  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 $\mathbb{T}^2\subset S^3\times S^1$, it is
$\frac12\mathbb{Z}$-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 $S^3\times S^1$.
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.
