9. GAUGE THEORIES 25 the renormalization group flow. In addition, one requires that the effective actions Seff[Λ] satisfy a locality axiom, as before. This picture also solves the problem that the low energy gauge symme- tries are not a group. The energy Λ effective action Seff[Λ], satisfying the energy Λ quantum master equation, gives the extended space of low-energy fields a certain homotopical algebraic structure, which has the following in- terpretation: (1) The space of low-energy infinitesimal gauge symmetries has a Lie bracket. (2) This Lie algebra acts on the space of low-energy fields. (3) The space of low-energy fields has a functional, invariant under the bracket. (4) The action of the Lie algebra on the space of fields, and on itself, preserves the Lebesgue measure. However, these axioms don’t hold on the nose, but hold up to a sequence of coherent higher homotopies. 9.5. Let us now formalize our definition of a gauge theory. As we have seen, whenever we have the data of a classical gauge theory, we get an extended space of fields, that we will denote by E . This is always the space of sections of a graded vector bundle on the manifold M. As before, let E≤Λ denote the space of low-energy extended fields. Definition 9.5.1. A theory in the BV formalism consists of a set of low-energy effective actions Seff[Λ] : E≤Λ → R[[ ]], which is a formal series both in E≤Λ and , and which is such that: (1) The renormalization group equation is satisfied. (2) Each Seff[Λ] satisfies the energy Λ quantum master equation. (3) The same locality axiom as before holds. (4) There is one more technical restriction : modulo , each Seff[Λ] is of the form Seff[Λ](e) = e, Qe + cubic and higher terms where −, − is a certain canonical pairing on E , and Q : E → E satisfies certain ellipticity conditions. As before, the locality axiom needs to be expressed in length-scale terms. The main theorem holds in this context also, but in a slightly modified form. If we remove the requirement that the effective actions satisfy the quantum master equation, we find a bijection between theories and local action functionals, depending on the choice of a renormalization scheme, as before. Requiring that the effective actions satisfy the QME leads to a constraint on the corresponding local action functional, which is called the

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