10. CONSTRUCTING LOCAL COUNTERTERMS 63 Figure 5. Explanation of why ICT 1,1 (ε, L) is independent of L. Since ICT 1,1 (ε, L) is independent of L, to verify that it is local we only need to examine the behaviour of W1,1 (P (ε, L),I) at small L. Theorem 9.5.1 implies that Singε W1,1 (P (ε, L),I) has a small L asymptotic expansion in terms of local action functionals. Therefore, since we know ICT 1,1 (ε, L) is independent of L, it follows that it is local. Now that we know ICT 1,1 (ε, L) is independent of L, we will normally drop L from the notation. 10.3. The next step is to construct I1,2 CT (ε, L). However, it is just as simple to construct directly the general counterterm ICT i,k (ε, L). Let us lex- icographically order the set Z≥0 × Z≥0, so that (i, k) (j, l) if i j or if i = j and k l. Let us write W (i,k) (P, I) = (j,l) (i,k) j Wj,l (P, I) . We can write this expression in terms of stable graphs, as follows. Let Γ (i,k) denote the set of stable graphs with genus smaller than i, or with genus equal to i and fewer than k external edges. Then, W (i,k) (P, I) = γ∈Γ (i,k) g(γ) |Aut γ| wγ(P, I). Let us assume, by induction, we have constructed counterterms ICT j,l (ε) for all (j, l) (i, k), with the following properties.

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