Chern-Simons-Rozansky-Witten topological field theory

TL;DR

This paper introduces a novel three-dimensional topological field theory combining Chern-Simons and Rozansky-Witten theories, revealing new links between gauge theories, hyper-Kahler geometry, and superalgebra structures.

Contribution

It constructs a new topological field theory that generalizes known models and connects supergroup Chern-Simons theory with hyper-Kahler geometry and categorical structures.

Findings

Model reduces to supergroup Chern-Simons theory for affine space X.

Identifies Wilson loops with objects in a quantum-deformed category.

Provides insights into the role of Lie superalgebras in topological field theories.

Abstract

We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern-Simons and Rozansky-Witten theory and can be regarded as a topologically-twisted version of the N=4 d=3 supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group G and a hyper-Kahler manifold X with a tri-holomorphic action of G. In the case when X is an affine space, we show that the model is equivalent to Chern-Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witten. For general X, our model appears to be new. We describe some of its properties, focusing on the case when G is simple and X is the cotangent bundle of the flag variety of G. In particular, we show that Wilson loops are labeled by objects of a certain category which is a quantum deformation of the equivariant derived category of coherent sheaves on X.