ,

(10.55)

where I is the unit matrix and ε = (εab ) is the anti-symmetric matrix with ε12 = −ε21 = 1. This function can be easily computed for τ = 1 and q = 0. We have O a b (1; 0, ω, y) =

∂ yˆ a (1; 0, ω, y) = cos ωδab − sin ωεab . ∂y b

(10.56)

Therefore, θ (1; 0, ω, y) = −ω.

(10.57)

10.2 Algebraic Method for S 2 and H 2

163

To obtain the heat kernel, we need to compute the action of the isometries on the delta function V (q, ω, y) = exp [q, P + ωL] δ(y).

(10.58)

Proposition 10.5 For y = 0, the function V has the form  V (q, ω, 0) =

ω/2 sin(ω/2)

2 exp (ωG) δ(q).

(10.59)

Proof The action of the isometries is determined by the pullback   V (q, ω, y) = (ψ1∗ δ)(y) = exp (−θ G) δ(y) ˆ 

τ =1

.

(10.60)

This delta function picks the values of q¯ such that the trajectory satisfies the zero terminal condition at τ = 1, y(1; ˆ q, ¯ ω, y) = 0.

(10.61)

This implicitly defines functions q¯k = q¯k (ω, y). For the initial condition y(0) ˆ = y = 0, this means that the functions q¯k (ω, 0) satisfy the equation rˆ (1; q¯k , ω, 0) = 0, that is, ⎛ ⎞ s¯k2 + a 2 ω2 ⎠ = 0, s¯k sin ⎝ 2a

(10.62)

where s¯k = |q¯k (ω, 0)|. This equation has the obvious solution s¯ (ω) = 0.

(10.63)

If ω is real, it also has other multiple solutions that lie on the spheres S 2 of radius 2π ka with k = 1, 2, . . . in the q-space R3 ,

s¯k (ω) = a 4π 2 k 2 − ω2 .

(10.64)

However, our parameter ω has an infinitesimal positive imaginary part since we integrate over ω over the contour C+ in the complex plane. In this case there are no zeros for the real values of the parameters q. Therefore, q¯ = s¯ = 0 is the only solution of Eq. (10.62). Therefore, by using the transformation properties of the delta function and Eqs. (10.57) and (10.45), we obtain

164

10 Algebraic Method for the Heat Kernel

 V (0, ω, y) = exp(ωG) det

∂ yˆ a ∂q b

−1

  δ(q)

τ =1,q=y=0

,

(10.65)

and, finally, by using (10.45), we obtain the result (10.59). This result enables one to obtain the heat kernel diagonal. Proposition 10.6 The heat kernel diagonal on S 2 has the form   t 1 − G2 4 a2  2 2   ω/2 a ω × dω exp − + ωG . sin(ω/2) 4t

S Udiag (t) = a(4π t)−3/2 exp 2



(10.66)

C+

Proof This is obtained by using Eqs.(10.53) and (10.59) and integrating over q. This result should be compared with (8.26) obtained by a completely different method as well as with the scalar case (6.74). The calculation for the hyperbolic space H 2 is very similar, with the simple substitution √ a → ib. The radial coordinate r = y a ya is ranging over 0 ≤ r ≤ ∞. The Killing vectors have the form (4.75) and (4.76) with h(r) = b1 coth(r/b). The Lie derivatives are (10.10)–(10.12) and form a representation of the isometry algebra so(1, 2) (10.13) with the substitution a → ib. The Laplacian has the same form (10.16), with a = ib. Proposition 10.7 The heat semigroup on H 2 can be represented in form of an integral  

   t 1 2 − G2 exp tΔH = b(4π t)−3/2 exp − (10.67) 4 b2     ρ2 sinh (ρ/2b) exp − exp [q, P + iωL] , dq × dω ρ/2b 4t R

where ρ =

√

R2

s 2 + b2 ω2 with s = |q|.

Proof This follows from Eq. (3.73) or (2.187) and (3.69) and (2.185), by using the isometry algebra and the representation of the Laplacian in terms of the twisted Lie derivatives. Let ψτ : M → M be the one-parameter diffeomorphism generated by the Killing vector field ξ = q a Pa + iωL and yˆ = ψτ (y). These functions satisfy the equations

10.2 Algebraic Method for S 2 and H 2

165

d yˆ a 1 − h(ˆr )ˆr = qa h(ˆr )ˆr + qb yˆ b yˆ a − iωεab yˆ b , dτ rˆ 2 where rˆ =

(10.68)

yˆa yˆa , with the initial condition yˆ a (0) = y a .

