Hopf subalgebras from Green’s functions

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

Contents

1 Basic definitions 7

1.1 Hopf algebras . . . 7

1.2 Rooted trees . . . 10

1.3 Operads . . . 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 21 3.1 Single equation . . . 21

3.2 The operadic approach . . . 23

3.2.1 Operadic proof of theorem 4 . . . 24

3.3 Main theorem . . . 26

4 Application in physics 29

3

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 Dyson- Schwinger 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 graph- theoretical notions (graphs, rooted trees) needed for the rest of the work, with some examples. Chapter 2 introduces the Hopf algebraH_D, the grafting oper- ator, and DSEs in H_D. 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

Chapter 1

Basic definitions

1.1 Hopf algebras

Letkbe a field. For two vector spacesV1andV2, letτV₁,V₂ :V1⊗V2−→V2⊗V1

denote the twist mapv⊗w7→w⊗v¹.

Definition 1. An (associative) k-algebra (A, m, u) is a k-vector space A to- gether with two linear maps,m:A⊗_kA−→A(multiplication) andu:k−→A (unit) such that:

m◦(id⊗m) =m◦(m⊗id), (1.1)

m◦(u⊗id) =m◦(id⊗u).

The multiplication is commutative ifm=m◦τ.

Definition 2. A (coassociative)k-coalgebra(C,∆, )is a k-vector spaceC to- gether 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 ifτ◦∆ = ∆.

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).

1The tensor product always written⊗should be read as⊗_k(⊗_Qin chapters 2 and 3).

7

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 withk-algebra structure(m, u) andk-coalgebra structure(∆, ). Then(m, u) are coalgebra morphisms if and only if(∆, ) are algebra morphisms.

Proof. The fieldkhas a coalgebra structure given by the isomorphisms ∆(1) = 1⊗1 and =id_k. Substitute φ= ∆ :B −→B×B and φ=:B −→kinto 1.3, andψ=m andψ=uinto 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 inS∈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∈Hom_k(H, H), ε=u◦.

Then the antipode of H can be defined by S ? idH =idH? S=ε.

Proposition 1. For any Hopf algebraH, the antipodeS is unique.

Proof. Let S1, S2 be two possible antipodes. Then, since (Homk(H, H), ?, ε) satisfies 1.1,

S₁=S₁? ε=S₁? id_H? S₂=ε ? S₂=S₂.

1.1. HOPF ALGEBRAS 9 Definition 6. A Hopf algebra over kis graded and connected if there exist Hi

such that:

H=

∞

M

i=0

H_i, (1.6)

H₀'k, (1.7)

m(Hi⊗Hj)⊆Hi+j, (1.8)

∆(Hi)⊆ M

j+k=i

Hj⊗Hk. (1.9)

Example 1. For a multiplicative group (G,·) the group algebra kG is the k- vector space generated by the elements of G. An element x∈kG has the form of a finite sumx=P

g∈Gα_gg, with α_g∈k. One can construct a Hopf algebra (kG, m, u,∆, , S)as follows:

• m(α₁g₁⊗α₂g₂) = (α₁α₂)(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 iffGis abelian.

Example 2. For a vector space V over a field k, the tensor algebra of V is defined as

T(V) =

∞

M

n=0

Tⁿ(V), whereTⁿ(V) =V^⊗n.

Consider arbitrary elements x∈V,

v=v1⊗...⊗vn ∈Tⁿ(V), w=w₁⊗...⊗w_m∈T^m(V)

ofT(V). Define a Hopf algebra(T(V), m, u,∆, , S)by:

• m(v, w) =v1⊗...⊗vn⊗w1⊗...wm∈T^n+m(V).

• u(v) = 1_kv

• ∆(x) = 1k⊗x+x⊗1k.

• e(x) =

(x, x∈V

0, f v∈Tⁿ(V), n >1 .

• S(x) =−x.

This Hopf algebra is cocommutative but not commutative, and is graded con- nected with homogeneous componentsTⁱ(V): T⁰(V)'k andT¹(V) =V.

1.2 Rooted trees

Definition 7. A graph Gconsists 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 ande₂=wx, forv, w, x∈V(G).

• Two verticesv, w∈V(G) are neighbours ifvx∈E(G).

