Hopf subalgebras from Green’s functions [version 15 Dec 2014 ed.]

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Hopf subalgebras from Green’s functions Masterarbeit Mathematisch-Naturwissenschaftliches Fakult¨at Instut¨ ut f¨ ur Mathematik Humboldt-Universit¨at zu Berlin.

Lucia Rotheray Geboren am 18.01.1990 in Beverley. Betreuer: Prof. Dr. Dirk Kreimer

2

Contents 1 Basic definitions 1.1 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 10 12

2 The Hopf algebra of rooted trees 13 2.1 Hopf subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Dyson-Schwinger equations . . . . . . . . . . . . . . . . . . . . . 18 3 Hopf subalgebras from Dyson-Schwinger equations 3.1 Single equation . . . . . . . . . . . . . . . . . . . . . 3.2 The operadic approach . . . . . . . . . . . . . . . . . 3.2.1 Operadic proof of theorem 4 . . . . . . . . . 3.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . 4 Application in physics

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21 21 23 24 26 29

3

4

CONTENTS

Introduction Since the late 1990s ([3],[4], 1998) there has been much work on the study of Hopf algebras as the underlying mathematical structure of local quantum field theory (QFT). In QFT, Green’s functions are developed as series in coupling constants indexed by Feynman graphs, which have a Hopf algebra structure. Here we work not in Feynman graphs but in the closely related Hopf algebra of decorated rooted trees. The Green’s functions appear as solutions to combinatorial DysonSchwinger equations (DSEs), which build families of graphs through the action of a combinatorial grafting operator. The set of graphs constructed in this way generates a subalgebra which, for certain forms of DSE, is itself a Hopf algebra. The work of Foissy classifies DSEs in Hopf algebras of trees which give rise to Hopf subalgebras ([13],[9],[10],[11],[12]), and in [1], Bergbauer and Kreimer show that the solution of a certain DSE with a coupling constant gives rise to Hopf subalgebras. Here we generalise this result to a finite system of DSEs with a finite number of coupling constants. Chapter 1 reviews algebraic (bialgebras, Hopf algebras, operads) and graphtheoretical notions (graphs, rooted trees) needed for the rest of the work, with some examples. Chapter 2 introduces the Hopf algebra HD , the grafting operator, and DSEs in HD . Chapter 3 presents and compares the two approaches given for the proof of Theorem 3 in [1], then uses the second approach to extend the theorem from a single Dyson-Schwinger equation to a system of DSEs. The final chapter introduces Feynman graphs and their relation to rooted trees, and gives some examples of DSEs in Feyman graphs.

5

6

CONTENTS

Chapter 1

Basic definitions 1.1

Hopf algebras

Let k be a field. For two vector spaces V1 and V2 , let τV1 ,V2 : V1 ⊗ V2 −→ V2 ⊗ V1 denote the twist map v ⊗ w 7→ w ⊗ v 1 . Definition 1. An (associative) k-algebra (A, m, u) is a k-vector space A together with two linear maps, m : A ⊗k A −→ A (multiplication) and u : k −→ A (unit) such that: m ◦ (id ⊗ m) = m ◦ (m ⊗ id),

(1.1)

m ◦ (u ⊗ id) = m ◦ (id ⊗ u). The multiplication is commutative if m = m ◦ τ . Definition 2. A (coassociative) k-coalgebra (C, ∆, ) is a k-vector space C together with two linear maps ∆ : C −→ C ⊗ C (comultiplication) and  : C −→ k (counit) such that: (id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆,

(1.2)

(id ⊗ ) ◦ ∆ = τC,k ( ⊗ id) ◦ ∆. The coproduct is cocommutative if τ ◦ ∆ = ∆. Remark 1. A common notation for the coproduct of an element x ∈ C which will be used here is the Sweedler notation ∆(x) = x0 ⊗ x00 . Definition 3. For two algebras (A1 , m1 , u1 ) and (A2 , m2 , u2 ), a linear map φ : A1 −→ A2 is an algebra morphism if φ ◦ u1 = u2 ,

(1.3)

φ ◦ m1 = m2 ◦ (φ ⊗ φ). 1 The

tensor product always written ⊗ should be read as ⊗k (⊗Q in chapters 2 and 3).

7

8

CHAPTER 1. BASIC DEFINITIONS

For two coalgebras (C1 , ∆1 , 1 ) and (C2 , ∆2 , 2 ), a linear map ψ : C1 −→ C2 is a coalgebra morphism if 2 ◦ ψ = 1 ,

(1.4)

∆2 ◦ ψ = (ψ ⊗ ψ) ◦ ∆1 . Definition 4. A k-bialgebra (B, m, u, ∆, ) is a k-vector space B which has a k-algebra structure (m, u) and a k-coalgebra structure (∆, ) such that (m, u) are coalgebra morphisms and (∆, ) are algebra morphisms. Theorem 1. Let B be a space with k-algebra structure (m, u) and k-coalgebra structure (∆, ). Then (m, u) are coalgebra morphisms if and only if (∆, ) are algebra morphisms. Proof. The field k has a coalgebra structure given by the isomorphisms ∆(1) = 1 ⊗ 1 and  = idk . Substitute φ = ∆ : B −→ B × B and φ =  : B −→ k into 1.3, and ψ = m and ψ = u into 1.4. The resulting four equations are the same in each case: ∆ ◦ u = (u ⊗ u), ∆ ◦ m = (m ⊗ m) ◦ (∆ ⊗ ∆),  ◦ u = 1,  ◦ m =  ⊗ .

Definition 5. A Hopf algebra (H, m, u, ∆, , S) is a bialgebra (H, m, u, ∆, ) endowed with an antipode: a map in S ∈ Homk (H, H) which satisfies m ◦ (S ⊗ id)∆ = m ◦ (id ⊗ S)∆ = u ◦ e.

