\Omega-deformation and quantization

We formulate a deformation of Rozansky-Witten theory analogous to the $\mathrm{\Omega }$-deformation. It is applicable when the target space $X$ is hyperk"ahler and the spacetime is of the form $\mathbb{R}\times \mathrm{\Sigma }$, with $\mathrm{\Sigma }$ being a Riemann surface. In the case that $\mathrm{\Sigma }$ is a disk, the $\mathrm{\Omega }$-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 $\mathrm{\Omega }$-deformation and quantization of integrable systems. The other concerns supersymmetric loop operators and quantization of the algebra of holomorphic functions on a hyperk"ahler manifold.