Ω-deformation and quantization

Abstract

We formulate a deformation of Rozansky-Witten theory analogous to the $\Omega$-deformation. It is applicable when the target space $X$ is hyperkähler 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 hyperkähler manifold.

24 pages. v2: minor changes, references added; v3: minor changes, a reference added, published version

References in corpus (2)

• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Comments on N=(2,2) Supersymmetry on Two-Manifolds

Cited by in corpus (3)

• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence