6. THE MAIN THEOREM 17 scheme is a way to extract the singular part of certain functions of one variable. We construct a certain subalgebra P((0, 1)) ⊂ C∞((0, 1)) consisting of functions f(ε) of a “motivic” nature. Functions in P((0, 1)) arise as the periods of families of algebraic varieties over Zariski open subsets U ⊂ A1 Q , such that U(R) contains (0, 1). (For more details, see Chapter 2, Section 9). Definition 6.0.1. A renormalization scheme is a subspace P((0, 1)) 0 ⊂ P((0, 1)) of “purely singular” functions, complementary to the subspace P((0, 1))≥0 ⊂ P((0, 1)) of functions whose r → ∞ limit exists. The choice of a renormalization scheme gives us a way to extract the singular part of functions in P((0, 1)). The variant theorem is the following. Theorem B. The choice of a renormalization scheme leads to a bijec- tion between the space of theories and the space of local action functionals S : C∞(M) → R[[ ]]. Equivalently, there is a bijection between the space of theories and the space of Lagrangians up to the addition of a Lagrangian which is a total derivative. Theorem B implies theorem A, but theorem A is the more natural for- mulation. There are certain caveats: (1) Like the effective actions Seff[Λ], the local action functional S is a formal power series both in φ ∈ C∞(M) and in . (2) Modulo , we require that S is of the form S(φ) = − 1 2 M φ D φ + cubic and higher terms. There is a more general formulation of this theorem, where the space of fields is allowed to be the space of sections of a graded vector bundle. In the more general formulation, the action functional S must have a quadratic term which is elliptic in a certain sense. 6.1. Let me sketch how to prove theorem A. Given the action S, we construct the low-energy effective action Seff[Λ] by renormalizing a certain functional integral. The formula for the functional integral is Seff[Λ](φ L ) = log φH∈C∞(M)(Λ,∞) eS(φL+φH)/ .

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