• The degree of a vertexv 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 e_i, e_i+1∈P are adjacent.

• A path {e1, ..., ek} ⊆E(G) will be denoted by (v1, v2) where e1 =v1w, e2=wu, ... ,ek−1=tv2,ek=xv2 for somet, 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 anyv, 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 vertexr∈V(T) which is called the root (here the root will always be drawn at the top of the tree).

• A vertexv∈V(T)\ {r} with degree one is called a leaf ofT.

• w∈V(T) is a child of v∈V(T) if any path (r, w) contains the edgewv.

• 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 treesT and the set of all rooted forestsF.

Definition 9. A decorated rooted tree consists of a rooted tree T, a countable set Dand a surjective map c:D −→V(T), which assigns an element of D to each vertex ofT. For anyv∈V(T), we denote c⁻¹(v) =d(v).

LetTDandFDdenote respectively the sets of rooted trees and forests decorated by a setD.

Definition 10. An admissible cut of T is a nonempty subset c (E(T) such that

|c∩(r, v)|≤1 ∀v∈V(T).

2We 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 byC^∗(T) the unionC(T)∪ ∅of admissible cuts and the empty cut.

• Given a cutc∈ C(T), we construct a disconnected graphGwithE(G) = E(T)−candV(G) =V(T).

• The connected component of G which contains the root of T is a tree which we callR^c(T).

• The component(s) ofGnot connected to the root ofT 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 ofG₂, we label the edges:

1 2 3

C G P^C(G₂) R^C(G₂) C Admissible?

{1} • • Yes

{2} ^• • Yes

{3} ^• • Yes

{1,2} •• × × No

{1,3} •• × × No

{2,3} ^{• •} •• Yes

{1,2,3} • • • × × No

1.3 Operads

Definition 11. An operadO³ 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 setX,M ap(Xⁿ, X)is the set of functions fromXⁿ toX. The endomorphism operad End(X)consists of the set of objects

{M ap(Xⁿ, X)}_n∈N^∗ and maps◦i defined by

(f◦ig)(x1, ..., x_m+n−1) =f(x1, ..., x_i−1, g(xi, ..., x_i+m−1), xi+m, ..., x_m+n−1), wheref ∈M ap(Xⁿ, X), g∈M ap(X^m, X)andxj ∈X ∀j.

Example 5 (The rooted trees operad). Take the set T(n) of rooted trees T withn leaves {l1, ..., l_n} ⊂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 l_i ∈ V(T1)”.

3For a formal introduction to and definition of algebraic operads, see [7].

Chapter 2

The Hopf algebra of rooted trees

We consider a Hopf algebra H_D overQ generated byT_D, whereD={d_n}_n∈N is a set which we may take, without loss of generality, to beN. Define a Hopf algebra structure (H_D, m,I,∆,ˆ

I, S):

• T₁T₂:=m(T₁⊗T₂) is given by the forestT₁∪T₂.

• The unit map sendsq∈QtoqI∈ HD1.

• The coproduct onHD is defined on trees using admissible cuts:

∆(T) =I⊗T +T⊗I+ X

c∈C(T)

P^c(T)⊗R^c(T) (2.1)

and extended over all of HD by ∆(T₁T₂) = ∆(T₁)∆(T₂).

• The counit ˆIis given by ˆI(T) :=

(1, T =I 0, T 6=I. .

• Using these definitions ofm and ∆ along with the defining property 1.5 for the antipode, andS(I) =Iobtain an expression forSacting on a tree T ∈ H:

m(S⊗idH)(∆(T)) =S(I)T+S(T)I+X

c

S(P^c(T))R^c(T) =I(ˆI(T)) = 0

⇒S(T) =−T−X

c

S(P^c(T))R^c(T). (2.2)

This can be extended to all of HD byS(T₁, ..., T_k) =S(T_k)...S(T₁).

Example 6.

∆(

1 2

3 4) =

1 2

3 4⊗I+I⊗

1 2

3 4+

2

3 4⊗•₁+•₃⊗

1 2 4+•₄⊗

1 2

3+•₃•₄⊗ 1 2 ^,

1It is a standard abuse of notation to useIfor both the unit map and the “empty tree”

defined byV(I) =∅.

13

S(

1 2

3 4) =−

1 2

3 4+

2

3 4•1−•3 4 2

−•4 2

3+•3•4•2+•3 1 2 4 +•4

1 2 3−•3•4

1 2 ^.

For any treeT ∈ H_D, definen(T) = (n₁, n₂, ....) by ni :=| {v∈V(T) :d(v) =dn∈ D} |.

For a forest F =T1...Tk, ni(F) =Pk

l=1ni(Tl). Define Hⁿ =span_Q{F ∈ F_D : n(F) =n}. Then

H_D= M

n∈N^∞

Hⁿ is a grading onH_D.

Aug^HD :=L

n∈N^∞\{0}Hⁿis the augmentation ideal ofH_D. It contains all elements inH_D which vanish under the action of the counit.

Theorem 2. (HD, m,I,∆,ˆI, S)as defined above gives a graded connected, com- mutative, 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 allFi and Ti be arbitrary forests and trees (respectively) inHD, and q∈Q.

• m is associative: m(id⊗m)(F1⊗F2⊗F3) = F1(F2F3) = (F1F2)F3 = m(m⊗id)(F1⊗F2⊗F3)

• Iis a unit: m(I⊗id)(q⊗F1) =qIF1=F1qI=m(id⊗I)(F1⊗q).

• ˆIis 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)).

• ˆ

Iis an algebra morphism:

ˆI(I(q)) = ˆI(qI) =q= 1q=u_Qq,

ˆI(m(F1⊗F2)) = ˆI(F1F2) =m(ˆI(F1)⊗ˆI(F2)) =m(ˆI⊗ˆI)(F1⊗F2).

• S satisfies 1.5 by construction.

• H=L

Hi gives a grading:

H⁰=span_Q{F∈ FD :V(F) =∅}=QI'Q, m(H^j⊗H^k)⊆ {F ∈ F_D:n_i=j_i+k_i}=H^j+k,

∆(H^j) ={F :F =P^c(T), T ∈H^} ⊗ {T⁰:T⁰ =R^c(T), T ∈H^}

⊆ M

k_i+l_i=j_i

H^k⊗H^l.

15

• Commutativity: F1F2:=F1∪F2=F2∪F1=:F2F1.

• Non-cocommutativity: in general τ_HD,HD(∆(F)) 6= ∆(F), as ∆(T) ∈ H_D⊗ T_D and ∆◦τ_HD,HD(T)∈ T_D⊗ H_D for any treeT.

For proof that ∆ is coassociative, see proposition 3.

Definition 12. For each d∈ D, the grafting operator B₊^d on H_D is a linear map B₊^d :H_D −→span_Q(T_D). B₊^d acts on a rooted forestT1, ..., Tk by joining the roots ofT1, ..., Tk to a new root decorated bydgiving a tree:

B₊^d(T1...Tk) =

T1 T2 T3

d

...

.

Example 7. LetD=N. B₊¹(

2 3•₂) =

2 3

2 1

.

Proposition 2. Each grafting operatorB₊^d satisfies

∆◦B₊^d =B₊^d ⊗I+ (id⊗B₊^d)◦∆. (2.3)

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

For any cutc∈ C(T), the restriction ofcto any subtreeTiis either inC^∗(Ti) or is the total cut onTi. Consider any set of cutsc1, ..., ck such thatci ∈ C^∗(ti).

Then there exists a unique cutc ∈ C^∗(T) such that the restriction ofc to any T_i gives the correspondingc_i.

For this cut, we have

P^c(T) =

k

Y

i=1

P^ci(Ti),

R^c(T) =B^d₊ⁿ

k

Y

i=1

R^ci(T_i)

! .

So:

∆(T) =T⊗I+I⊗T + X

c∈C(T)

P^c(T)⊗R^c(T)

=B^d₊(F)⊗I+ X

c∈C^∗(T)

P^c(T)⊗R^c(T)

=B^d₊(F)⊗I+ X

c_i∈C^∗(T_i) k

Y

i=1

P^ci(T_i)⊗B₊^d(R^ci(T_i))

!