(10.69)

In spherical coordinates, the dynamical equations are d rˆ = s cos(ϕˆ − α). dτ   d ϕˆ s rˆ = iω − coth sin(ϕˆ − α), dτ b b

(10.70) (10.71)

The solution of these equations is found similarly to the case of S 2 and has the form C − biω cosh(ˆr /b) , s sinh(ˆr /b)   ρ   ρ  biω rˆ cosh = A cosh τ + B sinh τ + 2 C, b b b ρ

sin(ϕˆ − α) =

(10.72) (10.73)

where  r  r  s  − biω sin(ϕ − α) sinh , s cosh b b ρ2 r  s B = − sinh cos(ϕ − α), ρ b r  r  + s sin(ϕ − α) sinh . C = biω cosh b b A=

(10.74) (10.75)

In the diagonal case, r = 0, the solution has the form −1

rˆ (τ ; q, ω, 0) = 2b sinh

 ρ  s sinh τ . ρ 2b

(10.76)

For q = 0, the solution has the form y(τ ˆ ) = exp(−iτ ωε)y.

(10.77)

This means, in particular, that the function θ defined by Eq. (10.55) for τ = 1 and q = 0 has the form θ (1; 0, ω, y) = −iω.

(10.78)

166

10 Algebraic Method for the Heat Kernel

Then the action of the isometries on the delta function has the form V (q, ω, y) = exp [q, P + iωL] δ(y)   = exp (−θ G) δ(y) ˆ  . τ =1

(10.79)

This delta function picks the values of q¯ such that y(1; ˆ q, ¯ ω, y) = 0, that is, it implicitly ¯ y), determined from the equation rˆ (1; q, ¯ ω, y) = 0. defines a function q¯ = q(ω, For simplicity, we consider the diagonal case y = r = 0 in more detail. Then this equation takes a very simple form ρ s sinh = 0, ρ 2b

(10.80)

s¯ (ω) = 0.

(10.81)

which has the unique solution

The Taylor series of the solution in s has the form (with τ = 1) rˆ (1; q, ω, 0) = s

sinh(ω/2) + O(s 2 ). ω/2

(10.82)

Therefore, we obtain 

ω/2 V (q, ω, 0) = exp (iωG) sinh(ω/2)

2 δ(q).

(10.83)

Proposition 10.8 The heat kernel diagonal on H 2 has the form  

 t 1 H2 − G2 Udiag (t) = b(4π t)−3/2 exp − 4 b2  2 2   b ω ω/2 exp − + iωG . × dω sinh(ω/2) 4t

(10.84)

R

Proof This is obtained by using (10.67) and (10.83). This result should be compared with (8.46) obtained by a completely different method as well as with the scalar case (6.98).

10.3 Heat Kernel Diagonal on S n and H n

10.3

167

Heat Kernel Diagonal on S n and H n

We follow our paper [6]. The sphere S n is the homogeneous space S n = SO(n+1)/SO(n) with the isometry group SO(n + 1) and the isotropy group SO(n). We consider a spintensor vector bundle realizing a representation G of the spin group Spin(n) with the generators Gab satisfying the commutation relations (2.130) and the Casimir operator G2 = 12 Gab Gab . Let Pa and Lab be the Lie derivatives defined by (4.102) and (4.103). The Laplacian is given by (4.123)  1  2 L − G2 . 2 a

Δ = P2 +

(10.85)

We use the notation of Sect. 2.5. We split the canonical coordinates on SO(n + 1) according to (kAB = (qa , ωcd ), where the indices a and c < d range over 1, 2, . . . , n, and, by using the twisted Lie derivatives (4.102) and (4.103), define the operators 1 − cos(r/a) a b q y Gab , ar sin(r/a)

q, P = q b Pb + ω, L =

(10.86)

1 1 ab ω Lab + ωab Gab , 2 2

(10.87)

where Pb and Lab the Killing vector fields defined by (4.75), (4.76). Proposition 10.9 The heat semigroup on S n can be represented as an integral     t n exp tΔS = exp − 2 G2 a



 dω

dq ΦSO(n+1) (t; q, ω) exp [q, P+ω, L] ,

Rn

Rn(n−1)/2

(10.88)

where the function ΦSO(n+1) (t; q, ω) is defined in (2.184). Proof This is proved by using Eq. (2.186). The heat kernel is obtained by acting on the delta function. Proposition 10.10 The heat kernel on the sphere S n has the form U S (t; y) = (4π t)−n/2 exp n



t a2



n(n2 − 1) − G2 24



 Ψn

t y , a2 a

 ,

(10.89)

168

10 Algebraic Method for the Heat Kernel

where Ψn (t, y) = (4π t)

−n(n−1)/4



 dω

Rn(n−1)/2

dq JSO(n+1) (q, ω)

Rn

  |q|2 + |ω|2 V (q, ω; y), × exp − 4t

(10.90)

with JSO(n+1) being the function defined by (2.67), and V (q, ω; y) = exp [q, P + ω, L] δ(y).

(10.91)

Proof This is obtained by using the above representation of the heat semigroup and the explicit form of the function ΦSO(n+1) , (2.184). The Lie derivatives L(k) are the generators of the twisted isometries. Let ψτ : M → M be the one-parameter diffeomorphism generated by the Killing vector field ξ = P(q) + L(ω) and yˆ = ψτ (y).

(10.92)

We will omit the argument τ where it will not cause a confusion, that is, we will just write yˆ = y(q, ˆ ω, y) and set τ = 1 at the end of the calculation. Therefore, for any section f of the vector bundle   exp (τ ξ ) f (y) = ψτ∗ f (y).

(10.93)

The action of isometries is computed as follows. We define the Jacobian matrix O = O(q, ω, y) = (O a b ) by Oab =

∂ yˆ a . ∂y b

(10.94)

Proposition 10.11 The matrix O is orthogonal and, therefore, has the form O = exp θ, where θ = θ (q, ω, y) is an anti-symmetric matrix. Proof This follows from the fact that ψτ is the isometry (for details, see [6]).

(10.95)

10.3 Heat Kernel Diagonal on S n and H n

169

Proposition 10.12 The function V (q, ω, y) has the form  1 ab det N −1 δ(q − q), V (q, ω, y) = exp − θab G ¯ 2 

(10.96)

where q¯ = q(ω, ¯ y) is implicitly defined by y(1; ˆ q, ω, y) = 0,

(10.97)

and the matrix N = N(q, ω, y) = (N a b ) is defined by Nab =

∂ yˆ a . ∂q b

(10.98)

Proof By using the well known property for any section f of the vector bundle (see [6]) (ψτ∗ f )(y)

 1 ab f (y), ˆ = exp − θab G 2 

(10.99)

we obtain    1  ˆ  . V (q, ω, y) = exp − θab Gab δ(y) τ =1 2

(10.100)

The delta function in (10.96) picks the values of q satisfying Eq. (10.97). Therefore, δ(y) ˆ = det N −1 δ(q − q). ¯

(10.101)

Thus, the calculation of the heat kernel is reduced to the calculation of the functions yˆ and q, ¯ the matrix θab , and the Jacobian N . Then the heat kernel is reduced to the calculation ¯ If of the integral (10.90). We assumed here that Eq. (10.97) has the unique solution q. there are multiple solutions, then we should add the summation over all such solutions in (10.101). This integral should be regularized by considering complex values of the variables ωab to avoid singularities on the real line due to the presence of closed geodesics (more on this later). The heat kernel diagonal is obtained from here by setting y = 0. For q = y = 0, the matrices O and N can be computed exactly. Proposition 10.13 There holds θ (0, ω, 0) = −τ ω,

(10.102)

170

10 Algebraic Method for the Heat Kernel

and  sinh(τ ω/2) . ω/2

(10.103)

Y = N(0, ω, 0).

(10.104)

 det N(0, ω, 0) = det Proof Let X = O(0, ω, 0), These matrices satisfy the same equation

d X + ωX = 0, dτ d Y + ωY = 0, dτ

(10.105) (10.106)

with different initial conditions X(0) = I,

Y (0) = 0.

(10.107)

The solutions of these equations are O(0, ω, 0) = exp(−τ ω), N(0, ω, 0) =

exp(−τ ω) − I . ω

(10.108) (10.109)

This immediately gives (10.102), and by using the anti-symmetry of the matrix ω we get (10.103). Proposition 10.14 There holds  V (q, ω; 0) = det

sinh(ω/2) ω/2

−1 expω, Gδ(q),

(10.110)

where ω, G =

1 ab ω Gab . 2

(10.111)

Proof First, one can show that for y = 0 the function q(ω, ¯ y) vanishes, that is, q(ω, ¯ 0) = 0.

(10.112)

10.3 Heat Kernel Diagonal on S n and H n

171

Next, by using (10.96), (10.103), and (10.102), we get the result. Notice that this becomes singular for real values of ωab . Proposition 10.15 The heat kernel diagonal on S n has the form Sn Udiag (t)

= (4π t)

−n/2



t exp 2 a



n(n2 − 1) − G2 24



 Ψn

 t ,0 , a2

(10.113)

where Ψn (t, 0) = (4π t)

−n(n−1)/4





|ω|2 dω exp − 4t

Rn(n−1)/2



sinh(ω/2) χ (ω) = det ω/2

−1/2





sinh[F (ω)/2] det F (ω)/2

χ (ω) expω, G , (10.114) 1/2 (10.115)

,

where F (ω) = (F ab cd ) is the matrix defined by (2.158).

Proof This is obtained by setting y = 0 in (10.90). By using (10.110), the integral over q becomes trivial and we get (10.113), (10.114) with  χ (ω) = det

sinh(ω/2) ω/2

−1 (10.116)

JSO(n+1) (0, ω).

Finally, by using (2.182) and (2.181), we compute 

sinh(ω/2) JSO(n+1) (0, ω) = det ω/2

1/2



sinh[F (ω)/2] det F (ω)/2

1/2 .

