Chiral algebras from Ω-deformation

TL;DR

The article demonstrates that the chiral algebras of unitary four-dimensional ${ m N}=2$ SCFTs can be realized as the ${ m Q}^ obreak ext{hbar}$-cohomology of local operators in an Omega-deformed, topological-holomorphic twist, and these algebras are isomorphic to the Beem et al. chiral algebras. By first analyzing Omega-deformations in two dimensions and then lifting to four dimensions, the authors show that the deformation replaces the nilpotent supercharge with a deformed one ${ m Q}^ obreak ext{hbar}$ that squares to a twisted rotation ${ m L}_{V}$, yielding a holomorphic structure along the complex plane while localizing operators to the origin on ${ m R}^2$. Localization reduces the four-dimensional path integral to a gauged beta-gamma system on the holomorphic curve, and, for vector multiplets and hypermultiplets, this reproduces Beem’s chiral algebra, modulo a BRST analysis that imposes anomaly cancellation conditions. The framework extends to nonconformal theories via surface defects, which induce compatible two-dimensional theories on the defect and allow a consistent chiral algebra after BRST reduction, with potential theta-like terms restoring twisted rotational symmetry when needed. Finally, the paper extends the discussion to three-dimensional ${ m N}=4$ theories, where Omega-deformation yields a topological quantum mechanics that quantizes the Rozansky–Witten chiral ring, aligning with known results on deformation quantization in this setting.

Abstract

In the presence of an $Ω$-deformation, local operators generate a chiral algebra in the topological-holomorphic twist of a four-dimensional $\mathcal{N} = 2$ supersymmetric field theory. We show that for a unitary $\mathcal{N} = 2$ superconformal field theory, the chiral algebra thus defined is isomorphic to the one introduced by Beem et al. Our definition of the chiral algebra covers nonconformal theories with insertions of suitable surface defects.