=B^d₊(F)⊗I+ (id⊗B₊^d)

k

Y

i=1













 X

c_i∈C^∗

P^ci(Ti)⊗R^ci(Ti)

| {z }

= ∆(T₁)...∆(T_k)















=B^d₊(F)⊗I+ (id⊗B₊^d)∆(F).

Remark 3. This proposition is, in fact, saying the the operator B₊ is a 1- cocycle in the Hochschild cohomology ofH_D ².

Proposition 3. The coproduct∆ is coassociative.

Proof. Show that ∆ satisfies 1.2. Clearly

(∆⊗id_H)∆(I) = ∆(I)⊗I=I⊗I⊗I=I⊗∆(I) = (id_H⊗∆)∆(I).

We proceed by induction on|V(F)|for anyF ∈ H_D.

Assume that for all|V(F)|< n, the action of ∆ on F is coassociative, and considerF ∈ 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 onF.

2. If F is a tree, then F = B₊^d(X) for some X ∈ H and d ∈ D, with

|V(X)|=n−1. By assumption ∆ acts coassoiatively onX, and applying equation 2.3 gives:

(id⊗∆)∆(B₊^d(X)) = (id⊗∆)(B^d₊(X)⊗I+ (id⊗B₊^d)∆(X))

=B₊^d(X)⊗I⊗I+ (id⊗∆◦B₊^d)∆(X)

=B₊^d(X)⊗I⊗I+ (id⊗B^d₊)∆(X)⊗I

| {z }

?

+ (id⊗id⊗B₊^d)(id⊗∆)∆(X)

| {z }

??

.

(∆⊗id)∆(B₊^d(X)) = (∆⊗id)(B^d₊(X)⊗I+ (id⊗B₊^d)∆(X))

= ∆(B₊^d(X))⊗I+ (∆⊗B₊^d)∆(X)

=B₊^d(X)⊗I⊗I+ (id⊗B^d₊)∆(X)⊗I

| {z }

?

+ (id⊗id⊗B₊^d)(∆⊗id)∆(X)

| {z }

??

.

The terms labelled ? are immediately equal, the terms marked?? cancel under the assumption onX.

2for more information see, e.g. [3], [8].

2.1. HOPF SUBALGEBRAS 17

2.1 Hopf subalgebras

Definition 13. A sub-Hopf algebra of a graded Hopf algebra (H, m, u,∆, , S), H =L

iHi, is a subspaceH⁰⊂H with Hopf algebra structure(H⁰, m, u,∆, , S) such thatH⁰=L

i(H⁰∩H_i) gives a grading onH⁰. Example 8. The ladder onnvertices is the tree given by

λ_n:= (B₊)ⁿ(I).

λ0=I, λ1=•, λ2= , λ3= , λ4= .

LetH^L be the subalgebra ofHgenerated by the set of all ladders, andH^L_n ⊂H_n those elements which have n vertices. ∆(λ_n) = P

m≤nλ_m⊗λ_n−m: H^L is a Hopf subalgebra of H.

Example 9. More generally, the subalgebra generated by all trees whose vertices have fertility bounded from above bynis a Hopf subalgebra (the ladders are the casen= 1).

Example 10. For any d ∈ D, the set of forests with ni(F) =

(1, di=d 0di6=d forms a Hopf subalgebra. It is isomorphic to the Hopf algebraHof undecorated rooted trees, which has m,I,∆,ˆ

I and S ad H_D, but only one grafting operator B₊.

Example 11 (The Connes-Moscovici Hopf algebra). The natural growth oper- ator is a linear mapN :H −→ Hdefined byN(I) =• and

N(T) := X

v∈V(T)

Tv,

where Tv is the tree obtained by adding one extra vertex w to V(T), and the edge vw toE(T). N acts as a derivation on forests.

Defineδn:=Nⁿ(I).

δ₀=I δ1=• δ2=

δ₃= +

δ4= 3 + + +

Theorem 3. The elements{δn}_n∈N generate a Hopf subalgebraHCM ofH.