(10.117)

This gives the result (10.115). Notice that the only difference with the scalar case is the presence of the exponential expω, G. Therefore, when G = 0, this result should coincide with (6.114) giving a nontrivial relation   ∞  n(n2 − 1)  n + 1 n/2 t exp − t dk (n) exp {−tk(k + n − 1)} , 2 24 k=0 (10.118) where the dk (n) are defined by (6.112). This idea works also for the hyperbolic space H n which is the homogeneous space n H = SO(1, n)/SO(n) with the isometry group SO(1, n) and the isotropy group SO(n). 2n−1 Ψn (t, 0) = √ Γ π



172

10 Algebraic Method for the Heat Kernel

The Killing vectors, Pa and La , the Lie derivatives, Pa and La , and the Laplacian ΔH for H n are obtained from the same objects for S n by replacing a → ib.

n

Proposition 10.16 The heat semigroup on H n can be represented as an integral     t 2 Hn = exp 2 G exp tΔ b



 dω

dq Φ˜ SO(n+1) (t; q, ω) exp [q, P + iω, L] ,

Rn

Rn(n−1)/2

(10.119)

where the function Φ˜ SO(n+1) (t; q, ω) is defined in (2.185). Proof This is proved by using Eq. (2.187). Proposition 10.17 The heat kernel on the hyperbolic space H n has the form      t y t n(n2 − 1) n − G2 Ψ˜ n , U H (t; y) = (4π t)−n/2 exp − 2 , 24 b b2 b

(10.120)

where Ψ˜ n (t, y) = (4π t)−n(n−1)/4





Rn(n−1)/2

dω

dq J˜SO(n+1) (q, ω)

Rn

  |q|2 + |ω|2 ˜ × exp − V (q, ω; y), 4t

(10.121)

with J˜SO(n+1) being the function defined by (2.183) and V˜ (q, ω; y) = exp [q, P + iω, L] δ(y).

(10.122)

Proof This is obtained by using the above representation of the heat semigroup and the explicit form of the function Φ˜ SO(n+1) (2.185). The function V˜ (q, ω, y) is obtained from the function V (q, ω, y) by replacing ω → iω and a → ib; in particular, we obtain   sin(ω/2) −1 expiω, Gδ(q). V˜ (q, ω; 0) = det ω/2

(10.123)

Proposition 10.18 The heat kernel diagonal on H n has the form Hn Udiag (t)

= (4π t)

−n/2



t exp − 2 b



n(n2 − 1) − G2 24



Ψ˜ n



 t ,0 , b2

(10.124)

10.3 Heat Kernel Diagonal on S n and H n

173

where Ψ˜ n (t, 0) = (4π t)−n(n−1)/4 

sin(ω/2) χ˜ (ω) = det ω/2

  |ω|2 χ˜ (ω) expiω, G , (10.125) dω exp − 4t



Rn(n−1)/2

−1/2



sin[F (ω)/2] det F (ω)/2

1/2 .

(10.126)

Proof This is obtained by setting y = 0 in (10.121). By using (10.123), the integral over q becomes trivial and we get (10.124) and (10.125) with 

sin(ω/2) χ(ω) ˜ = det ω/2

−1

J˜SO(n+1) (0, ω).

(10.127)

Finally, by using (2.183) and (2.181), we compute     sin(ω/2) 1/2 sin[F (ω)/2] 1/2 det . J˜SO(n+1) (0, ω) = det ω/2 F (ω)/2

(10.128)

This gives the result (10.126). When G = 0, this result should coincide with (6.141) giving a non-trivial relation Ψ˜ n (t, 0) =

 ∞      n−1 2t n/2 (n − 1)(n − 2)(n − 3)  n  exp t exp −tν 2 , dν T ν, 24 2 Γ 2 0

(10.129) where T (ν, α) is the function defined by (6.135).

A

Integrals, Series, and Special Functions

A.1

Integrals

Gaussian integrals are integrals of the form  dx exp [−S(x)] ϕ(x),

(A.1)

Rn

where S(x) is a quadratic polynomial with a positive real part for any x and ϕ(x) is a polynomial. All such integrals can be computed exactly for any n. The most fundamental integral is 

  √ dx exp −x 2 = π .

(A.2)

R

By changing the integration variable, one gets the more general integral (with Re a > 0)  R

 2   √ b dx exp −ax 2 + bx = π a −1/2 exp . 4a

(A.3)

Then by differentiating with respect to b, one obtains  R





√   n  (2k)! dx exp −ax + bx x = π 2k k!22k 2

n/2

n

k=0

 2  n−2k b b −1/2−n+k a exp ; 2 4a

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0

(A.4)

175

176

A Integrals, Series, and Special Functions

in particular,  R

  √ (2k)! dx exp −ax 2 x 2k = π 2k a −1/2−k . 2 k!

