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).