(1.5)

Remark 2. One can define an algebra (Homk (H, H), ?, ε), with multiplication and unit defined by φ ? ψ = m ◦ (φ ⊗ ψ) ◦ ∆

∀φ, ψ ∈ Homk (H, H),

ε = u ◦ . Then the antipode of H can be defined by S ? idH = idH ? S = ε. Proposition 1. For any Hopf algebra H, the antipode S is unique. Proof. Let S1 , S2 be two possible antipodes. Then, since (Homk (H, H), ?, ε) satisfies 1.1, S1 = S1 ? ε = S1 ? idH ? S2 = ε ? S2 = S2 .

1.1. HOPF ALGEBRAS

9

Definition 6. A Hopf algebra over k is graded and connected if there exist Hi such that: H=

∞ M

Hi ,

(1.6)

i=0

H0 ' k,

(1.7)

m(Hi ⊗ Hj ) ⊆ Hi+j , M ∆(Hi ) ⊆ Hj ⊗ Hk .

(1.8) (1.9)

j+k=i

Example 1. For a multiplicative group (G, ·) the group algebra kG is the kvector space generated P by the elements of G. An element x ∈ kG has the form of a finite sum x = g∈G αg g, with αg ∈ k. One can construct a Hopf algebra (kG, m, u, ∆, , S) as follows: • m(α1 g1 ⊗ α2 g2 ) = (α1 α2 )(g1 · g2 ). • u(α) = α1G ∀ α ∈ k. • ∆(g) = g ⊗ g ∀g ∈ G. • (g) = 1G ∀g ∈ G. • G: S(g) = g −1 ∈ G ∀g ∈ G.. This Hopf algebra is cocommutative, and is commutative iff G is abelian. Example 2. For a vector space V over a field k, the tensor algebra of V is defined as T (V ) =

∞ M

T n (V ),

n=0

where T (V ) = V ⊗n . Consider arbitrary elements n

x ∈ V, v = v1 ⊗ ... ⊗ vn ∈ T n (V ), w = w1 ⊗ ... ⊗ wm ∈ T m (V ) of T (V ). Define a Hopf algebra (T (V ), m, u, ∆, , S) by: • m(v, w) = v1 ⊗ ... ⊗ vn ⊗ w1 ⊗ ...wm ∈ T n+m (V ). • u(v) = 1k v • ∆(x) = 1k ⊗ x + x ⊗ 1k . ( x, x ∈ V • e(x) = 0, f v ∈ T n (V ), n > 1

.

• S(x) = −x. This Hopf algebra is cocommutative but not commutative, and is graded connected with homogeneous components T i (V ): T 0 (V ) ' k and T 1 (V ) = V .

10

CHAPTER 1. BASIC DEFINITIONS

1.2

Rooted trees

Definition 7.
A graph G consists of a set of vertices V (G) and a set of edges 2 E(G) ⊆ V2 . • v ∈ V (G) and e ∈ E(G) are said to be incident if e = vw for some w ∈ V (G). • Two edges e1 , e2 ∈ E(G) are adjacent if they have a common incident vertex: e1 = vw and e2 = wx, for v, w, x ∈ V (G). • Two vertices v, w ∈ V (G) are neighbours if vx ∈ E(G). • The degree of a vertex v is the number of neighbours it has (equivalently, the number of edges incident to it). • A path is a subset P = {e1 , e2 , e3 , ...} ⊆ E(G) such that any two edges ei , ei+1 ∈ P are adjacent. • A path {e1 , ..., ek } ⊆ E(G) will be denoted by (v1 , v2 ) where e1 = v1 w, e2 = wu, ... , ek−1 = tv2 , ek = xv2 for some t, u, v1 , v2 , w, x ∈ V (G). • A cycle is a path of the form (v, v). • A graph is connected if there exists a path (v, w) for any v, w ∈ V (G). Definition 8. A connected graph with no cycles is called a tree. • A rooted tree is a tree T along with some distinguished vertex r ∈ V (T ) which is called the root (here the root will always be drawn at the top of the tree). • A vertex v ∈ V (T ) \ {r} with degree one is called a leaf of T . • w ∈ V (T ) is a child of v ∈ V (T ) if any path (r, w) contains the edge wv. • The fertility of a vertex is the number of children it has (fertility(v)=degree(v)1). • A union of (rooted) trees, a graph which has no cycles but is not necessarily connected, is called a (rooted) forest. • We call the set of all rooted trees T and the set of all rooted forests F. Definition 9. A decorated rooted tree consists of a rooted tree T , a countable set D and a surjective map c : D −→ V (T ), which assigns an element of D to each vertex of T . For any v ∈ V (T ), we denote c−1 (v) = d(v). Let TD and FD denote respectively the sets of rooted trees and forests decorated by a set D. Definition 10. An admissible cut of T is a nonempty subset c ( E(T ) such that | c ∩ (r, v) |≤ 1 2 We

∀v ∈ V (T ).

assume graphs here to be simple, i.e. the edges are distinct.

1.2. ROOTED TREES

11

• Call the set of all admissible cuts C(T ). • Denote by C ∗ (T ) the union C(T ) ∪ ∅ of admissible cuts and the empty cut. • Given a cut c ∈ C(T ), we construct a disconnected graph G with E(G) = E(T ) − c and V (G) = V (T ). • The connected component of G which contains the root of T is a tree which we call Rc (T ). • The component(s) of G not connected to the root of T give a forest which we call P c (T ). Example 3.

G1 = is a graph which is not connected and contains one cycle.

G2 = is a subgraph of G1 which is a tree with two leaves. To look at the cuts of G2 , we label the edges:

1 3 C

P C (G2 )

G

{1} {2}

•

•

{3} {1,2}

•

••

{1,3} {2,3} {1,2,3}

••

•

•

•••

2

RC (G2 )

C Admissible?

