Four-Dimensional Chern-Simons and Gauged Sigma Models

TL;DR

This paper presents a novel method for constructing integrable gauged sigma models using four-dimensional Chern-Simons theory, extending previous work with new classes of defects and explicit examples like gauged Wess-Zumino-Witten models.

Contribution

It introduces a unified approach to generate gauged sigma models from 4d Chern-Simons theory with new defect configurations, demonstrating their integrability and providing explicit examples.

Findings

Constructed gauged sigma models from 4d CS with new defect types.

Identified models with gauge fields equivalent to Lax connections, ensuring integrability.

Derived conformal Toda models from nilpotent gauged WZW models.

Abstract

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.