3d $\mathcal{N}=4$ Mirror Symmetry, TQFTs, and 't Hooft Anomaly Matching

• URL: http://arxiv.org/abs/2412.21066v1
• Date: Mon, 30 Dec 2024 16:34:54 GMT
• Title: 3d $\mathcal{N}=4$ Mirror Symmetry, TQFTs, and 't Hooft Anomaly Matching
• Authors: Mahesh K. N. Balasubramanian, Anindya Banerjee, Matthew Buican, Zhihao Duan, Andrea E. V. Ferrari, Hongliang Jiang,
• Abstract summary: Local unitary 3d $mathcalN=4$ superconformal field theory has a corresponding "universal" relevant deformation that takes it to a gapped phase. This deformation preserves all continuous internal symmetries, $mathcalS$, and therefore also preserves any 't Hooft anomalies supported purely in $mathcalS$. We argue that the universal deformations take these QFTs to Abelian fractional quantum Hall states in the infrared (IR), and we explain how to match 't Hooft anomalies between the non-topological ultraviolet theories and
• Score: 1.950885067119349
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• Abstract: Any local unitary 3d $\mathcal{N}=4$ superconformal field theory (SCFT) has a corresponding "universal" relevant deformation that takes it to a gapped phase. This deformation preserves all continuous internal symmetries, $\mathcal{S}$, and therefore also preserves any 't Hooft anomalies supported purely in $\mathcal{S}$. We describe the resulting phase diagram in the case of SCFTs that arise as the endpoints of renormalization group flows from 3d $\mathcal{N}=4$ Abelian gauge theories with any number of $U(1)$ gauge group factors and arbitrary integer charges for the matter fields. We argue that the universal deformations take these QFTs to Abelian fractional quantum Hall states in the infrared (IR), and we explain how to match 't Hooft anomalies between the non-topological ultraviolet theories and the IR topological quantum field theories (TQFTs). Along the way, we give a proof that 3d $\mathcal{N}=4$ mirror symmetry of our Abelian gauge theories descends to a duality of these TQFTs. Finally, using our anomaly matching discussion, we describe how to connect, via the renormalization group, abstract local unitary 3d $\mathcal{N}=4$ SCFTs with certain 't Hooft anomalies for their internal symmetries to IR phases (partially) described by Abelian spin Chern-Simons theories.

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