9. GAUGE THEORIES 23

9.1. In mathematical parlance, a gauge theory is a field theory where

the space of fields is a stack. A typical example is Yang-Mills theory, where

the space of fields is the space of connections on some principal G-bundle

on space-time, modulo gauge equivalence.

It is important to emphasize the difference between gauge theories and

field theories equipped with some symmetry group. In a gauge theory, the

gauge group is not a group of symmetries of the theory. The theory does

not make any sense before taking the quotient by the gauge group.

One can see this even at the classical level. In classical U(1) Yang-Mills

theory on a 4-manifold M, the space of fields (before quotienting by the

gauge group) is

Ω1(M).

The action is S(α) =

M

dα∗dα. The highly degen-

erate nature of this action means that the classical theory is not predictive:

a solution to the equations of motion is not determined by its behaviour on

a space-like hypersurface. Thus, classical Yang-Mills theory is not a sensible

theory before taking the quotient by the gauge group.

9.2. Let us now discuss gauge theories in effective field theory. Naively,

one could imagine that to give a gauge theory would be to give an effective

gauge theory at every energy level, in a way related by the renormalization

group flow.

One immediate problem with this idea is that the space of low-energy

gauge symmetries is not a group. The product of low-energy gauge sym-

metries is no longer low-energy; and if we project this product onto its

low-energy part, the resulting multiplication on the set of low-energy gauge

symmetries is not associative.

For example, if g is a Lie algebra, then the Lie algebra of infinitesi-

mal gauge symmetries on a manifold M is

C∞(M)

⊗ g. The space of low-

energy infinitesimal gauge symmetries is then

C∞(M)≤Λ

⊗ g. In general,

the product of two functions in

C∞(M)≤Λ

can have arbitrary energy; so

that

C∞(M)

Λ

⊗ g is not closed under the Lie bracket.

This problem is solved by a very natural union of the Batalin-Vilkovisky

formalism and the effective action philosophy.

9.3. The Batalin-Vilkovisky formalism is widely regarded as being the

most powerful and general way to quantize gauge theories. The first step

in the BV procedure is to introduce extra fields – ghosts, corresponding

to infinitesimal gauge symmetries; anti-fields dual to fields; and anti-ghosts

dual to ghosts – and then write down an extended classical action functional

on this extended space of fields.

This extended space of fields has a very natural interpretation in ho-

mological algebra: it describes the derived moduli space of solutions to the

Euler-Lagrange equations of the theory. The derived moduli space is ob-

tained by first taking a derived quotient of the space of fields by the gauge

group, and then imposing the Euler-Lagrange equations of the theory in a

derived way. The extended classical action functional on the extended space

of fields arises from the differential on this derived moduli space.