13. VECTOR-BUNDLE VALUED FIELD THEORIES 77 We will assume that I(0,k) = 0 if k 3. The argument is essentially the same as the argument we gave earlier. We will perform induction on the set Z≥0 × Z≥0 with the lexicographical order: (i, k) (r, s) if i r or if i = r and k s. Suppose, by induction, we have constructed counterterms ICT (i,k) (ε) Oloc(E , C∞((0, ∞)ε) A ) for all (i, k) (I, K). The ICT (i,k) are supposed, by induction, to have the following properties: (1) Each ICT (i,k) (ε) is homogeneous of degree k as a function of the field e E . (2) Each ICT (i,k) (ε) is required to be a finite sum ICT (i,k) (ε) = gr(ε)Φr where gr(ε) C∞((0, ∞)ε) and Φr Oloc(E , A ). Each gr(ε) is required to have a finite order pole at 0 that is, limε→0 εkgr(ε) = 0 for some k 0. (3) Recall that A is the space of global sections of a vector bundle A on X. For any element α A, let αx Ax denote its value at x X. We require that, for all L (0, ∞) and all x X, the limit lim ε→0 W(r,s) ⎝P (ε, L),I (i,k)≤(r,s) i ICT (i,k) (ε)⎠ x exists in the topological vector space Hom(E ⊗r , R) Ax. Here, Hom(E ⊗r , R) is given the strong topology (i.e. the topology of uni- form convergence on bounded subsets). Now we need to construct the next counterterm ICT (I,K) (ε). We would like to define ICT (I,K) (ε) = Sing ε W(I,K) ⎝P (ε, L),I (r,s) (I,K) r I(r,s)CT (ε)⎠ . In order to be able to define the singular part like this, we need to know that W(I,K) ⎝P (ε, L),I (r,s) (I,K) r I(r,s)CT (ε)⎠ has a nice small ε asymptotic expansion. The required asymptotic expansion is provided by the following theorem, proved in Appendix 1. Theorem 13.4.4. For all graphs γ, and all I Oloc(E , A )[[ ]], there exist local action functionals Φr Oloc(E , A ⊗C∞((0, ∞)L) and functions gr
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