14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 85 The resulting linear map E ⊗T (γ) → A f → wγ(P, {Iv})(f) is easily seen to have proper support. Note that this lemma is false if the graph γ has no tails. This is the reason why we only consider functionals on E modulo constants, or equivalently, without a constant term. Corollary 14.5.2. The renormalization group operator Op + (Ec, A )[[ ]] → Op + (Ec, A )[[ ]] I → W (P (ε, L),I) is well-defined. As always, O+(Ec, p A )[[ ]] refers to the subspace of Op(Ec, A )[[ ]] con- sisting of elements which are at least cubic modulo and the ideal Γ(X, m) ⊂ A . Proof. The renormalization group operator is defined by W (P (ε, L),I) = γ 1 Aut)γ) g(γ) wγ(P (ε, L),I). The sum is over connected stable graphs and, as we are working with func- tionals on E modulo constants, we only consider graphs with at least one tail. Lemma 14.5.1 shows that each wγ(P (ε, L),I) is well-defined. 14.6. Now we can define the notion of theory on the manifold M, using the fake propagator P (ε, L). Definition 14.6.1. A theory is a collection {I[L] | L ∈ R 0 } of ele- ments of O+(Ec, p A )[[ ]] which satisfy (1) The renormalization group equation, I[L] = W (P (ε, L),I[ε]) (2) The asymptotic locality axiom: there a small L asymptotic expan- sion I[L] fi(L)Ψi in terms of local action functionals Ψi ∈ O+ loc (Ec, A )[[ ]]. We will assume that the functions fi(L) appearing in this expansion have at most a finite order pole at L = 0 that is, we can find some n such that limL→0 Lnfi(L) = 0. Let T (n) (E , A ) denote the set of theories defined modulo n+1 , and let T (∞) (E , A ) denote the set of theories defined to all orders in .

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