•

Yes

•

Yes

•

Yes

×

×

No

×

×

No

•• ×

×

Yes No

12

CHAPTER 1. BASIC DEFINITIONS

1.3

Operads

Definition 11. An operad O

3

consists of:

• A collection of objects {O(n)}n∈N • A collection of maps {◦i }i , ◦i : O(m) × O(n) −→ O(m + n − 1) Example 4 (The endomorphism operad). For a set X, M ap(X n , X) is the set of functions from X n to X. The endomorphism operad End(X) consists of the set of objects {M ap(X n , X)}n∈N∗ and maps ◦i defined by (f ◦i g)(x1 , ..., xm+n−1 ) = f (x1 , ..., xi−1 , g(xi , ..., xi+m−1 ), xi+m , ..., xm+n−1 ), where f ∈ M ap(X n , X), g ∈ M ap(X m , X) and xj ∈ X ∀j. Example 5 (The rooted trees operad). Take the set T (n) of rooted trees T with n leaves {l1 , ..., ln } ⊂ V (T ). Then there exists an operad T with the set of objects {T (n)}n∈N∗ and the maps ◦i , where T1 ◦i T2 means “graft the root of T2 to the root li ∈ V (T1 )”.

3 For

a formal introduction to and definition of algebraic operads, see [7].

Chapter 2

The Hopf algebra of rooted trees We consider a Hopf algebra HD over Q generated by TD , where D = {dn }n∈N is a set which we may take, without loss of generality, to be N. Define a Hopf algebra structure (HD , m, I, ∆, ˆI, S): • T1 T2 := m(T1 ⊗ T2 ) is given by the forest T1 ∪ T2 . • The unit map sends q ∈ Q to qI ∈ HD 1 . • The coproduct on HD is defined on trees using admissible cuts: X ∆(T ) = I ⊗ T + T ⊗ I + P c (T ) ⊗ Rc (T )

(2.1)

c∈C(T )

and extended over all of HD by ∆(T1 T2 ) = ∆(T1 )∆(T2 ). ( 1, T = I • The counit ˆI is given by ˆI(T ) := . 0, T 6= I. • Using these definitions of m and ∆ along with the defining property 1.5 for the antipode, and S(I) = I obtain an expression for S acting on a tree T ∈ H: X m(S ⊗idH )(∆(T )) = S(I)T +S(T )I+ S(P c (T ))Rc (T ) = I(ˆI(T )) = 0 c

⇒ S(T ) = −T −

X

S(P c (T ))Rc (T ).

(2.2)

c

This can be extended to all of HD by S(T1 , ..., Tk ) = S(Tk )...S(T1 ). Example 6.

∆( 3

1

1

1

2

2

2

4

)=

3

4

⊗I+I⊗ 3

2 4

+3

4 ⊗•1 +•3 ⊗

1

1

2

2

4

+•4 ⊗

3

1 +•3 •4 ⊗

1 It is a standard abuse of notation to use I for both the unit map and the “empty tree” defined by V (I) = ∅.

13

2,

14

CHAPTER 2. THE HOPF ALGEBRA OF ROOTED TREES 1

1

2

S( 3

2

2 4

)=−3

2

4

+3

4 •1 −•3

4

2

−•4

3

+•3 •4 •2 +•3

1

1

2

2

4

+•4

3

1 −•3 •4

For any tree T ∈ HD , define n(T ) = (n1 , n2 , ....) by ni :=| {v ∈ V (T ) : d(v) = dn ∈ D} | . Pk For a forest F = T1 ...Tk , ni (F ) = l=1 ni (Tl ). Define H n = spanQ {F ∈ FD : n(F ) = n}. Then M Hn HD = n∈N∞

is a grading onL HD . Aug HD := n∈N∞ \{0} H n is the augmentation ideal of HD . It contains all elements in HD which vanish under the action of the counit. Theorem 2. (HD , m, I, ∆, ˆI, S) as defined above gives a graded connected, commutative, non-cocommutative Hopf algebra. Proof. We need to show that this construction satisfies all the defining equations 1.1 -1.9 from the previous chapter. Let all Fi and Ti be arbitrary forests and trees (respectively) in HD , and q ∈ Q. • m is associative: m(id ⊗ m)(F1 ⊗ F2 ⊗ F3 ) = F1 (F2 F3 ) = (F1 F2 )F3 = m(m ⊗ id)(F1 ⊗ F2 ⊗ F3 ) • I is a unit: m(I ⊗ id)(q ⊗ F1 ) = qIF1 = F1 qI = m(id ⊗ I)(F1 ⊗ q). • ˆI is a counit: (id ⊗ ˆI)∆(T ) = T ⊗ 1 = τC,k (1 ⊗ T ) = (ˆI ⊗ id)∆(T ). • ∆ is an algebra morphism: ∆(I(q)) = qI ⊗ I = (I ⊗ I)(qI ⊗ I), ∆(m(F1 ⊗ F2 )) = ∆(F1 )∆(F2 ) = m((∆ ⊗ ∆)(F1 ⊗ F2 )). • ˆI is an algebra morphism: ˆI(I(q)) = ˆI(qI) = q = 1q = uQ q, ˆI(m(F1 ⊗ F2 )) = ˆI(F1 F2 ) = m(ˆI(F1 ) ⊗ ˆI(F2 )) = m(ˆI ⊗ ˆI)(F1 ⊗ F2 ). • S satisfies 1.5 by construction. L • H= Hi gives a grading: H 0 = spanQ {F ∈ FD : V (F ) = ∅} = QI ' Q, m(H j ⊗ H k ) ⊆ {F ∈ FD : ni = ji + ki } = H j+k , ∆(H j ) = {F : F = P c (T ), T ∈ H  } ⊗ {T 0 : T 0 = Rc (T ), T ∈ H  } M ⊆ H k ⊗ H l. ki +li =ji

2.

15 • Commutativity: F1 F2 := F1 ∪ F2 = F2 ∪ F1 =: F2 F1 .

• Non-cocommutativity: in general τHD ,HD (∆(F )) 6= ∆(F ), as ∆(T ) ∈ HD ⊗ TD and ∆ ◦ τHD ,HD (T ) ∈ TD ⊗ HD for any tree T . For proof that ∆ is coassociative, see proposition 3.

d Definition 12. For each d ∈ D, the grafting operator B+ on HD is a linear d d map B+ : HD −→ spanQ (TD ). B+ acts on a rooted forest T1 , ..., Tk by joining the roots of T1 , ..., Tk to a new root decorated by d giving a tree:

d d B+ (T1 ...Tk )

.

= T1 T2

.......

T3

1 2

2 1 Example 7. Let D = N. B+ (

• ) 3 2

= 3

2 .

d Proposition 2. Each grafting operator B+ satisfies

d d d ∆ ◦ B+ = B+ ⊗ I + (id ⊗ B+ ) ◦ ∆.

(2.3)

d Proof. Consider a forest F = T1 ...Tk ∈ HD . B+ (F ) = T is a tree in H. Let ∗ C (T ) = C(T ) ∪ ∅ for any tree.

For any cut c ∈ C(T ), the restriction of c to any subtree Ti is either in C ∗ (Ti ) or is the total cut on Ti . Consider any set of cuts c1 , ..., ck such that ci ∈ C ∗ (ti ). Then there exists a unique cut c ∈ C ∗ (T ) such that the restriction of c to any Ti gives the corresponding ci . For this cut, we have

P c (T ) =

k Y

P ci (Ti ),

i=1 c

R (T ) =

dn B+

k Y i=1

! ci

R (Ti ) .

16

CHAPTER 2. THE HOPF ALGEBRA OF ROOTED TREES

So: ∆(T ) = T ⊗ I + I ⊗ T +

X

P c (T ) ⊗ Rc (T )

c∈C(T )

=

d B+ (F )

X

⊗I+

P c (T ) ⊗ Rc (T )

c∈C ∗ (T )

=

d B+ (F )

⊗I+ ci

X

k Y

∈C ∗ (T

i=1

i)

! ci

P (Ti ) ⊗

d B+ (Rci (Ti ))



 =

d B+ (F )

=

d B+ (F )

  k  X  Y   ci ci P (Ti ) ⊗ R (Ti )    i=1 ci ∈C ∗  | {z }

⊗ I + (id ⊗

d B+ )

⊗ I + (id ⊗

d B+ )∆(F ).

= ∆(T1 )...∆(Tk )

Remark 3. This proposition is, in fact, saying the the operator B+ is a 1cocycle in the Hochschild cohomology of HD 2 . Proposition 3. The coproduct ∆ is coassociative. Proof. Show that ∆ satisfies 1.2. Clearly (∆ ⊗ idH )∆(I) = ∆(I) ⊗ I = I ⊗ I ⊗ I = I ⊗ ∆(I) = (idH ⊗ ∆)∆(I). We proceed by induction on | V (F ) | for any F ∈ HD . Assume that for all | V (F ) |< n, the action of ∆ on F is coassociative, and consider F ∈ HD with | V (F ) |= n. 1. If F is not a tree, F = T1 ...Tk and each | V (Tk ) |< n, so by assumption ∆ is coassociative on each Ti and therefore on F . d (X) for some X ∈ H and d ∈ D, with 2. If F is a tree, then F = B+ | V (X) |= n − 1. By assumption ∆ acts coassoiatively on X, and applying equation 2.3 gives: d d d (id ⊗ ∆)∆(B+ (X)) = (id ⊗ ∆)(B+ (X) ⊗ I + (id ⊗ B+ )∆(X)) d d = B+ (X) ⊗ I ⊗ I + (id ⊗ ∆ ◦ B+ )∆(X) d d d )(id ⊗ ∆)∆(X) . = B+ (X) ⊗ I ⊗ I + (id ⊗ B+ )∆(X) ⊗ I + (id ⊗ id ⊗ B+ {z } | {z } | ?

(∆ ⊗

?? d d = (∆ ⊗ id)(B+ (X) ⊗ I + (id ⊗ B+ )∆(X)) d d = ∆(B+ (X)) ⊗ I + (∆ ⊗ B+ )∆(X) d d B+ )∆(X) ⊗ I + (id ⊗ id ⊗ B+ )(∆ ⊗ id)∆(X) .

d id)∆(B+ (X))

d = B+ (X) ⊗ I ⊗ I + (id ⊗ {z | ?

}

|

{z ??

The terms labelled ? are immediately equal, the terms marked ?? cancel under the assumption on X. 2 for

more information see, e.g. [3], [8].

}

2.1. HOPF SUBALGEBRAS

2.1

17

Hopf subalgebras

Definition 13. A sub-Hopf algebra of a graded Hopf algebra (H, m, u, ∆, , S), L H = i Hi , is aL subspace H 0 ⊂ H with Hopf algebra structure (H 0 , m, u, ∆, , S) 0 such that H = i (H 0 ∩ Hi ) gives a grading on H 0 . Example 8. The ladder on n vertices is the tree given by λn := (B+ )n (I).

λ0 = I, λ1 = •, λ2 =

, λ3 =

, λ4 =

.

L

Let H be the subalgebra of H generated by the setP of all ladders, and HnL ⊂ Hn those elements which have n vertices. ∆(λn ) = m≤n λm ⊗ λn−m : HL is a Hopf subalgebra of H. Example 9. More generally, the subalgebra generated by all trees whose vertices have fertility bounded from above by n is a Hopf subalgebra (the ladders are the case n = 1). ( 1, di = d Example 10. For any d ∈ D, the set of forests with ni (F ) = 0 di 6= d forms a Hopf subalgebra. It is isomorphic to the Hopf algebra H of undecorated rooted trees, which has m, I, ∆, ˆI and S ad HD , but only one grafting operator B+ . Example 11 (The Connes-Moscovici Hopf algebra). The natural growth operator is a linear map N : H −→ H defined by N (I) = • and X N (T ) := Tv , v∈V (T )

where Tv is the tree obtained by adding one extra vertex w to V (T ), and the edge vw to E(T ). N acts as a derivation on forests. Define δn := N n (I). δ0 = I δ1 = • δ2 = δ3 =

