62 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Theorem 10.1.1. There exists a unique series of local counterterms ICT i,k (ε) ∈ Oloc(C∞(M )) ⊗alg P((0, ∞)) 0 , for all i 0,k ≥ 0, with ICT i,k homogeneous of degree k as a function of a ∈ C∞(M), such that, for all L ∈ (0, ∞], the limit lim ε→0 W ⎛ ⎝P (ε, L),I − i,k i ICT i,k (ε)⎠ ⎞ exists. Here the symbol ⊗alg denotes the algebraic tensor product, so only finite sums are allowed. 10.2. We will construct our counterterms using induction on the genus and number of external edges of the Feynman graphs. Later, we will see a very short (though unilluminating) construction of the counterterms, which does not use Feynman graphs. For reasons of exposition, we will introduce the Feynman graph picture first. Let Γi,k denote the set of all stable graphs of genus i with k external edges. Let Wi,k (P, I) = γ∈Γi,k wγ(P (ε, L),I). Thus, W (P, I) = i Wi,k (P, I) . If the graph γ is of genus zero, and so is a tree, then limε→0 wγ(P (ε, L),I) converges. Thus, the first counterterms we need to construct are those from graphs with one loop and one external edge. Let us define ICT 1,1 (ε, L) = Sing ε W1,1 (P (ε, L),I) . Section 9 explains the meaning of the singular part Singε of W1,1 (P (ε, L),I). We need to check that this has the desired properties. It is immediate from the definition that W1,1 ( P (ε, L),I − ICT 1,1 (ε, L) ) = W1,1 (P (ε, L),I) − ICT 1,1 (ε, L) and so the limit limε→0 W1,1 ( P (ε, L),I − ICT 1,1 (ε, L) ) exists. Next, we need to check that Lemma. ICT 1,1 (ε, L) is local. First we will show that Lemma. ICT 1,1 (ε, L) is independent of L. Figure 5 illustrates d dL W1,1 (P (ε, L),I). This expression is non-singular, as it is obtained by contracting the distribution I0,3 on C∞(M 3 ) with the smooth function KL ∈ C∞(M 2 ). Therefore ICT 1,1 (ε, L), which we defined to be the singular part of W1,1 (P (ε, L),I), is independent of L.

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