Proof. We need to show that ∆(HCM)⊆ HCM⊗ HCM andS(HCM)⊆ HCM. If we writeδn=P

iTi andδn+1=P

i⁰Ti⁰, we have

∆(δ0) =δ0⊗δ0,

∆(δ1) =δ1⊗δ0+δ0⊗δ1,

∆(δn) =δn⊗δ0+δ0⊗δn+X

i



 X

c∈C(Ti)

P^c(Ti)⊗R^c(Ti)



,

∆(δn+1) =δn+1⊗δ0+δ0⊗δn+1+nδ1⊗δn

+X

i



 X

c∈C(Ti)

N(P^c(Ti))⊗R^c(Ti) +P^c(Ti)⊗N(R^c(Ti))





+X

i



 X

c∈C(Ti)

δ1|R^c(Ti|P^c(Ti)⊗R^c(Ti)



.

By the assumption ∆(δ_n)∈ HCM⊗HCM, it immediately follows that ∆(δ_n+1)∈ HCM⊗ HCM andS(HCM)∈ HCM.

2.2 Dyson-Schwinger equations

A (combinatorial) Dyson-Schwinger equation builds a family of forests through the action of the grafting operators{B₊^d}d∈D on elements ofHD.

Example 12. Consider the Hopf algebraHof undecorated rooted trees, and the equation

X=αB₊(X²)

for a parameter α (which is called a coupling constant), andX ∈ H[[α]]. This equation has unique solutionX(α) =P∞

n=0c_nαⁿ, where the family {c_n}_n∈Nare defined recursively by c₀=Iand

c_n=B₊



 X

m≤n−1

c_mc_n



. The first five members of this family are

c₀=I c1=• c2= 2 c₃= 4 +

c4= 8 + + 2

2.2. DYSON-SCHWINGER EQUATIONS 19 In the next chapter we study a more general DSE

X =I+

∞

X

n=1

ωnαⁿB₊^dn(Xⁿ⁺¹),

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

ρ

ωⁱ_ρα₁^ζ1...α^ζ_nⁿB^d

i ρ

+

X₁^ζ1s1...X_i^ζisi+1...X_n^ζnsn ,

withXi∈ H_D[[{α1, ..., αn}]].

Chapter 3

Hopf subalgebras from

Dyson-Schwinger equations

In this chapter we consider solutions X(α) =X

n

αⁿc_n or, more generally,

X_i(α₁, ..., α_m) = X

k1,...,km

α^k₁¹...α^k_m^mcⁱ_k

1,...,k_m

to DSEs, and the algebraH_Dc generated by the coefficientscⁱ_k

1,...,k_m ∈ H_D. Let H_cⁿ= Hⁿ∩ H_Dc. The goal of this chapter is to show that these sub- algebrasH_Dc⊂ H_D are Hopf. That is, thatH_Dc is closed under ∆ andS, and respects the grading onH_D:

∆(cⁱ_k)∈ M

n_i+m_i=k_i

H_cⁿ⊗H_c^m, (3.1)

S(cⁱ_k)∈H_c^k.

To this end, we show thatR^c(cⁱ_k)∈ H_D^l_candP^c(cⁱ_k)∈ H_D^k−l_c for somel. Then 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

n=1

ω_nαⁿB₊^dn(Xⁿ⁺¹) (3.2)

with solutionsX(α)∈ H_D[[α]].

Lemma 1. Equation 3.2 has a unique solution X =X

n

αⁿcn, (3.3)

withcn∈ HD for alln∈N.

21

Proof. Substitute the ansatz 3.3 into 3.2 to find

c₀=I, (3.4)

c_n=

n

X

m=1

ω_nB₊^dm





X

k1+...+km+1=n−m

c_k1...c_km+1



.

Theorem 4. The coefficients c_n generate a Hopf subalgebra ofH_D:

∆(cn) =

n

X

k=0

P_kⁿ⊗ck, (3.5)

whereP_kⁿ is a homogeneous polynomial of degreen−kin theck (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+...+k_m+1 = n−m. Using the 1-cocycle property (2.3) of the grafting operators, see that

∆(c_n) =

n

X

m=1

ω_m∆ B₊^dm X

?

c_k1...c_km+1

!!

=c_n⊗I+X

m

ω_m(id⊗B₊^dm) X

?

∆(c_k1)...∆(c_km+1)

! , and using the assumption onci, i≤n:

∆(c_n) =c_n⊗I+X

m

ω_m(id⊗B₊^dm)



 X

?

