Chern-Simons three-dimensional and Yang-Mills-Higgs two-dimensional systems as four-dimensional topological quantum field theories

Abstract

We show that the second Chern class considered as a classical topological action ∫ M4 TrFF over the four-dimensional manifold M 3×[0, ∞] can be gauge fixed into a supersymmetric action defined over a three-dimensional submanifold M 3 and containing the Chern-Simons action ∫ M3 Tr(A dA+ 2 3 AAA) plus other metric dependent terms including a regularizing ordinary Yang-Mills action ∫ M3 TrF ∗F . We use a temporal gauge in M 4 and choose a gauge in the three-dimensional quantum field theory to probe the moduli space of solutions to the equation ∗F=0 in M 3. Ward identities should explain the metric independence of expectation values of Wilson loops computed from this action. We also show that another gauge fixing of the second Chern class gives a two-dimensional Yang-Mills-Higgs quantum field theory.