3D TFTs and boundary VOAs from BPS spectra of $(G,G')$ Argyres-Douglas theories

TL;DR

The work provides a concrete, computational bridge from 4d N=2 Argyres-Douglas SCFTs to 2d boundary VOAs by constructing 3d topological field theories via a wall-crossing invariant trace formula that encodes the Coulomb-branch BPS spectrum. By translating BPS quiver mutations into 3d ellipsoid partition functions and applying 3d A-model localization, the authors extract partial modular data for the boundary VOAs, including explicit matches with W-algebras and coset models and clear demonstrations of A-/B-twist correspondences and Galois conjugates. The approach yields detailed VOA data for a broad class of (G,G') theories, verifies known SCFT/VOA correspondences, and even provides new partial data for previously unexplored cases like (A3,D4) and (A2,E8) via the 3d TFT perspective. These results deepen the understanding of the 4d SCFT/2d VOA correspondence and illustrate how semisimple TFTs can serve as computationally tractable intermediaries between higher- and lower-dimensional quantum field theories with rich algebraic structures. Overall, the paper offers a powerful framework for systematically deriving VOA modular data from 4d SCFTs using 3d topological intermediates, with potential extensions to Schur indices, line defects, and non-semisimple TFTs.

Abstract

We explore 3d $ \mathcal{N}=4 $ theories arising from twisted compactification of 4d $ \mathcal{N}=2 $ $ (G, G') $ Argyres-Douglas superconformal field theories (SCFTs), together with the 2d vertex operator algebras (VOAs) supported on the holomorphic boundary of their topologically twisted sector. Starting from the Coulomb branch BPS spectra of the $ (G,G') $ Argyres-Douglas theories, we develop a systematic and efficient method to obtain the ellipsoid partition functions of associated 3d theories using quiver mutations and wall-crossing invariants. This allows us to extract the modular data of the boundary VOAs, which are related to the Schur sectors of the 4d theories through the 4d SCFT/2d VOA correspondence. Our results provide a useful computational bridge between 4d SCFTs and 2d VOAs through interpolating 3d topological field theories.

Figures (10)

• Figure 1: Central charges in two distinct chambers on the Coulomb branch of the $(A_1,A_2)$ Argyres-Douglas theory. In the LHS chamber, there are two BPS (anti-)particles, represented by solid(dashed) vectors on the complex plane, whereas in the RHS chamber there are three. The trace formula \ref{['eq: trace formula']} reads the Faddeev's quantum dilogarithms in the counterclockwise order of the central charge phases, indicated by the red and blue circular arrows in the LHS/RHS chambers, respectively.
• Figure 2: A mutation $\mathfrak{M}_i$ at the $i$-th node, where the central charge ${\cal Z}_{\gamma_i}$ of the associated basis charge $\gamma_i$ lies on the left boundary of the cone of BPS particles, corresponds to a clockwise rotation of the upper-half ${\cal Z}$-plane. Under this rotation, ${\cal Z}_{\gamma_i}$ exits the half-plane and becomes an anti-particle, while its CPT conjugate ${\cal Z}_{-\gamma_i}$ enters the half-plane and becomes a particle. In the figure, solid vectors represent BPS particles and dotted vectors represent anti-particles.
• Figure 3: Two sequences of mutations that flip the signs of all basis charges of $(A_1,A_2)$ Argyres-Douglas theory. The sequence shown in blue corresponds to the chamber with three BPS particles, while the sequence shown in red corresponds to chamber with two BPS particles.
• Figure 4: An example of the tensor product and the square product of the canonical quivers $A_3$ and $D_4$. The purple arrows in $A_3 \otimes D_4$ indicate the arrows belonging to the subquivers $\{I\}\otimes D_4$ and $A_3 \otimes \{J\}$, where $I$ and $J$ are sinks of $A_3$ and sources of $D_4$, respectively. By flipping these arrows, the square product quiver $A_3 \mathbin{\text{$\square$}} D_4$ is obtained whose nodes can be decomposed into two disjoint sets $\Sigma_\pm$ as in \ref{['eq: disjoint set of square']}. The nodes in $\Sigma_+$ and $\Sigma_-$ are colored red and blue, respectively.
• Figure 5: Examples of the square product $G \mathbin{\text{$\square$}} G'$ of BPS quivers. The vertical and horizontal directions correspond to the Dynkin diagrams of $G$ and $G'$, respectively. Each of the four arrows surrounding a single plaquette circulates around it. In the depicted BPS quivers, the nodes $\sigma \in \Sigma_\pm$ are colored red and blue, respectively. Note that there are no arrows between any two nodes of the same color, which implies that the Dirac pairing between any two charges assigned to nodes of the same color is trivial.