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 )
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,
· · ·
wγ(P, I) =
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.