SCFT/VOA correspondence via $\Omega$-deformation

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 $\Omega$-deformation of supersymmetric gauge theories. The two-dimensional $\Omega$-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 $\Omega$-deformation point of view on the correspondence.