δ4 = 3

+

+

+

+

18

CHAPTER 2. THE HOPF ALGEBRA OF ROOTED TREES

Theorem 3. The elements {δn }n∈N generate a Hopf subalgebra HCM of H. Proof. We need to P show that ∆(HCM P ) ⊆ HCM ⊗ HCM and S(HCM ) ⊆ HCM . If we write δn = i Ti and δn+1 = i0 Ti0 , we have ∆(δ0 ) = δ0 ⊗ δ0 , ∆(δ1 ) = δ1 ⊗ δ0 + δ0 ⊗ δ1 ,  ∆(δn ) = δn ⊗ δ0 + δ0 ⊗ δn +

X

 X

 i

P c (Ti ) ⊗ Rc (Ti ) ,

c∈C(Ti )

∆(δn+1 ) = δn+1 ⊗ δ0 + δ0 ⊗ δn+1 + nδ1 ⊗ δn   X X  + N (P c (Ti )) ⊗ Rc (Ti ) + P c (Ti ) ⊗ N (Rc (Ti )) i

c∈C(Ti )

 +

 X

X  i

δ1 | Rc (Ti | P c (Ti ) ⊗ Rc (Ti ) .

c∈C(Ti )

By the assumption ∆(δn ) ∈ HCM ⊗HCM , it immediately follows that ∆(δn+1 ) ∈ HCM ⊗ HCM and S(HCM ) ∈ HCM .

2.2

Dyson-Schwinger equations

A (combinatorial) Dyson-Schwinger equation builds a family of forests through d the action of the grafting operators {B+ }d∈D on elements of HD . Example 12. Consider the Hopf algebra H of undecorated rooted trees, and the equation X = αB+ (X 2 ) for a parameter α (which is called a coupling constant), and X ∈ H[[α]]. This P∞ equation has unique solution X(α) = n=0 cn αn , where the family {cn }n∈N are defined recursively by c0 = I and   X cn = B +  cm cn  . m≤n−1

The first five members of this family are c0 = I c1 = • c2 = 2 c3 = 4

c4 = 8

+

+

+2

2.2. DYSON-SCHWINGER EQUATIONS In the next chapter we study a more general DSE X =I+

∞ X

dn ωn αn B+ (X n+1 ),

n=1

with X ∈ HD [[α]], and then a system of DSEs given by Xi = I +

X

  di ωρi α1ζ1 ...αnζn B+ρ X1ζ1 s1 ...Xiζi si +1 ...Xnζn sn ,

ρ

with Xi ∈ HD [[{α1 , ..., αn }]].

19

20

CHAPTER 2. THE HOPF ALGEBRA OF ROOTED TREES

Chapter 3

Hopf subalgebras from Dyson-Schwinger equations In this chapter we consider solutions X X(α) = αn cn n

or, more generally, X

Xi (α1 , ..., αm ) =

km i α1k1 ...αm ck1 ,...,km

k1 ,...,km

to DSEs, and the algebra HD c generated by the coefficients cik1 ,...,km ∈ HD . Let Hcn = H n ∩ HD c . The goal of this chapter is to show that these subalgebras HD c ⊂ HD are Hopf. That is, that HD c is closed under ∆ and S, and respects the grading on HD : M ∆(cik ) ∈ Hcn ⊗ Hcm , (3.1) ni +mi =ki

S(cik ) ∈ Hck . To this end, we show that Rc (cik ) ∈ HD lc and P c (cik ) ∈ HD k−l for some l. Then c the conditions 3.1 will follow immediately from the definitions 2.1 and 2.2.

3.1

Single equation

Following the example of [1], we begin by studying the DSE X =I+

∞ X

dn ωn αn B+ (X n+1 )

(3.2)

n=1

with solutions X(α) ∈ HD [[α]]. Lemma 1. Equation 3.2 has a unique solution X X= αn cn , n

with cn ∈ HD for all n ∈ N. 21

(3.3)

22CHAPTER 3. HOPF SUBALGEBRAS FROM DYSON-SCHWINGER EQUATIONS Proof. Substitute the ansatz 3.3 into 3.2 to find

cn =

c0 = I, 



n X

X

dm  ωn B+

m=1

(3.4)

ck1 ...ckm+1  .

k1 +...+km+1 =n−m

Theorem 4. The coefficients cn generate a Hopf subalgebra of HD :

∆(cn ) =

n X

Pkn ⊗ ck ,

(3.5)

k=0

where Pkn is a homogeneous polynomial of degree n − k in the ck (k ≤ n). Proof. ∆(c0 ) = c0 ⊗ c0 , ∆(c1 ) = c1 ⊗ c0 + c0 ⊗ c1 , Now proceed by induction: assume that 3.5 holds for all ci , i ≤ n − 1. Let ? denote the condition k1 + ... + km+1 = n − m. Using the 1-cocycle property (2.3) of the grafting operators, see that !! n X X dm ck1 ...ckm+1 ∆(cn ) = ωm ∆ B+ ?

m=1

! = cn ⊗ I +

X

ωm (id ⊗

X

dm B+ )

m

∆(ck1 )...∆(ckm+1 ) ,

?

and using the assumption on ci , i ≤ n:

∆(cn ) = cn ⊗ I +

X

 X X dm  ωm (id ⊗ B+ ) ( Plk1 ⊗ cl )...(

m

?

l1 ≤k1

 X

km+1

Pl

lm+1 ≤km+1



= cn ⊗ I +

X m≤n−1

   X   km+1 dm ωm  Plk11 ...Plm+1 ⊗ B+ (cl1 ...clm+1 )  . ?, li ≤ki

For any fixed m, let l1 , ..., lm+1 vary. ∆(cn ) has a term  X km+1 dm Plk11 ...Plm+1 ⊗ B+ (cl1 ...clm+1 ) . ? k

m+1 By assumption, each product Plk11 ...Plm+1 is a polynomial of degree n − m − (l1 + ... + lm+1 ), and therefore so is the left hand side of this tensor product. The corresponding right hand side is nothing butPa single term in the expression 3.4 for cn−m . Therefore summing over m gives k≤n Plk ⊗ ck , since k = n − m by definition.

