Chern-Simons-Rozansky-Witten topological field theory

TL;DR

CSRW constructs a new 3d topological field theory that interpolates between Chern–Simons and Rozansky–Witten theories via a topological twist of the Gaiotto–Witten N=4 theory. The model depends on a gauge group $G$ and a hyper-Kähler target $X$, becoming a supergroup Chern–Simons theory for flat $X$ and a gauged Rozansky–Witten theory for curved $X$. Local observables in the flat case reproduce Lie superalgebra cohomology, while Wilson loops in the curved case are labeled by objects in a deformed equivariant derived category of $X$, with a deformation controlled by the Chern–Simons level through a parameter $oldsymbol{ abla}$. The work yields concrete constructions of BRST-invariant Wilson loops and suggests connections to quantum groups and new 3-manifold invariants, revealing a rich categorical structure underpinning CSRW 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.