New T-duality for Chern-Simons Theory

It has recently pointed out that a four-dimensional analog of Chern-Simons theory provides an elegant framework for understanding integrable models with spectral parameters. The goal of this short note is to better understand the relation of this theory to the more standard three-dimensional Chern-Simons theory. We point out that two Chern-Simons theories, in four dimensions and three dimensions, are related by a novel T-duality in field theory. We then discuss this T-duality in string theory. Our T-duality prescription applies to a more general class of topological quantum field theories, producing mixed topological/holomorphic theories. This paper is motivated by the observation by C. Vafa.