32 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Feynman graphs. Section 10 uses this to construct the local counterterms as- sociated to a Lagrangian, which are needed to render the functional integral finite. Section 11 gives the proof of theorems A and B above. Finally, we turn to generalizations of the main results. Section 13 shows how everything generalizes, mutatis mutandis, to the case when our fields are no longer just functions, but sections of some vector bundle. Section 14 shows how we can further generalize to deal with theories on non-compact manifolds, as long as an appropriate infrared cut-off is introduced. 2. The effective interaction and background field functional integrals As in the introduction, a quantum field theory in our Wilsonian defini- tion will be given by a collection of effective actions, related by the renor- malization group flow. In this section we will write down a version of the renormalization group flow, based on the effective interaction, which we will use throughout the book. 2.1. Let us assume that our energy Λ effective action can be written as S[Λ](φ) = − 1 2 φ, (D +m2)φ + I[Λ](φ) where: (1) The function I[Λ] is a formal series in , I[Λ] = I0[Λ]+ I1[Λ]+··· , where the leading term I0 is at least cubic. Each Ii is a formal power series on the vector space C∞(M) of fields (later, I will explain what this means more precisely). The function I[Λ] will be called the effective interaction. (2) , denotes the L2 inner product on C∞(M, R) defined by φ, ψ = M φψ. (3) D denotes1 the Laplacian on M, with signs chosen so that the eigenvalues of D are non-negative and m ∈ R 0 . Recall that the renormalization group equation relating S[Λ] and S[Λ ] can be written S[Λ ](φL) = log φH∈C∞(M)[Λ ,Λ) eS[Λ](φL+φH)/ . We can rewrite this in terms of the effective interactions, as follows. The spaces C∞(M) Λ and C∞(M)[Λ ,Λ) are orthogonal. It follows that S[Λ](φL + φH) = − 1 2 φL, (D +m2)φL − 1 2 φH, (D +m2)φH + I[Λ](φL + φH). 1 The symbol Δ will be reserved for the BV Laplacian

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