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
For a graph γ ∈ Γn, and smooth functions f1,...,fn ∈
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
Let f1,...,fn ∈
be smooth functions on M. The external vertex
vi of γ will be labelled by the distribution
Any edge e joining two internal vertices will be labelled by
P (L, ∞) ∈
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
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.