From BPS Spectra of Argyres-Douglas Theories to Families of 3d TFTs

Vertex operator algebras (VOAs) arise in protected subsectors of supersymmetric quantum field theories, notably in 4d ${\mathcal N}=2$ superconformal field theories (SCFT) via the Schur sector and in twisted 3d ${\mathcal N}=4$ theories via boundary algebras. These constructions are connected through twisted circle compactifications, which can be best understood from the dynamics of BPS particles in the Coulomb branch of the 4d SCFT. This data is encoded in an operator $\hat\Phi$ acting on the Hilbert space of an auxiliary quantum mechanics of BPS particles, whose trace yields the partition functions of a 3d topological field theory (TFT) bounding the VOA. We generalize this trace formula by considering higher powers of $\hat\Phi$, leading to a finite family of VOAs associated with a given 4d SCFT. Applying this framework to Argyres-Douglas theories labeled by $(A_1, G)$, where $G$ is an ADE-type group of rank up to 8, we extract the modular data of the family of boundary VOAs via TFT partition function calculations on Seifert manifolds. Our results suggest that the modular data obtained from different powers of $\hat\Phi$ are related by Galois transformations.