(A.5)

By using these integrals and by diagonalizing the quadratic form, one computes the Gaussian integrals over Rn (with a symmetric matrix A with a positive real part, Re A > 0) 





dx exp − x, Ax + B, x = π

n/2

(det A)

−1/2

Rn



 1  −1 B, A B . (A.6) exp 4

Now, by expanding this integral in Bi , we compute  Rn

  (2k)! dx exp − x, Ax x i1 · · · x i2k = π n/2 (det A)−1/2 2k A(i1 i2 · · · Ai2k−1 i2k ) . 2 k! (A.7)

Another useful integral used in the book is (with |Im b| < 1)  dx R

π  sin(bx) = π tanh b . sinh x 2

(A.8)

Finally, we used the dual relationship (with t > 0)  R

A.2

 2    x 2 −3/2 x 2 . dx exp −tx x tanh(π x) = √ t dx exp − t sinh x π

(A.9)

R

Poisson Summation Formula

We use Poisson summation formulas to obtain duality transformations. Let f be a function on R decreasing sufficiently fast at infinity and fˆ(x) =

 R

dp −ipx e f (p) 2π

(A.10)

its Fourier transform. Then there holds  n∈Z

f (n) = 2π

 k∈Z

fˆ(2π k).

(A.11)

A

Integrals, Series, and Special Functions

177

Also 

 f

n+

n∈Z

1 2



 (−1)k fˆ(2π k).

= 2π

(A.12)

k∈Z

In particular, let   f (p; t, y) = exp −tp 2 cos(py),   g(p; t, y) = p exp −tp 2 sin(py),

(A.13) (A.14)

with real parameters y and t > 0. Then

  (y − x)2 (y + x)2 1 −1/2 ˆ exp − + exp − , (A.15) f (x; t, y) = √ t 4t 4t 4 π

  (y − x)2 (y + x)2 1 −3/2 (y − x) exp − + (y + x) exp − . g(x; ˆ t, y) = √ t 4t 4t 8 π (A.16)

A.3

Bernoulli Polynomials

Bernoulli numbers, Bk , are defined by the generating function ∞

 zk zez = Bk , ez − 1 k!

(A.17)

k=0

so that B0 = 1,

1 B1 = − , 2

B2 =

1 . 6

(A.18)

By using the relation z  z z = coth − 1 , ez − 1 2 2

(A.19)

it is easy to see that the odd order Bernoulli numbers, B2k+1 , except B1 = −1/2, vanish B2k+1 = 0,

k = 1, 2, . . . .

(A.20)

178

A

Integrals, Series, and Special Functions

Other related series are ∞  z z2k 2(1 − 22k−1 )B2k = , sinh z (2k)!

(A.21)

k=0

tanh z =

∞ 

22k (22k − 1)B2k

k=1

z2k−1 . (2k)!

(A.22)

Bernoulli numbers can be expressed in terms of the integrals B2k = (−1)

k+1

π 1 − 21−2k

∞ dt 0

t 2k cosh2 (π t)

.

(A.23)

Bernoulli polynomials are defined by the generating function ∞

 zexz zk B (x) = k ez − 1 k!

(A.24)

k=0

and have the form Bn (x) =

n    n k=0

k

Bn−k x k .

(A.25)

There is a formula for the doubling of the argument    1 , Bn (2x) = 2n−1 Bn (x) + Bn x + 2

(A.26)

which enables one to compute the values B2k

A.4

    1 = − 1 − 21−2k B2k . 2

(A.27)

Gamma Function

Gamma function is a function of a complex variable z defined for Re z > 0 by ∞ Γ (z) = 0

dt t z−1 e−t .

(A.28)

A.5 Hypergeometric Function

179

It is easy to see that it satisfies the equation Γ (z + 1) = zΓ (z)

(A.29)

and for positive integer values n ∈ Z+ is equal to Γ (n) = (n − 1)!. The gamma function Γ (z) can be analytically continued to a meromorphic function of z with simple poles at the nonpositive integers, z = −k, with k = 0, 1, 2, . . . , with the residues (−1)k . k!

Res {Γ (z), −k} =

(A.30)

The gamma function satisfies the functional equation Γ (z)Γ (1 − z) =

π sin(π z)

(A.31)

and a useful formula for the doubling of the argument 22z−1 Γ (2z) = √ Γ (z)Γ π

  1 , z+ 2

which gives, in particular, the values at the half-integer points, Γ (1/2) = √ π /2, etc. The logarithmic derivative of the gamma function defines the function ψ(z) =

Γ  (z) , Γ (z)

(A.32) √ π , Γ (3/2) =

(A.33)

which, in turn, defines the Euler constant γ = −ψ(1).

A.5

Hypergeometric Function

We list below some of the properties of the hypergeometric function heavily used throughout the book (see, e.g., [25]). The hypergeometric function is a function of a complex variable z defined by the series F (a, b; c; z) =

∞  Γ (a + n) Γ (b + n) n=0

Γ (a)

Γ (b)

Γ (c) zn , Γ (c + n) n!

(A.34)

with complex parameters a, b, c. It satisfies the following identity: F (a, b; c; z) = (1 − z)c−a−b F (c − a, c − b; c; z).

