\Omega-deformation and quantization

TL;DR

This paper introduces an $oldsymbol{ ext{Ω}}$-deformation of Rozansky-Witten theory for hyperk"ahler targets, linking it to quantization of symplectic submanifolds and providing new insights into 4D gauge theories and integrable systems.

Contribution

It formulates a novel $oldsymbol{ ext{Ω}}$-deformation of Rozansky-Witten theory applicable to hyperk"ahler targets on $oldsymbol{ ext{R}} imes oldsymbol{ ext{Σ}}$, connecting it to quantization and gauge theory phenomena.

Findings

Quantizes symplectic submanifolds via $oldsymbol{ ext{Ω}}$-deformed Rozansky-Witten theory.

Establishes a correspondence between $oldsymbol{ ext{Ω}}$-deformation and integrable system quantization.

Analyzes supersymmetric loop operators and holomorphic function algebra quantization.

Abstract

We formulate a deformation of Rozansky-Witten theory analogous to the $\Omega$-deformation. It is applicable when the target space $X$ is hyperk\"ahler and the spacetime is of the form $\mathbb{R} \times \Sigma$, 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 hyperk\"ahler manifold.