52 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Figure 4. The weight Cγ(I[ε],f1,f2,f3) attached to a graph

with 3 external vertices.

These correlation functions will be defined as a sum over graphs.

Let Γn denote the set of graphs γ with n univalent vertices, which are

labelled as v1,...,vn. These vertices will be referred to as the external

vertices. The remaining vertices will be called the internal vertices. The

internal vertices of a graph γ ∈ Γn can be of any valency, and are labelled

by a genus g(v) ∈ Z≥0. The internal vertices of genus 0 must be at least

trivalent.

For a graph γ ∈ Γn, and smooth functions f1,...,fn ∈

C∞(M),

we will

define

Cγ(I[L])(f1,...,fn) ∈ R

by contracting certain tensors attached to the edges and the vertices.

We will label each internal vertex v of genus i and valency k by

Ii,k[L] :

C∞(M)⊗H(v)

→ R.

Let f1,...,fn ∈

C∞(M)

be smooth functions on M. The external vertex

vi of γ will be labelled by the distribution

C∞(M)

→ R

φ →

M

fφ.

Any edge e joining two internal vertices will be labelled by

P (L, ∞) ∈

C∞(M 2).

The remaining edges, which join two external vertices or join an external

and an internal vertex, will be labelled by P (0, ∞), which is a distribution

on M

2.

As usual, we can contract all these tensors to define an element

Cγ(I[L])(f1,...,fn) ∈ R.

One may worry that because some of the edge are labelled by the distribution

P (0, ∞), this expression is not well defined. However, because the external

edges are labelled by smooth functions fi, there are no problems. Figure 4

describes Cγ(I[ε],f1,f2,f3) for a particular graph γ.