(A.35)

180

A

Integrals, Series, and Special Functions

The derivative of the hypergeometric function is given by ∂z F (a, b; c; z) =

ab F (a + 1, b + 1; c + 1; z), c

(A.36)

and therefore, for any positive integer n ∂zn F (a, b; c; z) =

Γ (a + n)Γ (b + n)Γ (c) F (a + n, b + n; c + n; z). Γ (a)Γ (b)Γ (c + n)

(A.37)

Also, it is easy to see that   ∂z zc−1 F (a, b; c; z) = (c − 1)zc−2 F (a, b; c − 1; z)

(A.38)

and, therefore,   ∂zn zc−1 F (a, b; c; z) =

Γ (c) c−1−n z F (a, b; c − n; z). Γ (c − n)

(A.39)

The hypergeometric function satisfies the differential equation   z(1 − z)∂z2 + [c − (a + b + 1)z]∂z − ab F (a, b; c; z) = 0.

(A.40)

Furthermore, the hypergeometric function satisfies the duality transformation F (a, b; c; 1 − z) =

Γ (c)Γ (c − a − b) F (a, b; a + b − c + 1; z) Γ (c − a)Γ (c − b) +zc−a−b

(A.41)

Γ (c)Γ (a + b − c) F (c − a, c − b; c − a − b + 1; z). Γ (a)Γ (b)

In the limit z → 0, this gives the finite value if Re(c − a − b) > 0 F (a, b; c; 1) =

Γ (c)Γ (c − a − b) Γ (c − a)Γ (c − b)

(A.42)

and a singularity if Re(c − a − b) < 0 F (a, b; c; 1 − z) = zc−a−b

  Γ (c)Γ (a + b − c) + O zc−a−b+1 . Γ (a)Γ (b)

(A.43)

A.5 Hypergeometric Function

181

In the critical case c = a + b, the duality transformation takes the form

Γ (a + b) − [log z + ψ(b) + ψ(a) − 2ψ(1)]F (a, b; 1; z) F (a, b; a + b; 1 − z) = Γ (a)Γ (b)

−∂a F (a, b; 1; z) − ∂b F (a, b; 1; z) , (A.44) which leads to the logarithmic singularity as z → 0 F (a, b; a + b; 1 − z) = −

Γ (a + b) log z + O(1). Γ (a)Γ (b)

(A.45)

The analytic continuation of the hypergeometric series defines an analytic function of z with a branch cut along the positive real axis. The jump across the cut in the complex plane on the positive real axis (x > 1) is F (a, b; c; x + iε) − F (a, b; c; x − iε)

(A.46)

Γ (c) (x − 1)c−a−b F (c − a, c − b; c − a − b + 1; 1 − x). = 2π i Γ (a)Γ (b)Γ (c − a − b + 1) If c = a + b, this takes the form F (a, b; a+b; x+iε)−F (a, b; a+b; x−iε) = 2π i

Γ (a + b) F (a, b; 1; 1−x). Γ (a)Γ (b)

(A.47)

The asymptotics of the hypergeometric function as z → ∞ with | arg(−z)| < π are given by F (a, b, c; z) =

Γ (c)Γ (b − a) Γ (c)Γ (a − b) (−z)−a + (−z)−b Γ (b)Γ (c − a) Γ (a)Γ (c − b)     +O |z|−a+1 + O |z|−b+1 .

(A.48)

For special values of parameters, the hypergeometric function can be expressed in terms of elementary functions. If a or b is a nonpositive integer, a = −k, with k = 0, 1, 2, . . . , then F (−k, b; c; z) is a polynomial F (−k, b; c; z) =

k  n=0

(−1)n

Γ (b + n) Γ (c) n k! z , (k − n)!n! Γ (b) Γ (c + n)

(A.49)

182

A

Integrals, Series, and Special Functions

satisfying the identity F (−k, b; c; 1 − z) =

Γ (c)Γ (c − b + k) F (−k, b; b − c + 1 − k; z). Γ (c + k)Γ (c − b)

(A.50)

Also, for the functions F (a, b; c; z) with c = 1/2 and c = 3/2, there holds   1 1 − cos x = cos(ax), F a, −a; ; 2 2   1 1 1 − cos x cos(ax) 1  , + a, − a; ; = F 2 2 2 2 cos 12 x   1 3 1 − cos x sin(ax) 1  , + a, − a; ; = F 2 2 2 2 2a sin 12 x   sin(ax) 3 1 − cos x = . F 1 + a, 1 − a; ; 2 2 a sin x

A.6

(A.51) (A.52)

(A.53)

(A.54)

Legendre Functions

Legendre functions of the first and second kinds are analytic functions of z with a cut along the real line from 1 to −∞ defined by   1−z , (A.55) Pν (z) = F −ν, ν + 1; 1; 2

    1+z π 1−z Qν (z) = e− iνπ F −ν, ν + 1; 1; − F −ν, ν + 1; 1, , 2 sin(νπ ) 2 2 (A.56) where ν is a complex parameter and = sign(Im z). Legendre functions satisfy the relations μ

μ

P−ν (z) = Pν−1 (z), Qν (z) − Q−ν−1 (z) = π cot(νπ )Pν (z),

(A.57) (A.58)

as well as, for real x ∈ (−1, 1), Pν (−x) = − cos(νπ )Pν (x) −

2 sin(νπ )Qν (x). π

(A.59)

