3. GENERALITIES ON FEYNMAN GRAPHS 35

a pair of maps H(g) : H(γ) → H(γ), V (g) : V (γ) → V (γ), such that H(g)

commutes with σ, and such that the diagram

H(γ)

H(g)

H(γ)

V (γ)

V (g)

V (γ)

commutes.

3.2. Let U be a finite-dimensional super vector space, over a ground

field K. Let O(U ) denote the completed symmetric algebra on the dual

vector space U

∨.

Thus, O(U ) is the ring of formal power series in a variable

u ∈ U.

Let

O+(U

)[[ ]] ⊂ O(U )[[ ]]

be the subspace of those functionals which are at least cubic modulo .

For an element I ∈ O(U )[[ ]], let us write

I =

i,k≥0

iIi,k

where Ii,k ∈ O(U ) is homogeneous of degree k as a function of u ∈ U.

If f ∈ O(U ) is homogeneous of degree k, then it defines an Sk-invariant

linear map

Dk

f : U

⊗k

→ K

u1 ⊗ · · · ⊗ uk →

∂

∂u1

· · ·

∂

∂uk

f (0).

Thus, if we expand I ∈ O(U )[[ ]] as a sum I =

∑

iIi,k

as above, then we

have collection of Sk invariant elements

Dk

Ii,k ∈ (U

∨)⊗k.

Let γ be a stable graph, with n tails. Let φ : {1,...,n}

∼

=

T (γ) be an

ordering of the set of tails of φ. Let P ∈

Sym2

U ⊂ U

⊗2,

let I ∈

O+(U

)[[ ]],

and let a1,...,an ∈ U By contracting the tensors P and ai with the dual

tensor I according to a rule given by γ, we will define

wγ,φ(P, I)(a1,...,aT (γ)) ∈ K.

The rule is as follows. Let H(γ), T (γ), E(γ), and V (γ) refer to the sets

of half-edges, tails, internal edges, and vertices of γ, respectively. Recall that

we have chosen an isomorphism φ : T (γ)

∼

=

{1,...,n}. Putting a propagator

P at each internal edge of γ, and putting ai at the ith tail of γ, gives an

element of

U

⊗E(γ)

⊗ U

⊗E(γ)

⊗ U

⊗T (γ)

∼

= U

⊗H(γ).

Putting

Dk

Ii,k at each vertex of valency k and genus i gives us an element

of

Hom(U

⊗H(γ),

K).