26 1. INTRODUCTION

renormalized quantum master equation. This renormalized QME replaces

the ill-defined QME appearing in the naive BV formalism.

Physicists often say that a theory satisfying the quantum master equa-

tion is free of “gauge anomalies”. In general, an anomaly is a symmetry of

the classical theory which fails to be a symmetry of the quantum theory. In

my opinion, this terminology is misleading: the gauge group action on the

space of fields is not a symmetry of the theory, but rather an inextricable

part of the theory. The presence of gauge anomalies means that the theory

does not exist in a meaningful way.

9.6. Renormalizing gauge theories. It is straightforward to gener-

alize the Wilsonian definition of renormalizability (Definition 7.2.1) to apply

to gauge theories in the BV formalism. As before, this definition only works

on

Rn,

because one needs to rescale space-time. This rescaling of space-time

leads to a flow on the space of theories, which we call the local renormal-

ization group flow. (This flow respects the quantum master equation). A

theory is defined to be renormalizable if it exhibits at most logarithmic

growth under the local renormalization group flow.

Now we are ready to state one of the main results of this book.

Theorem. Pure Yang-Mills theory on

R4,

with coeﬃcients in a simple

Lie algebra g, is perturbatively renormalizable.

That is, there exists a theory {SY

eff

M

[Λ]}, which is renormalizable, which

satisfies the quantum master equation, and which modulo is given by the

classical Yang-Mills action.

The moduli space of such theories is isomorphic to R[[ ]].

Let me state more precisely what I mean by this. At the classical level

(modulo ) there are no diﬃculties with renormalization, and it is straight-

forward to define pure Yang-Mills theory in the BV

formalism4.

Because

the classical Yang-Mills action is conformally invariant in four dimensions,

it is a fixed point of the local renormalization group flow.

One is then interested in quantizing this classical theory in a renormal-

izable way.

The theorem states that one can do this, and that the set of all such

renormalizable quantizations is isomorphic (non-canonically) to R[[ ]].

This theorem is proved by obstruction theory. A lengthy (but straight-

forward) calculation in Lie algebra cohomology shows that the group of

obstructions to finding a renormalizable quantization of Yang-Mills theory

vanishes; and that the corresponding deformation group is one-dimensional.

Standard obstruction theory arguments then imply that the moduli space

of quantizations is R[[ ]], as desired.

4For

technical reasons, we use a first-order formulation of Yang-Mills, which is equiv-

alent to the usual formulation.