A.7 Polynomials

183

We list useful integral representations of Legendre functions used in the book √ θ cos[(ν + 12 )t] 2 , Pν (cos θ ) = dt √ π cos t − cos θ

(A.60)

0

√ π sin[(k + 12 )t] 2 Pk (cos θ ) = dt √ , π cos θ − cos t

k ∈ Z,

(A.61)

θ

   √ x cosh ν + 1 s 2 2 Pν (cosh x) = ds √ , π cosh x − cosh s 0

√

2 tan(νπ ) Pν (cosh x) = − π

∞ x

   sinh ν + 12 s ds √ , cosh s − cosh x

    √ ∞ exp − ν + 1 s 2 2 Qν (cosh x) = ds √ . 2 cosh s − cosh x

(A.62)

(A.63)

(A.64)

x

A.7

Polynomials

Let ρ(x) = (1 − x)α (1 + x)β

(A.65)

with α, β > −1 and x ∈ [−1, 1]. Jacobi polynomials are defined for n ≥ 0 by Pn(α,β) (x) =

(−1)n 1 n ∂ ρ(x)(1 − x 2 )n . n!2n ρ(x) x

(A.66)

They satisfy the differential equation 

 (1 − x 2 )∂x2 + [β − α − (α + β + 2)x]∂x + n(n + α + β + 1) Pn(α,β) (x) = 0 (A.67) and are orthogonal on [−1, 1] with the weight function ρ. Jacobi polynomials can be expressed in terms of the hypergeometric function Pn(α,β) (x) =

  1−x Γ (α + n + 1) F −n, n + α + β + 1; α + 1; . n!Γ (α + 1) 2

(A.68)

184

A

Integrals, Series, and Special Functions

In a particular case α = β, the Jacobi polynomials define the Gegenbauer polynomials Cn(α) (x) =

Γ (2α + n) Γ (α + 12 ) (α− 1 ,α− 12 ) Pn 2 (x). 1 Γ (2α) Γ (α + 2 + n)

(A.69)

Furthermore, in even more special case (as α = β → −1/2 and α = β = 1/2,) they define Chebyshev polynomials Tn (x) = Un (x) =

√ 2 π (n − 1)! Γ (n + √ π n!

1 2)

(− 12 ,− 12 )

Pn

( 1 , 12 )

4Γ (n + 32 )

Pn 2

(x),

(A.70) (A.71)

(x),

and (in the case α = β = 0) Legendre polynomials Pn (x) = Pn(0,0) (x).

(A.72)

Of course, all these polynomials can be expressed in terms of the hypergeometric function as well   Γ (2α + n) 1 1−x Cn(α) (x) = (A.73) F −n, 2α + n; α + ; , n!Γ (2α) 2 2   1 1−x , (A.74) Tn (x) = F −n, n; ; 2 2   3 1−x . (A.75) Un (x) = (n + 1)F −n, n + 2; ; 2 2 Chebyshev polynomials can also be expressed in terms of the trigonometric functions as follows: Tn (cos x) = cos(nx), Un (cos x) =

(A.76)

sin[(n + 1)x] . sin x

(A.77)

Legendre polynomials have the following useful generating function: ∞  k=0

Pk (x)t k = √

1 1 − 2xt + t 2

,

|t| < 1.

(A.78)

A.8 Dirac Matrices

A.8

185

Dirac Matrices

For the algebra of Dirac matrices, see, e.g., [42]. The Dirac matrices are complex matrices of dimension 2[n/2] satisfying the anticommutation relations γa γb + γb γa = δab I,

(A.79)

where I is the unit matrix. We define the anti-symmetrized products of Dirac matrices γa1 ...ak = γ[a1 · · · γak ] .

(A.80)

These matrices form a basis in the space of complex matrices. In particular, all the products of Dirac matrices can be expressed as linear combinations of the anti-symmetric ones. Some of the products of Dirac matrices used in the book are γa γ bc = γa bc + 2δa[b γ c] , γab γ cd =

[c d] γab cd − 4δ[a [c γb] d] − 2δ[a δb] .

(A.81) (A.82)

Thus the matrices γab form a representation of the Lie algebra so(n) [γab , γ cd ] = −8δ[a [c γb] d] .

(A.83)

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Index

A Adjoint representation, 25, 27, 43, 47 Angular operators, 15, 16 B Bernoulli numbers, 102, 177, 178 Bernoulli polynomials, 130, 178 Bi-invariant metric, 32 C Canonical coordinates, 26, 28, 29, 34–36, 43, 45, 47, 49, 77, 150, 156, 167 Cartan metric, 25, 38, 42, 47, 68 Cartan subalgebra, 25, 41 Casimir operator, 9, 10, 16, 25, 27, 30, 35, 37, 39–41, 43, 45, 48, 49, 55, 56, 71, 72, 77, 80, 143–146, 150, 151, 156, 157, 167 Character of a representation, 49, 56, 148, 149 Clifford algebra, 9, 48 D Determinant, 5 Dirac matrices, 9–11, 113, 119, 185 F Fundamental representation, 40, 48 G Gamma function, 178 Gaussian average, 12, 13

