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