[Paper Review] Generalized Symmetries and Anomalies of 3d N=4 SCFTs

Lakshya Bhardwaj, Mathew Bullimore+2 more|arXiv (Cornell University)|Jan 5, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy98 references11 citations

TL;DR

This paper analyzes generalized global symmetries (0-form, 1-form, 2-group) and their ’t Hooft anomalies in 3d N=4 SCFTs, providing a UV-to-IR framework to determine Coulomb branch symmetry groups and their mixed anomalies from quiver data, with consistency checks against magnetic quivers, 4d Class S, and 5d SCFTs, including mirror symmetry.

ABSTRACT

We study generalized global symmetries and their 't Hooft anomalies in 3d N=4 superconformal field theories (SCFTs). Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes. While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed 't Hooft anomalies are more subtle due to potential symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and their mixed 't Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement 't Hooft anomaly matching. The final expression for the symmetry group and 't Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes. We provide consistency checks by applying our method to compute symmetry groups of 3d N=4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods. Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry.

Motivation & Objective

• Motivate the study of generalized global symmetries in 3d N=4 SCFTs and their IR enhancement.
• Develop a method to determine IR Coulomb branch 0-form symmetry groups from UV quiver data.
• Compute mixed ’t Hooft anomalies involving Coulomb, Higgs, and 1-form symmetries.
• Provide consistency checks by comparing to magnetic quivers of Class S theories and 5d SCFTs, and via 3d mirror symmetry.

Proposed method

• Classify A-type and B-type 0-form (Coulomb/Higgs) symmetries and their 2-group structure in 3d N=4 theories.
• Use center charges of monopole operators to deduce IR Coulomb 0-form symmetry and perform ’t Hooft anomaly matching.
• Describe how to read the IR symmetry group and anomalies directly from UV quiver data (positions of unbalanced and flavor nodes relative to balanced nodes).
• Introduce and utilize obstruction classes w2^A and w2^B for lifting flavor symmetries to extended structures in the 2-group framework.
• Analyze 1-form symmetries via equivalence classes of half-BPS line defects and their junctions to obtain Γ_A and Γ_B and their backgrounds.
• Apply consistency checks by comparing results with flavor symmetries of 4d Class S theories and 5d SCFTs, and with 3d mirror theories.

Experimental results

Research questions

• RQ1 How can the IR Coulomb branch 0-form symmetry group of 3d N=4 SCFTs be determined from UV quiver data?
• RQ2 What are the mixed ’t Hooft anomalies between Coulomb 0-form, Higgs 0-form, and 1-form symmetries in these theories?
• RQ3 How do unbalanced and flavor nodes in good quiver gauge theories encode the global form of IR symmetries?
• RQ4 Do the obtained symmetry groups and anomalies match those of magnetic quivers for Class S theories and 5d SCFTs, and agree under 3d mirror symmetry?

Key findings

• A concrete UV-to-IR method is provided to deduce the IR Coulomb 0-form symmetry group from monopole operator center charges.
• The final symmetry group and ’t Hooft anomalies can be read off from quiver data, especially node balance and flavor node placement.
• Consistency checks show IR Coulomb 0-form symmetries of magnetic quivers for Class S theories and 5d SCFTs match flavor symmetry groups of the higher-dimensional theories computed by other means.
• The framework reproduces the Higgs 0-form symmetry of the mirror theory and supports anomaly matching across mirror pairs.
• The analysis enables study of higher-form and 2-group symmetries and their mixed anomalies in a unified UV-quiver language.
• Extensions to non-invertible symmetries and discrete 0-form symmetries are discussed as future directions.