36 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Figure 1. The first few graphs in the expansion of W (P, I).

The variable a ∈ U is placed at each external edge.

Contracting these two elements yields the weight wγ,φ(P, I)(a1,...,an).

Define a function

wγ(P, I) ∈ O(U )

by

wγ(P, I)(a) = wγ,φ(P, I)(a,...,a)

where φ is any ordering of the set of tails of γ. Note that wγ(P, I) is

homogeneous of degree n, and has the property that, for all a1,...,an ∈ U,

∂

∂a1

· · ·

∂

∂an

wγ(P, I) =

φ:{1,...,n}∼T

= (γ)

wγ,φ(P, I)(a1,...,an)

where the sum is over ways of ordering the set of tails of γ.

Let vi,k denote the graph with one vertex of genus i and valency k, and

with no internal edges. Then our definition implies that

wvi,k (P, I) = k!Ii,k.