The fate of discrete 1-form symmetries in 6d

Fabio Apruzzi,Markus Dierigl,Ling Lin

doi:10.21468/scipostphys.12.2.047

Abstract

Recently introduced generalized global symmetries have been useful in order to understand non-perturbative aspects of quantum field theories in four and lower dimensions. In this paper we focus on 1-form symmetries of weakly coupled 6d supersymmetric gauge theories coupled to dynamical tensor multiplets. We study the consistency of global 1-form symmetries corresponding to the center of the gauge groups, or subgroups thereof, by activating their background fields, which makes the instanton density fractional. In 6d, an instanton background for a given gauge theory sources BPS strings via tadpole cancelation. The non-trivial 1-form symmetry background configurations contribute to the charge of the BPS strings. However, Dirac quantization imposes restrictions on the consistent 1-form backgrounds, since they can in general lead to and induce fractional charges, thus making (part of) the putative higher-form symmetry inconsistent. This gives explicit criteria to determine whether the discrete 1-form symmetries are realized. We implement these criteria in concrete examples originating from string compactifications. We also corroborate this by finding that a non-trivial fractional contribution is related to states which explicitly break the global 1-form symmetry appearing as massive excitations of the 6d BPS strings. For 6d theories consistently coupled to gravity, this hints at a symmetry breaking tower of states. When the fractional contributions are absent, the F-theory realization of the theories points to the gauging of the 1-form symmetry via the presence of non-trivial Mordell--Weil torsion.

Highlights

In this work we have studied discrete 1-form symmetries in 6d N = (1, 0) theories that act as a subgroup Z ⊂ Z(G) of the center of a non-Abelian gauge symmetry G = j G j
We have focused on the interplay between this discrete higher-form center symmetry and the U(1) 1-form symmetries of tensor fields Bi arising from the Green–Schwarz–West–Sagnotti coupling (8)
In the presence of a background field for the center 1-form symmetry of the gauge factor G j, specified by a Z(G j)-valued 2-cocycle C2j, the GSWS coupling leads to a term

Summary

Introduction

Global symmetries have always played a crucial role in the investigation of quantum field theories. The interpretation is that the violation of the Dirac quantization due to induced fractional charges on the BPS strings is a low-energy effect, which practically allows us to detect whether the theory contains symmetry breaking states becoming massless at some points in the moduli space Using this method our findings are perfectly consistent with the results obtained via circle reduction of 6d theories to 5d KK-theories [43] and recent geometric studies of discrete M-theory fluxes related to higher-form symmetries in 5d field theories [33, 34, 44]. Some more technical aspects can be found in the appendices

Discrete 1-form Symmetries in 6d

Activating Backgrounds for the Discrete 1-Form Symmetries

Circle Reduction Perspective

Explicit Examples

Conclusions and Outlook