⊗ cl )

3.2. THE OPERADIC APPROACH

3.2

23

The operadic approach

The next step is to extend theorem 4 to cover a system of DSEs. Continuing with a proof following the method above would be possible, but labourious (and error-prone), with huge products of sums and sets of indices. Instead, we present the operadic proof given by Bergbauer and Kreimer in [1] and use it to reduce the proofs of theorem 4 and the main theorem (theorem 5) to simple counting exercises. Outline of the approach: • Consider trees in HD c as operadic objects, with inputs at every vertex and an output at the root. • We consider these operadic objects to be planar, i.e. 1

1 2

2

1

1

1

1

6=

,

whereas the forests in HD are nonplanar: 1

1 1

2

1

=

1

2 1.

• The number of inputs at each vertex is determined by the particular form of the DSE, and there may be different types of in/outputs. • The operadic maps ◦i are given composition of the output and the input at the ith leaf (where we can number the leaves, without loss of generality, 1,2,... working left-to-right). For example, 2

1 1

2 ◦1

2 =

2

2

2

.

• Construct the solution (a series in the coupling constants with operadic coefficients ν) to an operadic version of the DSE. • Translate this solution back into the solution X in terms of the c ∈ HD c by removing the in/outputs and removing the planarity restriction to recover weights. Example 13. We have already seen the DSE X = αB+ (X 2 )

24CHAPTER 3. HOPF SUBALGEBRAS FROM DYSON-SCHWINGER EQUATIONS and its solution. The operadic version of the coefficient c3 is

+

ν3 =

3.2.1

+

+

+

.

Operadic proof of theorem 4

Consider the operadic fixpoint equation: G=I+

X

αn µn+1 (G⊗n+1 ).

(3.6)

n

Each µj is a map V ⊗j −→ V for some vector space V , and G(α) is a series in α with coefficients ν: G(α) = I +

X

α k νk .

(3.7)

k

Lemma 2. Each νk is the sum with unit weights over all maps V ⊗k+1 −→ V obtained by compositions of the “undecomposable” maps µj . Proof. By simply inserting 3.7 into 3.6, we find that ν0 = I and ν1 = µ2 ◦ I = µ2 . Assume for some n ∈ N that the lemma holds for all k < n. Then we look at the coefficients of αn and see νn = µn+1 ◦ I + µn ◦1 ν1 + µn ◦2 ν1 ... + µ1 ◦1 νn−1 + ... + µ1 ◦1 νn−1 , and by assumption on ν2 , ..., νn−1 the lemma holds for νn . For this equation each map µj is a single vertex with j + 1 inputs:

µ1 =

1

, µ2 =

2

, µ3 =

3

, ....

Remark 4. To compute a given νk , it is not necessary to know all νj for j < k, as for the coefficients ck above. One can simply construct νk by making the appropriate compositions of the undecomposable µi .

3.2. THE OPERADIC APPROACH

25

Example 14. Let ωn = 1 for all n, and let D = N. 2

+

1

1

1

1

1

1

+

1

1

+

+

2

+

2

+

1

1

+

1 1

+

2

1

3

ν3 =

2

1

+

1

⇒ c3 = •3 + 3

1

+a

+4

.

We look at the action of the coproduct ∆ on the νn : ∆(νn ) = νn0 ⊗ νn00 . νn00 is the sum of Rc (νn ) terms (the admissible cuts of νn are exactly the admissible cuts of cn ). Lemma 2 tells us that νn is the sum over all maps with unit weight from V ⊗n+1 to V . As ∆ takes all admissible cuts, νn0 contains all maps with unit weight from any V ⊗k+1 , with k ≤ n − 1 to V , and therefore sums over every νk : V ⊗k+1 −→ V, with k ≤ n. νn0 contains all the terms P c (νn ). The left hand side corresponding to a νk on the right hand side of the tensor product is a sum over all forests (νi1 )r1 ...(νit )rt : V ⊗n+1 −→ V ⊗k+1 , with conditions: r = r1 + ... + rt = k + 1,

(3.8)

r1 i1 + ... + rt it + r = n + 1, ⇒ r1 i1 + ... + rt it = n − k. So we have ∆(νn ) =

X

Qnk ⊗ νk ,

k≤n

with Qnk =

X

(νl1 )r1 ...(νlt )rt .

3.8

What changes upon translating this equation back into the nonplanar form 3.5? Each term in the sum over k ≤ n will pick up some factor Fkn , and to avoid over-counting any monomial (ci1 )r1 ...(cit )rt we add in the condition that 1 ≤ i1 < ... < it (which we denote by ∗). To compute Fkn , we need to count how many planar forests (νi1 )r1 ...(νit )rt will map to the same nonplanar forest when the planarity restirction is lifted. This is the same as counting how many ways a given (νi1 )r1 ...(νit )rt : V ⊗n+1 −→ V ⊗k+1 can be composed with a the corresponding νk : V ⊗k+1 −→ V .

1

26CHAPTER 3. HOPF SUBALGEBRAS FROM DYSON-SCHWINGER EQUATIONS There are (k + 1)! possibilities for switching the positions of any the νl appearing in a monomial, and r1 !r2 !...rt ! of them give the same result, even in the planar setting. So (k + 1)! . Fkn = r1 !...rt ! Pn This means that ∆(cn ) = k=0 Pkn ⊗ ck , with Pkn =

X (k + 1)! r1 !...rt !

?,∗

3.3

cri11 ...critt .

