On the 4d/3d/2d view of the SCFT/VOA correspondence

TL;DR

This work develops a unifying, Omega-deformation–based approach to the SCFT/VOA correspondence by embedding the Beem et al. 4d VOA in a holomorphic-topological 4d theory on a cigar, then dimensionally reducing to a 3d theory with boundary data that encodes a boundary VOA. The Omega-deformation converts the boundary data into a deformed boundary condition $H_\varepsilon$, which matches Costello–Gaiotto's deformations and yields a 3d TQFT coupled to a holomorphic boundary, with a natural pathway to 2d $\mathcal{N}=(0,2)$ chiral algebras via interval reductions. The paper also clarifies how rank-zero 3d $\mathcal{N}=4$ SCFTs arise from twisted $S^1$ reductions of Argyres–Douglas theories and why many of these theories exhibit a match between 4d VOAs and boundary VOAs in 3d, extending to conjectures about module categories and potential MTC structures. Through Lagrangian and non-Lagrangian examples, the work connects boundary deformations, 2d chiral algebras, and 3d TQFTs, and outlines concrete open problems, including the fate of rank-zero theories under various twists and the precise dictionary between 3d lines and 2d VOA modules. Overall, this framework provides new control over the interdimensional VOA data and suggests feasible routes to probe non-unitary and non-rational VOAs via 3d boundary constructions and their 2d reductions.

Abstract

We start with the SCFT/VOA correspondence formulated in the Omega-background approach, and connect it to the boundary VOA in 3d $\mathcal{N}=4$ theories and chiral algebras of 2d $\mathcal{N}=(0,2)$ theories. This is done using the dimensional reduction of the 4d theory on the topologically twisted and Omega-deformed cigar, performed in two steps. This paves the way for many more interesting questions, and we offer quite a few. We also use this approach to explain some older observations on the TQFTs produced from the generalized Argyres-Douglas (AD) theories reduced on the circle with a discrete twist. In particular, we argue that many AD theories with trivial Higgs branch, upon reduction on $S^1$ with the $\mathbb{Z}_N$ twist (where $\mathbb{Z}_N$ is a global symmetry of the given AD theory), result in the rank-0 3d $\mathcal{N}=4$ SCFTs, which have been a subject of recent studies. A generic AD theory, by the same logic, leads to a 3d $\mathcal{N}=4$ SCFT with zero-dimensional Coulomb branch (and suggests that there are a lot of them). Our construction therefore puts various empirical observations on the firm ground, such as, among other things, the match between the 4d VOA and the boundary VOA for some 3d rank-0 SCFTs previously observed in the literature. We end with an extensive list of promising open problems.

Figures (13)

• Figure 1: The areas where Vertex Algebras play role, and which are unified by our construction relying on the Omega-deformed cigar.
• Figure 2: Degenerating the topologically twisted cigar engineers the boundary conditions $H$.
• Figure 3: $\mathcal{Q}_B$-exact deformation folding $C=\mathbb{R}^2$ into a cigar.
• Figure 4: State $\Psi$ in the TQFT provides conformal block for the boundary VOA.
• Figure 5: State $|0\rangle$ corresponding to the vacuum block is generated by the empty solid torus.