Generalized Symmetries and Anomalies of 3d N=4 SCFTs

TL;DR

The paper develops a comprehensive framework for generalized global symmetries in 3d N=4 SCFTs, focusing on good quiver gauge theories and identifying how Coulomb-branch symmetries, 1-form symmetries, and discrete data emerge and mix via ’t Hooft anomalies. It provides a concrete, quiver-dependent method to extract the IR global form and anomalies from UV Lagrangian data by analyzing center charges of monopole operators, with a transparent prescription read off from unbalanced and flavor nodes. The authors perform extensive consistency checks by applying the method to magnetic quivers of 4d Class S theories and 5d SCFTs, and by comparing with independent flavor-symmetry results and 3d mirror duals. They also explore generalizations including partial N=2 gaugings and discrete gauging, revealing new facets such as 2-group symmetries and mixed anomalies in broader contexts. The work thus offers a practical toolkit for anomaly matching and symmetry classification in lower-dimensional QFTs with eight supercharges, with potential geometric and non-invertible extensions discussed as future directions.

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.

Figures (1)

• Figure 1: The surface geometry for the marginal theory of type $D_{10}-I_1$, which gives rise to the rank 2 5d SCFTs discussed in this section (the labels for the curves is chosen in accord with the derivation of the geometry in Apruzzi:2019opn). The two compact surfaces are denoted by $S_i$ and the collection of rational curves and their intersections are shown in the figure. The self-intersection of the curves in each surface is either shown next to the sections $u_i$, $U, V$$\delta_i$, or is $-2$ for green and $-1$ for blue curves.