60 2. THEORIES, LAGRANGIANS AND COUNTERTERMS where the limit is taken in the topological vector space O(C∞(M ),C∞((0, ∞)L)). (2) The gi(ε) appearing in this asymptotic expansion have a finite order pole at zero: for each i there is a k such that limε→0 εkgi(ε) = 0. (3) Each Ψi appearing in the asymptotic expansion above has a small L asymptotic expansion of the form Ψi ∞ j=0 fi,j(L)Φi,j where the Φi,j are local action functionals, that is, elements of Ol(C∞(M )) and each fi,k(L) is a smooth function of L ∈ (0, ∞). This theorem is proved in Appendix 1. All of the hard work required to construct counterterms is encoded in this theorem. The theorem is proved by using the small t asymptotic expansion for the heat kernel to approximate each wγ(P (ε, L),I) for small ε. 9.4. These results allow us to extract the singular part of the finite- dimensional integral wγ(P (ε, L),I). Of course, the notion of singular part is not canonical, but depends on a choice. Definition 9.4.1. Let P((0, 1))≥0 ⊂ P((0, 1)) be the subspace of those functions f of ε which are periods and which admit a limit as ε → 0. A renormalization scheme is a complementary subspace P((0, 1)) 0 ⊂ P((0, 1)) to P((0, 1))≥0. Thus, once we have chosen a renormalization scheme we have a direct sum decomposition P((0, 1)) = P((0, 1))≥0 ⊕ P((0, 1)) 0 . A renormalization scheme is the data one needs to define the singular part of a function in P((0, 1)). Definition 9.4.2. If f ∈ P((0, 1)), define the singular Sing(f) of f to be the projection of f onto P((0, 1)) 0 . 9.5. We can now use this definition to extract the singular part of the functions wγ(P (ε, L),I). As before, let us think of wγ(P (ε, L),I) as a distri- bution on M k . Then, Theorem 9.3.1 shows that wγ(P (ε, L),I) has a small ε asymptotic expansion of the form wγ(P (ε, L),I) ∞ i=0 gi(ε)Φi where the gi(ε) are periods and the Φi ∈ Hom(C∞(M k ),C∞((0, ∞)L)). Theorem 9.3.1 also implies that there exists an N ∈ Z≥0 such that, for all n N, gn(ε) admits an ε → 0 limit.

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