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 wv i,k (P, I) = k!Ii,k.
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