Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 9

Local operators of $4d \mathcal{N}=2$ gauge theories from the affine Grassmannian

Pages: 3207 – 3247

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n9.a10

Author

Wenjun Niu (Department of Mathematics and Center for Quantum Mathematics and Physics, University of California, Davis, Calif., U.S.A.; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Abstract

We give a new, fully mathematical, construction of the space of local operators in the holomorphic-topological twist of $4d \: \mathcal{N} = 2$ gauge theories. It is based on computations of morphism spaces in the DG category of line operators, which (by work of Kapustin–Saulina and Cautis–Williams) may be represented as ind-coherent sheaves on the affine Grassmannian and, more generally, on the $\mathcal{R}_{G,V}$ spaces of Braverman–Finkelberg–Nakajima. We prove that characters of our morphisms spaces reproduce the Schur indices of $4d \: \mathcal{N}=2$ theories, and that the spaces themselves agree with the $4d \: \mathcal{N}=2$ vertex algebras of Beem–Lemos–Liendo–Peelaers–Rastelli–Van Rees, Oh-Yagi, Butson, and Jeong. We also generalize our construction to local operators at junctions of Wilson–’t Hooft lines, and compare the Euler character of the morphism spaces to the Schur indices in the work of Cordova–Gaiotto–Shao.

Published 30 October 2023