Monopole Breaking of Chern-Weil Symmetries

TL;DR

This paper investigates how magnetic monopoles and dyonic excitations in gauge theories with nontrivial fundamental groups can explicitly break or restore Chern-Weil symmetries, especially in nonabelian groups and Higgsed phases.

Contribution

It generalizes the understanding of monopole-induced symmetry breaking from U(1) to nonabelian gauge groups and explores the effects of Higgsing on instantonic symmetries.

Findings

Magnetic monopoles violate instantonic symmetries in nonabelian gauge theories.

Dyon modes can restore broken instantonic symmetries.

Higgsing can explicitly break instantonic symmetries from subgroup to larger gauge groups.

Abstract

Gauge theories in $d$ dimensions with a nontrivial fundamental group admit a $(d-3)$-form magnetic symmetry and a $(d-5)$-form instantonic symmetry. These are examples of Chern-Weil symmetries, with conserved currents built out of the gauge field strength, which can only be explicitly broken through violations of the Bianchi identity. For U(1) gauge theory, it is clear that magnetic monopoles violate not only the $(d-3)$-form magnetic symmetry but also lower-form symmetries like the instantonic symmetry. It is also known that an improved instanton number symmetry current, which is conserved, can be constructed in the case that the magnetic monopole admits a dyonic excitation. We study the generalization to other gauge groups, showing that magnetic monopoles also violate instantonic symmetries for nonabelian groups like PSU($n$), and that dyon modes can restore such symmetries. Furthermore, we show that in many (but not all) examples where a gauge group $G$ is Higgsed to a gauge group $H$, the structure of monopoles and dyons emerging from the Higgsing process explicitly breaks the instantonic symmetries of $H$ to those of $G$. The meaning of explicit breaking of a $(d-5)$-form symmetry is clearest for $d > 4$, but these results also extend to $d = 4$, where the breaking is interpreted as an obstruction to coupling the theory to a background axion field.