Main theorem

For ease of notation, we may write multi-indices as k = k1 , ..., kn and so on. Consider a system of n DSEs, with n coupling constants α1 , ..., αn given by:   X di (3.9) ωρi α1ζ1 ...αnζn B+ρ X1ζ1 s1 ...Xiζi si +1 ...Xnζn sn , Xi = I + ρ

where • The vector ρ is defined by ρk = ζk sk . • Xi = Xi (α1 , ..., αn ) ∈ HD [[{α1 , ..., αn }]] • The powers ζ1 , ..., ζn ∈ Z vary • s1 , ..., sn ∈ Z, si is fixed for each Xi . Lemma 3. The equation 3.9 has a unique solution X Xi = α1k1 ...αnkn cik1 ,...,kn

(3.10)

kj ∈N

Theorem 5. The coefficients cik generate a Hopf subalgebra of HD : ∆(cik ) =

X

i Pkk;i 0 ⊗ ck0 ,

ki0 ≤ki

Remark 5. This new result, appearing here for the first time, is a direct generalisation of work by Lo¨ıc Foissy, allowing for several coupling constants. Proof. Consider the operadic equation X ζ 1 s1 i si +1 Gi = I + α11 ...αnζn µiρ (G⊗ζ ...G⊗ζ ...Gn⊗ζn sn ), 1 i

(3.11)

ρ

with irreducible maps µiζ1 ,...,ζn : V1⊗ζ1 ⊗...⊗Vn⊗ζn −→ Vi , for some vector spaces V1 , ..., Vn . This has solutions X Gi = Gi = I + α1k1 ...αnkn νki . k∈Zn

3.3. MAIN THEOREM

27

Lemma 4. Each νki is the sum with unit weights over all maps V1⊗k1 s1 ⊗ ... ⊗ Vi⊗ki si +1 ⊗ ... ⊗ Vn⊗kn sn −→ V i obtained by compositions of undecomposable maps µiζ . As for lemma 2, this can be proved by induction. This structure of the νki is the basis for the proof of theorem 5. Consider the form of ∆(νki ) = ν 0 ⊗ ν 00 . The side ν 00 contains all possible trees obtained by removing some (non-root) forest in νki (which by lemma 4 means the sum of all νli : V1⊗l1 s1 ⊗ ... ⊗ Vi⊗li si +1 ⊗ ... ⊗ Vn⊗ln sn −→ V i where li ≤ ki ). The part of ν 0 corresponding to each νli contains the forests removed from νki to obtain νli , products which form maps V1⊗k1 s1 ⊗ ... ⊗ Vi⊗ki si +1 ⊗ ... ⊗ Vn⊗kn sn −→ V1⊗l1 s1 ⊗ ... ⊗ Vi⊗li si +1 ⊗ ... ⊗ Vn⊗ln sn . Consider one such term: a monomial(νli11 )r1 ...(νlitt )rt , where ( sj kj , j 6= i 1 t t (3.12) r1 lj + ... + r lj = si ki + 1, j = i ( X sj lj , j 6= i j r = rk = . (3.13) si li + 1, j = i i =j k

Now we have an expression X i ∆(νki ) = Qk;i l ⊗ (νl ), lj ≤kj j=1,..,m

where Qkn;i =

X

(νli11 )r1 ...(νlitt )rt .

3.12,3.13

Again, translation back to the planar picture adds a factor Flk;i to each term and the constraint (call it ∗) that the upper indices ij are distinct. To compute Flk;i , count the number of terms in Qn;i monomial k which translate to the same Q Q in Pkn;i : there are j rj ! ways to switch the order of the ν, and j rj ! are equivalent even in the operadic setting. This gives: Flk;i =

r1 !...rt ! , r1 !...rt !

so that Plk;i =

r1 !...rt ! i1 r1 (cl1 ) ...(ciltt )rt . r !...r ! 1 t 3.12,3.13,∗ X

28CHAPTER 3. HOPF SUBALGEBRAS FROM DYSON-SCHWINGER EQUATIONS

Chapter 4

Application in physics Definition 14. A Feynman graph is a connected multigraph1 Γ which has two kinds of vertices, called internal and external. We write this as V (Γ) = V int ∪ V ext . An external vertex is simply any vertex of degree 1. The unique edge indcident to an external vertex is called an external edge. All other vertices and edges are called internal. A Feynman graph has the following properties: • | V int |≥ 2 • There are several types of edge and vertex (i.e. there are decorations on both V and E). The different edge types represent different types of particle, while the vertex types represent possible interactions of the theory in question. • Γ must be “1-particle-irreducible”(1P I): Γ remains connected after deletion of any single internal edge. • The internal edges and vertices may be weighted. We refer to the sets V ext and E ext as the external structure of Γ. The set of Feynman graphs forms a graded Hopf algebra (HF G , m, I, ∆, ˆI, S) ([5]). Just as the coproduct of HD sums over ways to decompose a forest into smaller forests, the coproduct of HF G keeps track of all cycles in a Feynman graph: ∆(Γ) = Γ ⊗ I + I ⊗ Γ +

X γ

γ⊗

Γ , γ

where γ are the subgraphs of Γ whose connected components are 1P I (and whose external structure and weight may be constrained), and Γγ is the graph obtained by shrinking γ in Γ to a single vertex. The insertion operator Bγ on Feynman graphs is a linear map in Bγ : HF G −→ HF G which acts on Γ by replacing some internal edge of Γ by the graph γ 2 . In perturbative QFT, Feynman graphs are 1 Edges

are not necessarily distinct. d Hochschild 1-cocycles B+i of HD and Bγ of HF G are linked by a Hopf algebra morphism, see theorem 5 of [13]. 2 The

29

30

CHAPTER 4. APPLICATION IN PHYSICS

associated via diffeomorphisms (called Feynman rules) to integrals 3 which form a perturbation series describing a given interaction. Unfortunately for quantum field theorists, the integrals encountered are often divergent (a divergence in an integral corresponds to a cycle in its associated Feynman graph). The art of renormalisation 4 exists to tame these integrals and produce a usable field theory . In the Hopf-algebraic setting, one associates a decorated rooted tree to each graph Γ, which represents the structure of the divergences of Γ ([8],[5],[6]). Example 15. In φ3 -theory, the graph Γ = γ1 =