(X

l1≤k1

P_l^k1⊗c_l)...( X

lm+1≤km+1

P_l^km+1⊗c_l)





=cn⊗I+ X

m≤n−1

ωm





 X

?, l_i≤ki

P_l^k1

1 ...P_l^km+1

m+1 ⊗B₊^dm(cl₁...cl_m+1)





. For any fixedm, letl1, ..., lm+1 vary. ∆(cn) has a term

X

?

P_l^k1

1 ...P_l^km+1

m+1 ⊗B^d₊^m(cl₁...cl_m+1) .

By assumption, each product P_l^k1

1 ...P_l^km+1

m+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 but a single term in the expression 3.4 forc_n−m. Therefore summing overmgivesP

k≤nP_l^k⊗ck, sincek=n−m by definition.

3.2. THE OPERADIC APPROACH 23

3.2 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 H_Dc 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

1 2

6=

1 1

1 2

, whereas the forests inHD are nonplanar:

1 1 1

2

=

1 1

1 2 .

• 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 thei^thleaf (where we can number the leaves, without loss of generality, 1,2,... working left-to-right). For example,

2 ^◦1

1 2 2

=

1 2 2 2

.

• Construct the solution (a series in the coupling constants with operadic coefficientsν) to an operadic version of the DSE.

• Translate this solution back into the solutionXin terms of thec∈ H_Dcby removing the in/outputs and removing the planarity restriction to recover weights.

Example 13. We have already seen the DSE X =αB+(X²)

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

n

αⁿµn+1(G^⊗n+1). (3.6)

Each µj is a map V^⊗j −→V for some vector spaceV, andG(α) is a series in αwith coefficientsν:

G(α) =I+X

k

α^kν_k. (3.7)

Lemma 2. Each νk is the sum with unit weights over all maps V^⊗k+1 −→V obtained by compositions of the “undecomposable” mapsµ_j.

Proof. By simply inserting 3.7 into 3.6, we find that ν0=I

and

ν₁=µ₂◦I=µ₂.

Assume for some n∈Nthat the lemma holds for all k < n. Then we look at the coefficients ofαⁿ and see

ν_n=µ_n+1◦I+µ_n◦1ν₁+µ_n◦2ν₁...+µ₁◦1ν_n−1+...+µ₁◦1ν_n−1, and by assumption onν₂, ..., ν_n−1the lemma holds forν_n.

For this equation each mapµ_j is a single vertex withj+ 1 inputs:

µ1= 1

, µ2= 2

, µ3= ³ , ....

Remark 4. To compute a givenνk, it is not necessary to know allνj forj < k, as for the coefficients ck above. One can simply construct νk by making the appropriate compositions of the undecomposableµi.

3.2. THE OPERADIC APPROACH 25 Example 14. Let ωn = 1for alln, and letD=N.

ν3= ³ +

2

1 +

2

1 +

2

1 + ²

1

+ ²

1

+

1 1

1 +

1 1

1

+

1 1

1 +

1 1

1

⇒c3=•3+ 3 +a + 4 .

We look at the action of the coproduct ∆ on theνn:

∆(νn) =ν_n⁰ ⊗ν_n⁰⁰.

ν_n⁰⁰ is the sum of R^c(νn) terms (the admissible cuts of νn are exactly the admissible cuts ofcn). Lemma 2 tells us thatνn is the sum overall maps with unit weight from V^⊗n+1 to V. As ∆ takes all admissible cuts,ν_n⁰ containsall maps with unit weight from any V^⊗k+1, withk ≤n−1 to V, and therefore sums over everyνk :V^⊗k+1−→V,withk≤n.

ν_n⁰ contains all the termsP^c(νn). The left hand side corresponding to aνkon the right hand side of the tensor product is a sum over all forests (νi₁)^r1...(νi_t)^rt : V^⊗n+1−→V^⊗k+1, with conditions:

r=r1+...+rt=k+ 1, (3.8)

r1i1+...+rtit+r=n+ 1,

⇒r1i1+...+rtit=n−k.

