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.
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