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FEYNMAN DIAGRAMS AND THE S-MATRIX, AND OUTER SPACE (SUMMER 2020)

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FEYNMAN DIAGRAMS AND THE S-MATRIX, AND OUTER SPACE (SUMMER 2020)

DIRK KREIMER (LECT. MAY 04, 2020)

1. The Hopf algebra of rooted trees

We follow Lo¨ıc FoissyAn introduction to Hopf algebras of trees (see link on course home- page).

2. General remarks on Hopf algebras and co-actions

We collect some material on Hopf algebras and co-actions. For simplicity, we only discuss vector spaces instead of modules, and we consider all vector spaces to be defined over Q.

A coalgebra is a vector space H together with a coproduct

∆ :H →H⊗H, that is coassociative,

(∆⊗id)∆ = (id⊗∆)∆

and is equipped with a counit, i.e., a map ˆI:H →Qsuch that (ˆI⊗id)∆ = (id⊗ˆI)∆ = id.

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A commutative Hopf algebra is a commutative algebra (with product·) that is at the same time a coalgebra (not necessarily co-commutative) such that the product and coproduct are compatible,

∆(a·b) = ∆(a)·∆(b), and it is equipped with an antipode

S :H →H such that

S(a·b) =S(b)·S(a) = S(a)·S(b), and

m(S⊗id)∆ =m(id⊗S)∆ = ˆI◦I,

where m denotes the multiplication in H and I : Q→ H is the unit (map), I(1) =I is the unit in H.

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A (left-)comodule over a coalgebraH is a vector spaceM together with a map (co-action) ρ:M →H⊗M

such that

(id⊗ρ)ρ= (∆⊗id)ρ, M →H⊗H⊗M,

and (ˆI⊗id)ρ = id. Our Hopf algebras are commutative and graded, H = ⊕j=0H(j) and connected H(0) ∼QI, the H(j) are finite-dimensional Q-vectorspaces.

The vectorspace HC of Cutkosky graphs forms a left comodule over the core Hopf algebra Hcore.

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3. The vectorspace HC

Consider a Cutkosky graph Gwith a corresponding vG-refinement P of its set of external edges LG. It is a maximal refinement of VG.

The core Hopf algebra co-acts on the vector-space of Cutkosky graphs HC.

(3.1) ∆core:HC →Hcore⊗HC.

We say G∈HC(n) ⇔ |G|=n and define AutC =⊕i=1HC(i).

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Note that the sub-vectorspaceHC(0)is rather large: it contains all graphsG= ((HG,VG,EG),(HG,VG,EH)) HC such that ||G||= 0. These are the graphs where the cuts leave no loop intact.

For any G∈HC there exists a largest integer corC(G)≥0 such that

∆˜corcoreC(G)(G)6= 0, ∆˜corcoreC(G)(G) :HC →Hcore⊗corC(G)⊗HC(0), whilst ˜∆corcoreC(G)+1(G) = 0.

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Proposition 3.1.

corC(G) =||G||.

Proof. The primitives of Hcore are one-loop graphs.

In particular there is a unique element g⊗G/g ∈Hcore⊗HC(0):

core(G)∩

Hcore⊗HC(0)

=g⊗G/g, with |g|=||G||.

For any graph G we let G =P

T∈TG(G, T). Here TG is the set of all spanning trees of G and we set for G= ˙∪iGi,TG = ˙∪iTGi.

The maximal refinement P induces for each partition P(i),0≤i≤vG a unique spanning forest fi of G/g. The set FG,P(i) of spanning forests of G compatible with P(i) is then determined by fi and the spanning trees in Tg.

DefineGi :=P

F∈FG,P(i)(G, F).

(3.2) ∆˜||G||G,FGi =X

i=1

G(1)i ⊗ · · · ⊗G(||G||+1)i . Note that|Gki|= 1, ∀k (||G||+ 1) and |G||G||+1i |= 0.

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3.1. The pre-Lie product and the cubical chain complex. So consider the pair (G, F) of a pre-Cutkosky graph with compatible forest F with ordered edges. Assume there are graphs G1, G2 and forests F1, F2 such that

(G, F) = (G1, F1)⋆(G2, F2).

Here, ⋆is the pre-Lie product which is induced by the co-product ∆GF by the Milnor–Moore theorem.

Theorem 3.2. [?] We can reduce the computation of the homology of the cubical chain complex for large graphs to computations for smaller graphs by a Leibniz rule:

d((G1, F1)⋆(G2, F2)) = (d(G1, F1))⋆(G2, F2) + (−1)|EF1|(G1, F1)⋆(d(G2, F2)).

Here, d=d0+d1 is the boundary operator which either shrinks edges or cuts a graph.

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4. Flags

4.1. Bamboo. The notion of flags of Feynman graphs was for example already used in [?,?].

4.2. Flags of necklaces. Here we use it based on the core Hopf algebra introduced above.

We introduce Sweedler’s notation for the reduced co-product in Hcore:

∆˜core(G) := ∆core(G)−I⊗G−G⊗I=:

XG⊗G′′.

We define a flag f ∈Aug⊗kcore of length k to be an element of the form f =γ1⊗ · · · ⊗γk,

where the γi ∈Augcore∩ hHcorei fulfill ˜∆corei) = 0, |γi|= 1.

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Here, hHcorei = {G ∈ Hcore| |H0(G)| = 1} is the linear Q-span of bridge-free connected graphs as generators.

Note that for elements G∈ hHcorei, we have ˜∆|G|−1core (G)6= 0.

We have ˜∆core := (P ⊗P)∆core for P : Hcore → Augcore the projection into the augmen- tation ideal Augcore.

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Define the flag associated to a graph G∈ hHcoreito be a sum of flags of length |G| where in each flag each element γi has unit degree,|γi|= 1:

F lG:= ˜∆|G|−1core (G)∈Aug⊗|G|core.

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Similarly, for a pair (G, F) we can define

F lG,F := ˜∆|G|−1GF ((G, F))∈Aug⊗|G|GF , as a sum of flags

F lG,F =X

i

1, f1)i⊗ · · ·(γ|G|, f|G|)i,

∆˜GF((γl, fl)i) = 0, ∀i, l, 1 ≤l ≤ |G|.

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Humboldt U. Berlin

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