3. GENERALITIES ON FEYNMAN GRAPHS 41

3.6. One can ask, how much of this picture holds if U is replaced by

an infinite dimensional vector space? We can’t define the Lebesgue measure

in this situation, thus we can’t define the integral directly. However, one

can still contract tensors using Feynman graphs, and one can still define

the expression W (P, I), as long as one is careful with tensor products and

dual spaces. (As we will see later, the singularities in Feynman graphs arise

because the inverse to the quadratic forms we will consider on infinite di-

mensional vector spaces do not lie in the correct completed tensor product.)

Let us work over a ground field K = R or C. Let M be a manifold and

E be a super vector bundle on M over K. Let E = Γ(M, E) be the super

nuclear Fr´ echet space of global sections of E. Let ⊗ denote the completed

projective tensor product, so that E ⊗ E = Γ(M ×M, E E). (Some details

of the symmetric monoidal category of nuclear spaces, equipped with the

completed projective tensor product, are presented in Appendix 2).

Let O(E ) denote the algebra of formal power series on E ,

O(E ) =

n≥0

Hom(E

⊗n,

K)Sn

where Hom denotes continuous linear maps and the subscript Sn denotes

coinvariants. Note that O(E ) is an algebra: direct product of distributions

defines a map

Hom(E

⊗n,

K) × Hom(E

⊗m,

K) → Hom(E

⊗n+m,

K).

These maps induce an algebra structure on O(E ).

We can also regard O(E ) as simply the completed symmetric algebra of

the dual space E

∨,

that is,

O(E ) = Sym

∗

(E

∨).

Here, E

∨

is the strong dual of E , and is again a nuclear space. The completed

symmetric algebra is taken in the symmetric monoidal category of nuclear

spaces, as detailed in Appendix 2.

As before, let

O+(E

)[[ ]] ⊂ O(E )[[ ]]

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

Let

Symn

E denote the Sn-invariants in E

⊗n.

If P ∈

Sym2

E and I ∈

O+(E

)[[ ]] then, for any stable graph γ, one can define

wγ(P, I) ∈ O(E ).

The definition is exactly the same as in the finite dimensional situation. Let

T (γ) be the set of tails of γ, H(γ) the set of half-edges of γ, V (γ) the set of

vertices of γ, and E(γ) the set of internal edges of γ. The tensor products

of interactions at the vertices of γ define an element of

Hom(E

⊗H(γ),

R).