Coulomb branches with complex singularities

TL;DR

This paper constructs 4d superconformal field theories with Coulomb branches exhibiting complex singularities, challenging the assumption of freely generated coordinate rings and providing new insights into moduli space geometry.

Contribution

It introduces a method to create SCFTs with singular Coulomb branches by gauging discrete symmetries of $ ext{N}=4$ super Yang-Mills theories, revealing novel moduli space structures.

Findings

Coulomb branch coordinate rings are not freely generated.

Different SCFTs can share identical moduli space geometries.

Constructed theories have Lagrangian descriptions with disconnected gauge groups.

Abstract

We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of $\mathcal N=4$ superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as $\mathcal N=4$ sYM theories with disconnected gauge groups.