, γ2 =

. This gives

∆(Γ) = Γ ⊗ I + I ⊗ Γ + γ1 ⊗ ⊗I+I⊗

=

has two sub-divergences:

Γ Γ Γ + γ2 ⊗ + γ1 ∪ γ2 ⊗ γ1 γ2 γ1 ∪ γ2

+

⊗

+

⊗

+

We can associate to each sub-divergence a vertex in a rooted tree T . The γ3 tree associated to Γ is

γ2 . γ1 Dyson-Schwinger equations are equations of motion for Green’s functions in QFT. They take the form of combinatorial equations based on the action γ of B+ in the Hopf-algebraic setting, and of analytic integral equations which result from applying Feynman rules to the combinatorial DSEs. The Green’s functions themselves appear as infinite sums in coupling constants, indexed by Feynman graphs with a certain external structure. Example 16. In quantum electrodynamics (QED) there are two possible edge types ,

,

and three possible external structures for any graph. We let X1 =

, X2 =

, X3 =

denote the infinite sums of graphs with each particular external sturcture, where denotes all possible internal structures. The Dyson-Schwinger equations of this theory are   (1 + X3 )2 X1 = B (1 − X1 )(1 − X2 ) (1 + X3 )2 X2 = B (1 − X1 )2   1+2l(γ) X (1 + X3 ) X3 = Bγ (1 − X1 )2l(γ) (1 − X2 )l(γ) γ 3 An explanation of the concepts and methods of perturbative QFT can be found in, for example, the early chapters of [15]. 4 Described in great detail in, for example, [2].

⊗

.

31 where l(γ) is the number of cycles in γ. Example 17. Consider a theory with two edge types ,

,

and three allowed vertex types

v1 =

, v2 =

, v3 =

.

The Feynman graphs of this theory with one cycle are

γ1 =

γ4 =

γ7 =

, γ2 =

, γ5 =

, γ8 =

+

, γ3 =

, γ6 =

, γ9 =

Associate to each vertex type vi a coupling constant αi . Again, denote

X1 =

, X2 =

, X3 =

.

After a truncation, the Dyson-Schwinger equations are X1 = I + α12 Bγ1 (X13 ) + α2 Bγ2 (X1 X2 ) + α3 Bγ3 (X1 X3 ), 1 X2 = I + α12 α3 Bγ4 (X12 X3 ) + α3 Bγ5 (X2 X3 ) + α12 Bγ6 (X12 X2 ), α2 1 X3 = I + α3 Bγ7 (X32 ) + α12 Bγ8 (X12 X3 ) + α14 Bγ9 (X14 ). α3 This system is of the form 3.9, so it does indeed generate a Hopf subalgebra of the algebra HF G generated by all Feynman graphs of this theory.

32

CHAPTER 4. APPLICATION IN PHYSICS

Bibliography [1] C. Bergbauer and D. Kreimer. Hopf algebras in renormalisation theory: locality and Dyson-Schwinger equations from Hochschild cohomology. IRMA Lect.Math.Theor.Phys. 10 (2006) 133-164, arXiv hep-th/0506190. [2] J.C. Collins. Renormalisation. Cambridge univeristy press 1984. [3] A. Connes and D. Kreimer. Hopf algebras, renormalisation and noncommutative geometry. Commun.Math.Phys. 199 (1998) 203-242, arXiv hepth/9808042. [4] D. Kreimer. On the Hopf algebra structure of perturbative quantum field theory. Adv.Theor.Math.Phys. 2 (1998) 303-334, arXiv q-alg/9707029. [5] D. Kreimer. Combinatorics of (perturbative) quantum field theory. Phys.Rept. 363 (2002) 387-424, arXiv hep-th/0010059. [6] D. Kreimer. New mathematical structures in renormalizable quantum field theories. Annals Phys. 303 (2003) 179-202, Erratum-ibid. 305 (2003) 79, arXiv hep-th/0211136. [7] J. Vatne. Introduction to operads. Lecture notes, University of Bergen 2004. [8] L. Foissy. An introduction to Hopf algebras of trees. (preprint) [9] L. Foissy. Fa‘a di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations. Adv. Math. 218 (2008), 136-162, arXiv abs/0707.1204. [10] L. Foissy. Hopf subalgebras of the Hopf algebra of rooted trees coming from Dyson-Schwinger equations and Lie algebras of Fa di Bruno type. Motives, QFT and PsDO, Clay Math. Proc. 12 (2010), 189-210. [11] L. Foissy. General Dyson-Schwinger equations and systems. (preprint) [12] L. Foissy. Lie algebras associated to systems of Dyson-Schwinger equations. Adv. Math.226 (2011), no. 6, 4702-4730, arXiv abs/0909.0358. [13] L. Foissy. Pre-Lie algebras and systems of Dyson-Schwinger equations. Proceedings of the DSFdB2011 Meeting, Strasbourg. [14] I. Mencattini. On the structure on the insertion-elimination Lie Algebra. Doctoral thesis, 2005. 33

34

BIBLIOGRAPHY

[15] M.E. Peskin and D.V. Schroeder. An introduction to quantum field theory. Levant books 1995. [16] J. Stasheff. What is an operad? American Mathematical Society notices, 2004. [17] S. Dˇ ascˇ alescu, C. Nˇastˇasescu, S¸. Raianu. Hopf Algebras: an introduction. Marcel Dekker 2001.

Selbstst¨ andigkeitserkl¨ arung Ich habe die vorliegende Arbeit selbstst¨andig und nur unter Verwendung der angegebenen Quellen und Hilfsmittel angefertigt, und ich reiche zum erstenmal eine Masterarbeit in diesem Studiengang ein.

Berlin,

Lucia Rotheray

35