Gauge theories from principally extended disconnected gauge groups

TL;DR

This work introduces principal extensions $\widetilde{G}=G\rtimes\Gamma$ as a principled way to gauge charge conjugation by enlarging the gauge group to a disconnected Lie group. Focusing on $\widetilde{\mathrm{SU}}(N)$, it develops the necessary group-theoretic machinery—representations, invariants, and Weyl integration formulas—and applies standard 4d $\mathcal{N}=2$ gauge-theory machinery to build SQCD-like theories with vector multiplets in the adjoint and real bifundamental-like hypermultiplets. A key result is that the Coulomb branch of $\widetilde{\mathrm{SU}}(N)$ is typically not freely generated (e.g., for $N\ge5$), as revealed by the Coulomb branch index and invariant theory, while the Higgs branch Hilbert series shows a modified global symmetry: $\mathrm{SO}(N_f)$ for even $N$ and $\mathrm{Sp}(\lfloor N_f/2\rfloor)$ for odd $N$. These findings highlight new IR structures, motivate extensions to quiver constructions and large-$N$ analyses, and suggest interesting avenues for string embeddings and class-$\mathcal{S}$ connections.

Abstract

We introduce gauge theories based on a class of disconnected gauge groups, called principal extensions. Although in this work we focus on 4d theories with N=2 SUSY, such construction is independent of spacetime dimensions and supersymmetry. These groups implement in a consistent way the discrete gauging of charge conjugation, for arbitrary rank. Focusing on the principal extension of SU(N), we explain how many of the exact methods for theories with 8 supercharges can be put into practice in that context. We then explore the physical consequences of having a disconnected gauge group: we find that the Coulomb branch is generically non-freely generated, and the global symmetry of the Higgs branch is modified in a non-trivial way.

Figures (2)

• Figure 1: Summary of the global symmetries of QCD with gauge group $\mathrm{SU}(N)$ and $\widetilde{\mathrm{SU}}(N)$ and $N_f$ flavors.
• Figure 2: Dynkin diagram automorphisms.