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Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality

Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal

Abstract

We study a 3d lattice gauge theory with gauge group $\mathrm{U}(1)^{N-1}\rtimes \mathrm{S}_N$, which is obtained by gauging the $\mathrm{S}_N$ global symmetry of a pure $\mathrm{U}(1)^{N-1}$ gauge theory, and we call it the semi-Abelian gauge theory. We compute mass gaps and string tensions for both theories using the monopole-gas description. We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire $\mathrm{SU}(N)$ root system. Even though the center symmetry of the semi-Abelian gauge theory is given by $\mathbb{Z}_N$, we observe that the string tensions do not obey the $N$-ality rule and carry more detailed information on the representations of the gauge group. We find that this refinement is due to the presence of non-invertible topological lines as a remnant of $\mathrm{U}(1)^{N-1}$ one-form symmetry in the original Abelian lattice theory. Upon adding charged particles corresponding to $W$-bosons, such non-invertible symmetries are explicitly broken so that the $N$-ality rule should emerge in the deep infrared regime.

Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality

Abstract

We study a 3d lattice gauge theory with gauge group , which is obtained by gauging the global symmetry of a pure gauge theory, and we call it the semi-Abelian gauge theory. We compute mass gaps and string tensions for both theories using the monopole-gas description. We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire root system. Even though the center symmetry of the semi-Abelian gauge theory is given by , we observe that the string tensions do not obey the -ality rule and carry more detailed information on the representations of the gauge group. We find that this refinement is due to the presence of non-invertible topological lines as a remnant of one-form symmetry in the original Abelian lattice theory. Upon adding charged particles corresponding to -bosons, such non-invertible symmetries are explicitly broken so that the -ality rule should emerge in the deep infrared regime.

Paper Structure

This paper contains 23 sections, 124 equations, 2 figures.

Table of Contents

  1. Introduction
  2. 3d $\mathrm{U}(1)^{N-1}$ lattice gauge theory with $\mathrm{S}_N$ global symmetry
  3. Description of the $\mathrm{U}(1)^{N-1}$ lattice gauge theory
  4. Villain formulation
  5. Global symmetries:
  6. Wilson formulation
  7. Mass gap and spectrum
  8. Multi-component Coulomb gas representation of the Villain form
  9. Long-distance effective theory
  10. Remarks:
  11. Wilson loops and string tensions
  12. Explicit evaluation of string tensions
  13. Semi-Abelian theory
  14. Gauging of the $\mathrm{S}_N$ global symmetry
  15. $\mathbb{Z}_N$ center symmetry
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Left: The dual of the plaquette $p$ is the link $*p$ on the dual lattice intersecting $p$ as shown. Right: $D$ is the shaded region bounded by the curve $C$. The Poincare dual $[D]$ is a bump 1-form on the dual lattice that is 1 on each of the red links shown, and 0 everywhere else.
  • Figure 2: Coboundary operator on a link in a 3d lattice: $\delta \ell = p_1 + p_2 + p_3 + p_4$