78 2. THEORIES, LAGRANGIANS AND COUNTERTERMS in the space P ((0, ∞)ε) ⊂ C∞((0, ∞)ε) of periods, such that for all l ∈ Z≥0, there is a small ε asymptotic expansion ∂l ∂lL wγ(P (ε, L),I) ∂l ∂lL r≥0 gr(ε)Φr. This means that for each l and each x ∈ X, there exists a non-decreasing sequence dR ∈ Z, tending to ∞, such that lim ε→0 ε−dR ∂l ∂lL ⎛ ⎝w γ (P (ε, L),I) − r≥0 gr(ε)Φr⎠ ⎞ x = 0 in the topological vector space Hom(E ⊗T (γ) , R) ⊗ Ax. Here, Hom(E ⊗T (γ) , R) is given the strong topology (i.e. the topology of uniform convergence on bounded subsets). Also, each gr has a finite order pole at ε = 0. This means that the limit limε→0 εkgr(ε) is 0 for some k 0. Further, each Φr(L, e) ∈ Hom(E ⊗T (γ) , C∞((0, ∞)L)) has a small L asymptotic expansion Φr hs(L)Ψr,s where each Ψr,s ∈ Oloc(E , A ). The definition of small L asymptotic expan- sion is in the same sense as before: there exists a non-decreasing sequence dS ∈ Z≥0, tending to ∞, such that, for all x ∈ X, and all S ∈ Z≥0, lim L→0 L−dS Φr(e) − S s=0 hs(L)Ψr,s(e) x = 0 in the topological vector space Hom(E ⊗T (γ) , R) ⊗ Ax. It follows from this theorem that it makes sense to define the next coun- terterm ICT (I,K) simply by ICT (I,K) (ε, L) = Sing ε ⎛ ⎝W (I,K) ⎛ ⎝P (ε, L),I − (r,s) (I,K) r ICT (r,s) (ε)⎠⎠ ⎞⎞ . We would like to show the following properties of ICT (I,K) (ε, L). (1) ICT (I,K) (ε, L) is independent of L. (2) ICT (I,K) (ε) is local, that is, it is an element of Oloc(E , A ) ⊗alg C∞((0, ∞)ε).

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