We formulate a deformation of Rozansky-Witten theory analogous to the Ω-deformation. It is applicable when the target space X is hyperkähler and the spacetime is of the form R×Σ, with Σ being a Riemann surface. In the case that Σ is a disk, the Ω-deformed Rozansky-Witten theory quantizes a symplectic submanifold of X, thereby providing a new perspective on quantization. As applications, we elucidate two phenomena in fourdimensional gauge theory from this point of view. One is a correspondence between the Ω-deformation and quantization of integrable systems. The other concerns supersymmetric loop operators and quantization of the algebra of holomorphic functions on a hyperkähler manifold.

The supersymmetric index of the 4d N = 1 theory realized by a brane tiling coincides with the partition function of an integrable 2d lattice model. We argue that a class of half-BPS surface defects in brane tiling models are represented on the lattice model side by transfer matrices constructed from L-operators. For the simplest surface defects in theories with SU(2) flavor groups, we identify the relevant L-operator as that discovered by Sklyanin in the context of the eight-vertex model. We verify this identification by computing the indices of class-S and -S k theories in the presence of the surface defects.A Definitions and useful formulas 53 A.1 Theta functions 53 A.2 Elliptic gamma function 54

Abstract:We study the six-dimensional (2, 0) superconformal field theory on S 1 ×S 2 ×M via compactification to five dimensions, where M is a three-manifold. Twisted along M , the five-dimensional theory has a half of N = (2, 2) supersymmetry on S 2 , the other half being broken by a superpotential. We show that in the limit where M is infinitely large, the twisted theory reduces to a three-dimensional topological quantum field theory which is closely related to Chern-Simons theory for the complexified gauge group.