Gaussian integral, 56, 102, 131, 150, 153, 175, 176 Generators of a representation, 8–10, 26, 37, 38, 40, 41, 44, 47, 48, 50, 55, 68, 69, 79, 82, 167, 168 Geodesic, 33, 37, 55, 58, 64, 65, 169 Geodesic distance, 6, 34, 54, 55, 58, 63, 64, 74, 91, 147, 148, 152, 153 Geodesic spherical coordinates, 65

H Heat kernel, 3–7, 23, 36, 37, 43, 46, 55, 74, 91–94, 96, 97, 99, 100, 103, 104, 106, 107, 110, 111, 118, 119, 122, 124, 126, 128, 129, 132, 135–139, 142, 144, 145, 147–153, 155, 156, 163, 164, 166, 167, 169, 171, 172 Heat kernel diagonal, 4, 5, 97, 100, 104, 107, 111, 122, 124, 128, 129, 132, 136, 137, 139, 142, 149, 153, 156, 164, 166, 169, 171, 172 Heat semigroup, 3, 7, 37, 55, 143–145, 147, 151, 152, 156, 162, 164, 167, 168, 172 Heat trace, 5, 97, 101, 107, 117, 122, 128, 136, 139, 144, 146, 149, 150 Hyperbolic space, 3–5, 7, 59, 62, 64, 65, 83, 85, 91–93, 107, 110, 111, 122–124, 135, 138, 139, 156, 164, 171, 172 Hypergeometric coordinates, 91 Hypergeometric function, 18, 20, 97, 105, 108, 111, 120, 179–181, 183, 184 Hypergeometric operator, 17, 116

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. G. Avramidi, Heat Kernel on Lie Groups and Maximally Symmetric Spaces, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-27451-0

189

190 I Irreducible representation, 42, 49, 55, 56, 125, 145, 149, 153 Isometry algebra, 67, 68, 70, 76, 84, 85, 157, 162, 164 Isometry group, 63, 156, 167, 171 Isotropy algebra, 67, 125, 157

J Jacobi polynomial, 21, 121, 122, 126, 183, 184

K Killing vector field, 67, 79, 80, 85, 144, 157, 167

L Laplacian, 3, 6, 11, 17, 35, 38, 39, 53, 71–74, 77, 80, 81, 83–87, 91, 93, 98, 105, 113, 114, 116, 117, 119, 121, 125–127, 143–146, 150, 151, 155–157, 162, 164, 167, 172 Left-invariant one-form, 52 Left-invariant vector field, 29, 31, 32, 39, 52, 75, 76, 150 Legendre function, 97, 98, 103, 182, 183 Levi-Civita connection, 8, 32, 33, 38, 53, 60, 61, 66, 77, 78, 83, 86 Lie algebra, 7, 15, 25, 27, 40, 41, 44, 48, 76 Lie derivative, 10, 11, 39, 69, 70, 72, 79–81, 85, 86, 143, 151, 156, 157, 162, 164, 167, 168, 172 Lie group, 6, 25, 39, 44, 75, 145, 156

M Maximally symmetric space, 7, 11, 59, 60, 67, 75, 91, 155 Metric tensor, 52, 57 Maximal torus, 25

N Normal coordinates, 58, 64, 156

O Operator of parallel transport, 6, 155 Orthogonal group, 39, 59, 63, 82

Index Orthogonal Lie algebra, 10, 42 Orthonormal frame, 7, 57, 60, 61, 66

P Pauli matrices, 48 Poisson summation formula, 94, 176 Pseudo-orthogonal group, 63

R Rank of Lie algebra, 25, 39, 41, 48, 145 Resolvent, 3–6, 23, 92–99, 102, 103, 105–108, 110, 118–123, 126, 127, 131, 132, 155 Ricci curvature tensor, 8, 33, 59 Riemann curvature tensor, 33 Right-invariant vector field, 30, 35, 38, 52 Root of Lie algebra, 26, 34, 40, 41, 43, 48 Ruse–Synge function, 58

S Scalar curvature, 8, 11, 33, 43, 45, 53, 59 Sphere, 3–5, 7, 12–14, 17, 56, 59, 62, 65, 72, 75–77, 83, 91, 97, 99–101, 105–107, 113, 120, 125, 128, 133, 136, 137, 139, 142, 143, 147, 149, 150, 163, 167 Spherical coordinates, 14, 47, 52, 56, 65, 165 Spin connection, 9, 38, 79, 84 Spin group, 8–10, 38, 69, 75, 113, 167 Spin-tensor vector bundle, 8, 38, 69, 79, 83, 156, 167 Stereographic coordinates, 63, 64 Structure constants, 25, 26, 42, 44, 47, 50, 68, 156

V Van Vleck–Morette determinant, 6, 58, 60, 61, 155 Volume form, 8, 31, 32, 34, 36, 53, 59, 61, 65

W Warped product, 65 Weyl-Cartan basis, 41

Z Zeta function, 5