13. VECTOR-BUNDLE VALUED FIELD THEORIES 75 Note that Ol(E , A ) is not a closed subspace. However, as we will see in Appendix 2, Oloc(E , A ) has a natural topology making it into a complete nuclear space, and a module over A in the symmetric monoidal category of nuclear spaces. Our interactions will be elements of Ol(E , A )[[ ]]. We would like to allow our interactions to have quadratic and linear terms modulo . However, we require that these quadratic terms are accompanied by elements of the nilpotent ideal I = Γ(X, I) ⊂ A (recall that A /I = C∞(X)). If we don’t impose this condition, we will encounter infinite sums. Thus, let us denote by O+(E , A )[[ ]] ⊂ O(E , A )[[ ]] the subset of those functionals which are at least cubic modulo the ideal generated by I and . Then, the renormalization group operator W (P (ε, L),I) = log ( exp( ∂P (ε,L) ) exp(I/ ) ) : O+(E , A )[[ ]] → O+(E , A )[[ ]] is well-defined. Because we now allow quadratic and linear interaction terms modulo , the Feynman graph expansion of this expression involves univalent and bivalent genus 0 vertices. However, each such vertex is accompanied by an element of the ideal I of A . Since this ideal is nilpotent, there is a uniform bound on the number of such vertices that can occur, so there are no infinite sums. Definition 13.4.2. A theory is given by a collection of even elements I[L] ∈ O+(E , C∞((0, ∞)L) ⊗ A )[[ ]], such that (1) The renormalization group equation I[L ] = W ( P (L, L ),I[L] ) holds. (2) Each I(i,k)[L] has a small L asymptotic expansion I(i,k)[L](e) Ψr(e)fr(L) where Ψr ∈ Ol(E ) are local action functionals. Let T (∞) (E ) denote the space of such theories, and let T (n) (E ) denote the space of theories defined modulo n+1 , so that T (∞) (E ) = lim ←− T (n) (E ).

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