Ω-deformation and quantization

TL;DR

This paper develops an ${\Omega}$-deformation for Rozansky-Witten theory by modeling RW on ${M = \mathbb{R} \times \Sigma}$ with hyperkähler target $X$ as an ${\Omega}$-deformed B-twisted Landau-Ginzburg theory with target $Y = {\rm Map}(\mathbb{R},X)$. For a disk boundary, the deformation leads to a quantum mechanical system on a symplectic submanifold $(L,\omega_J) \subset X$, with Planck constant $\hbar \propto {\varepsilon}$, and observables forming a noncommutative deformation of holomorphic functions restricted to $L$. The framework unifies two distinct 4d ${\mathcal N}=2$ phenomena: quantization of integrable systems via spacetime twisting and the quantization of holomorphic function algebras via supersymmetric loop operators, via a common 3d/2d topological construction and brane data. It also elucidates the relation to A-model branes and offers a path to generalizations, including gauged RW theories and broader two-dimensional dualities. Overall, the work provides a cohesive picture linking Omega-deformations, quantization, and four-dimensional gauge theory dynamics.

Abstract

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 $\mathbb{R} \times Σ$, 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 four-dimensional 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.