Four-Dimensional Chern-Simons and Gauged Sigma Models

In this paper, we introduce a new method for constructing gauged $\sigma$-models from four-dimensional Chern-Simons (4d CS) gauge theory. We begin with a review of recent work by several authors on the classical generation of integrable $\sigma$-models from 4d CS. In this approach, a gauge field is required to satisfy certain boundary conditions on two-dimensional defects inserted into the bulk. Using these boundary conditions, the equations of motion are solved, and the result is substituted back into the action. This yields a $\sigma$-model whose integrability is guaranteed because the 4d CS field is gauge equivalent to a Lax connection. Using a theory consisting of two 4d CS fields coupled together on new classes of ``gauged'' defects, we construct gauged $\sigma$-models and identify a unifying action. These models are conjectured to be integrable because the 4d CS fields remain gauge equivalent to two Lax connections. Finally, we consider two examples: the gauged Wess-Zumino-Witten model and the nilpotent gauged Wess-Zumino-Witten models. This latter model is of note as one can find the conformal Toda models from it.