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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 i Ii,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 = ∑ i Ii,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).