Time-reversal invariant TQFTs from self-mirror symmetric SCFTs

TL;DR

This paper links self-mirror symmetric 3D $ ext{SCFTs}$ with time-reversal invariant Abelian TQFTs, deriving constraints, formulas, and invariance properties that deepen understanding of their structure and symmetries.

Contribution

It provides new constraints on charge matrices, formulas for superconformal indices, and introduces the Gauss generating function, connecting self-mirror symmetry with time-reversal invariance in TQFTs.

Findings

Charge matrix constraints for self-mirror symmetric SCFTs

Superconformal index invariance under R-symmetry fugacity inversion

Gauss generating function invariance under complex conjugation

Abstract

We establish a connection between three-dimensional self-mirror symmetric $\mathcal N=4$ superconformal field theories (SCFTs) and time-reversal invariant topological quantum field theories (TQFTs) arising from universal mass deformations. Focusing on the Abelian case, the ultraviolet (UV) SCFT is characterized by the charge matrix $Q$, while the infrared (IR) TQFT corresponds to an Abelian Chern-Simons theory with level matrix $K=QQ ^T$. We derive constraints on the charge matrix for self-mirror symmetric SCFTs and demonstrate that the Coulomb and Higgs branch Hilbert series of these theories coincide. Additionally, we derive a general formula for the superconformal indices of Abelian $\mathcal N=4$ SCFTs with arbitrary charge matrices. For SCFT with the constrained charge matrix, the superconformal index is argued to exhibit invariance under the inversion of fugacity associated with R-symmetry, providing further evidence of self-mirror symmetry. We explore various properties of time-reversal invariant Abelian Chern-Simons theories in detail and establish their connections to self-mirror symmetry in SCFTs from multiple perspectives. In particular, we introduce a quantity, dubbed Gauss generating function, which is real and thus invariant under complex conjugation for time-reversal symmetric TQFTs, in parallel with the superconformal index, which is invariant under the inversion of R-symmetry fugacity for self-mirror symmetric SCFTs.