SCFT/VOA correspondence via $Ω$-deformation

TL;DR

The paper develops a localization-based, Ω-deformation framework to realize the 4D $\mathcal{N}=2$ SCFT/2D VOA correspondence. By applying the holomorphic-topological twist and an $Ω$-deformation, it localizes the theory to a 2D chiral CFT on $\mathcal{C}$, with the VOA recovered as the algebra of local operators and the noncommutative deformation parameter identified with the deformation parameter $\varepsilon$. The work further shows that the 4D Schur index matches the 2D VOA vacuum character at the level of path integrals, providing a concrete bridge between 4D indices and 2D CFT data. It also clarifies the origin of noncommutativity in the chiral algebra within this setup and outlines potential generalizations to non-Lagrangian SCFTs and refined indices. Overall, the Ω-deformation viewpoint offers a transparent, localization-based realization of the SCFT/VOA correspondence and a structured path from 4D indices to 2D CFT data.

Abstract

We investigate an alternative approach to the correspondence of four-dimensional $\mathcal{N}=2$ superconformal theories and two-dimensional vertex operator algebras, in the framework of the $Ω$-deformation of supersymmetric gauge theories. The two-dimensional $Ω$-deformation of the holomorphic-topological theory on the product four-manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the $\mathcal{N}=2$ superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our $Ω$-deformation point of view on the correspondence.