So we have

∆(νn) =X

k≤n

Qⁿ_k ⊗νk,

with

Qⁿ_k =X

3.8

(νl₁)^r1...(νl_t)^rt.

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 F_kⁿ, and to avoid over-counting any monomial (ci₁)^r1...(ci_t)^rt we add in the condition that 1≤i1< ... < it(which we denote by∗).

To compute F_kⁿ, we need to count how many planar forests (νi₁)^r1...(νi_t)^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...(νi_t)^rt :V^⊗n+1−→

V^⊗k+1 can be composed with a the correspondingνk:V^⊗k+1−→V.

There are (k+ 1)! possibilities for switching the positions of any the νl

appearing in a monomial, andr1!r2!...rt! of them give the same result, even in the planar setting. So

F_kⁿ= (k+ 1)!

r₁!...r_t!. This means that ∆(cn) =Pn

k=0P_kⁿ⊗ck, with P_kⁿ=X

?,∗

(k+ 1)!

r1!...rt!c^r_i1¹...c^r_it^t.

3.3 Main theorem

For ease of notation, we may write multi-indices as k =k1, ..., kn and so on.

Consider a system ofnDSEs, withncoupling constantsα1, ..., αn given by:

Xi=I+X

ρ

ω_ρⁱα^ζ₁¹...α^ζ_nⁿB^d

i ρ

+

X₁^ζ1s1...X_i^ζisi+1...X_n^ζnsn

, (3.9)

where

• The vectorρis defined byρ_k=ζ_ks_k.

• Xi=Xi(α1, ..., αn)∈ H_D[[{α1, ..., αn}]]

• The powers ζ1, ..., ζn∈Zvary

• s1, ..., sn∈Z,si is fixed for eachXi.

Lemma 3. The equation 3.9 has a unique solution Xi= X

k_j∈N

α^k₁¹...α^k_nⁿcⁱ_k1,...,kn (3.10)

Theorem 5. The coefficients cⁱ_k generate a Hopf subalgebra ofH_D:

∆(cⁱ_k) = X

k⁰_i≤ki

P_k^k;i0 ⊗cⁱ_k0,

Remark 5. This new result, appearing here for the first time, is a direct gen- eralisation of work by Lo¨ıc Foissy, allowing for several coupling constants.

Proof. Consider the operadic equation Gi=I+X

ρ

α^ζ₁¹...α^ζ_nⁿµⁱ_ρ(G^⊗ζ₁ ^1s1...G^⊗ζ_i ^isi+1...G^⊗ζ_n ^nsn), (3.11)

with irreducible mapsµⁱ_ζ

1,...,ζ_n:V₁^⊗ζ1⊗...⊗V_n^⊗ζn−→Vi, for some vector spaces V1, ..., Vn. This has solutions

G_i=G_i=I+ X

k∈Zⁿ

α^k₁¹...α^k_nⁿν_kⁱ.

3.3. MAIN THEOREM 27 Lemma 4. Each ν_kⁱ is the sum with unit weights over all maps V₁^⊗k1s1⊗...⊗ V_i^⊗kisi+1⊗...⊗V_n^⊗knsn −→ Vⁱ obtained by compositions of undecomposable mapsµⁱ_ζ.

As for lemma 2, this can be proved by induction. This structure of the νⁱ_k is the basis for the proof of theorem 5.

Consider the form of ∆(ν_kⁱ) =ν⁰⊗ν⁰⁰. The sideν⁰⁰contains all possible trees obtained by removing some (non-root) forest in ν_kⁱ (which by lemma 4 means the sum of allν_lⁱ:V₁^⊗l1s1⊗...⊗V_i^⊗lisi+1⊗...⊗V_n^⊗lnsn−→Vⁱ whereli≤ki).

The part ofν⁰corresponding to eachν_lⁱcontains the forests removed fromν_kⁱ to obtainν_lⁱ, products which form mapsV₁^⊗k1s1⊗...⊗V_i^⊗kisi+1⊗...⊗V_n^⊗knsn−→

V₁^⊗l1s1⊗...⊗V_i^⊗lisi+1⊗...⊗V_n^⊗lnsn.

Consider one such term: a monomial(ν_lⁱ1¹)^r1...(ν_lⁱt^t)^rt, where r1l¹_j+...+r^tl^t_j=

(sjkj, j6=i

siki+ 1, j=i (3.12)

r^j= X

i_k=j

rk=

(s_jl_j, j6=i

sili+ 1, j=i . (3.13)

Now we have an expression

∆(ν_kⁱ) = X

lj≤kj

j=1,..,m

Q^k;i_l ⊗(ν_lⁱ),

where

Q^n;i_k = X

3.12,3.13

(ν_lⁱ1¹)^r1...(ν_lⁱt^t)^rt.

Again, translation back to the planar picture adds a factorF_l^k;i to each term and the constraint (call it∗) that the upper indicesij are distinct. To compute F_l^k;i, count the number of terms inQ^n;i_k which translate to the same monomial in P_k^n;i: there are Q

jr^j! ways to switch the order of the ν, and Q

jr_j! are equivalent even in the operadic setting. This gives:

F_l^k;i =r¹!...r^t! r1!...rt!, so that

P_l^k;i = X

3.12,3.13,∗

r¹!...r^t!

r1!...rt!(cⁱ_l¹1)^r1...(cⁱ_l^tt)^rt.

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 andE). The different edge types represent different types of parti- cle, while the vertex types represent possible interactions of the theory in question.

• Γ must be “1-particle-irreducible”(1P I): Γ remains connected after dele- tion of any single internal edge.

• The internal edges and vertices may be weighted.

We refer to the sets V^ext andE^ext as the external structure ofΓ.

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 ofHF G keeps track of all cycles in a Feynman graph:

∆(Γ) = Γ⊗I+I⊗Γ +X

γ

γ⊗Γ γ,

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

1Edges are not necessarily distinct.

2The Hochschild 1-cocyclesB₊^diofHDandBγofH_{F G}are linked by a Hopf algebra mor- phism, see theorem 5 of [13].

29

associated via diffeomorphisms (called Feynman rules) to integrals³which 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 graphΓ = has two sub-divergences:

γ₁= , γ₂= . This gives

∆(Γ) = Γ⊗I+I⊗Γ +γ1⊗ Γ

γ₁ +γ2⊗ Γ

γ₂ +γ1∪γ2⊗ Γ γ₁∪γ₂

= ⊗I+I⊗ + ⊗ + ⊗ + ⊗ .

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

tree associated to Γis

γ₃

γ₁ γ₂^.

Dyson-Schwinger equations are equations of motion for Green’s functions in QFT. They take the form of combinatorial equations based on the action of B^γ₊ 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

X1=B

(1 +X3)² (1−X₁)(1−X₂)

X₂=B (1 +X₃)² (1−X1)² X3=X

γ

Bγ

(1 +X3)^1+2l(γ) (1−X₁)^2l(γ)(1−X₂)^l(γ)

3An explanation of the concepts and methods of perturbative QFT can be found in, for example, the early chapters of [15].

4Described in great detail in, for example, [2].

31 wherel(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= , γ₂= + , γ₃=

γ4= , γ5= , γ6=

γ7= , γ8= , γ9=

Associate to each vertex typevi a coupling constantαi. Again, denote

X1= , X2= , X3= .

After a truncation, the Dyson-Schwinger equations are

X1=I+α²₁Bγ1(X₁³) +α2Bγ2(X1X2) +α3Bγ3(X1X3), X2=I+α²₁α3

1 α2

Bγ4(X₁²X3) +α3Bγ₅(X2X3) +α²₁Bγ₆(X₁²X2), X3=I+α3Bγ₇(X₃²) +α₁²Bγ₈(X₁²X3) +α⁴₁ 1

α3

Bγ₉(X₁⁴).

This system is of the form 3.9, so it does indeed generate a Hopf subalgebra of the algebraHF G generated by all Feynman graphs of this theory.

Bibliography

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[2] J.C. Collins. Renormalisation.Cambridge univeristy press 1984.

[3] A. Connes and D. Kreimer. Hopf algebras, renormalisation and noncom- mutative geometry. Commun.Math.Phys. 199 (1998) 203-242, arXiv hep- th/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.

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Doctoral thesis, 2005.

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[15] M.E. Peskin and D.V. Schroeder. An introduction to quantum field theory.

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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