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Discrete gauge theories of charge conjugation

Guillermo Arias-Tamargo, Antoine Bourget, Alessandro Pini, Diego Rodriguez-Gomez

Abstract

We define gauge theories whose gauge group includes charge conjugation as well as standard $\mathrm{SU}(N)$ transformations. When combined, these transformations form a novel type of group with a semidirect product structure. For $N$ even, we show that there are exactly two possible such groups which we dub $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$. We construct the transformation rules for the fundamental and adjoint representations, allowing us to explicitly build four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories based on $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ and understand from first principles their global symmetry. We compute the Haar measure on the groups, which allows us to quantitatively study the operator content in protected sectors by means of the superconformal index. In particular, we find that both types of $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ groups lead to non-freely generated Coulomb branches.

Discrete gauge theories of charge conjugation

Abstract

We define gauge theories whose gauge group includes charge conjugation as well as standard transformations. When combined, these transformations form a novel type of group with a semidirect product structure. For even, we show that there are exactly two possible such groups which we dub . We construct the transformation rules for the fundamental and adjoint representations, allowing us to explicitly build four-dimensional supersymmetric gauge theories based on and understand from first principles their global symmetry. We compute the Haar measure on the groups, which allows us to quantitatively study the operator content in protected sectors by means of the superconformal index. In particular, we find that both types of groups lead to non-freely generated Coulomb branches.

Paper Structure

This paper contains 38 sections, 76 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Construction of two disconnected groups
  3. Explicit matrix realization
  4. Extensions and semidirect products
  5. Real forms and Antiinvolutions
  6. A construction of automorphisms
  7. General theory
  8. The $\mathfrak{sl}(N,\mathbb{C})$ case
  9. The $\mathfrak{sl}(4,\mathbb{C})$ example
  10. The flip involution
  11. Representations, invariants and integration measure
  12. Representations
  13. The fundamental representation
  14. The adjoint representation
  15. Invariants
  16. ...and 23 more sections

Figures (2)

  • Figure 1: Schematic representation of the method of classification of split extensions of $\mathbb{Z}_2$ by $\mathrm{SU}(N)$, covered in sections \ref{['secExtensions']} and \ref{['secRealForms']}.
  • Figure 2: Summary of the duality relations between homogeneous space $X$ and $X_c$.