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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