Chiral Algebra, Localization, Modularity, Surface defects, And All That

We study the 2D vertex operator algebra (VOA) construction in 4D $mathcal{N}=2$ superconformal field theories (SCFT) on $S^3 times S^1$, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus $mathbb{T}^2subset S^3times S^1$, it is $frac12mathbb{Z}$-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $mathbb{Z} +frac12$-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserve the VOA, we find that it admits discrete deformations, which allow access to the remaining spin structures (1,1), (0,1) and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle, the two-point functions, and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the $T$-transformation acts on our four partition functions and lifts to a large diffeomorphism on $S^3times S^1$. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled both by the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the $S$-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the $Omega$-background.