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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