76 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Let me explain more precisely what I mean by saying there is a small L asymptotic expansion Ii,k[L] j∈Z ≥0 gr(L)Φr. Without loss of generality, we can require that the local action functionals Φr appearing here are homogeneous of degree k in the field e ∈ E . Recall that A is the global sections of some bundle of algebras A on a manifold with corners X. Let Ax denote the fibre of A at x ∈ X. For every element α ∈ A , let αx ∈ Ax denote the value of α at x. The statement that there is such an asymptotic expansion means that there is a non-decreasing sequence dR ∈ Z, tending to infinity, such that for all R, for all fields e ∈ E , for all x ∈ X, lim L→0 L−dRαx Ii,k[L](e) − R r=0 gr(L)Φr(e) = 0 in the finite dimensional vector space Ax. Then, as before, the theorem is: Theorem 13.4.3. The space T (n+1) (E ) has the structure of a principal Oloc(E , A ) bundle over T (n) (E ), in a canonical way. Further, T (0) (E ) is canonically isomorphic to the space O+ loc (E , A ) of A -valued local action functionals on E which are at least cubic modulo the ideal I ⊂ A . Further, the choice of renormalization scheme gives rise to a section T (n) (E ) → T (n+1) (E ) of each torsor, and so a bijection between T (∞) (E ) and the space O+ loc (E , A )[[ ]] of local action functionals with values in A , which are at least cubic modulo and modulo the ideal I ⊂ A . Proof. The proof is essentially the same as before. The extra diﬃcul- ties are of two kinds: working with an auxiliary parameter space X intro- duces extra analytical diﬃculties, and working with quadratic terms in our interaction forces us to use Artinian induction with respect to the powers of the ideal I ⊂ A . For simplicity, I will only give the proof when the effective interactions I[L] are all at least cubic modulo . The argument in the general case is the same, except that we also must perform Artinian induction with respect to the powers of the ideal I ⊂ A . As before, we will prove the renormalization scheme dependent ver- sion of the theorem, saying that there is a bijection between T (∞) (E ) and O+ loc (E , A )[[ ]]. The renormalization scheme independent formulation is an easy corollary. Let us start by showing how to construct a theory associated to a local interaction I = i I(i,k) ∈ Oloc(E , A )[[ ]].

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