Ω-deformation and quantization

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