12. PROOF OF THE PARAMETRIX FORMULATION 69 The following lemma now shows that T (n+1) → T (n) is a torsor for Oloc(C∞(M )). Lemma 11.1.1. Let I, J ∈ T (n+1) be theories which agree in T (n) . Then, the functional I0[L] + δ −(n+1) (I[L] − J[L]) ∈ O(C∞(M )) satisfies the classical renormalization group equation modulo δ2, and so de- fines an element of TI 0 [L] (T (0) ) ∼ Oloc(C∞(M )). Note that −(n+1) (I[L] − J[L]) is well-defined as I[L] and J[L] agree modulo n+1 . Proof. This is a simple calculation. 12. Proof of the parametrix formulation of the main theorem In this page we will prove the equivalence of the definition of theory based on arbitrary parametrices, explained in Section 8 with the definition based on the heat kernel. Since this result is not used elsewhere in the book, I will not give all the details. Thus, suppose we have a theory in the heat kernel sense, given by a fam- ily I[L] of effective interactions satisfying the renormalization group equation and the locality axiom. If P is a parametrix, let us define a functional I[P ] by I[P ] = W (P − P (0,L),I[L]) ∈ O(C∞(M ))[[ ]]. Since P (0,L) and P are both parametrices for the operator D +m2, the difference between them is smooth. Thus, W (P − P (0,L),I[L]) is well- defined. Lemma 12.0.1. The collection of effective interactions {I[P ]}, defined for each parametrix P , defines a theory using the parametrix definition of theory. Proof. To prove this, we need to verify the following. (1) If P is another parametrix, then I[P ] = W ( P − P, I[P ] ) (this is the version of the renormalization group equation for the definition of theory based on parametrices. (2) By choosing a parametrix P with support close to the diagonal, we can make the distribution Ii,k[P ] ∈ D(M k )S k on M k supported as close as we like to the small diagonal.

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