Anomalies of Generalized Symmetries from Solitonic Defects

Abstract

We propose the general idea that 't Hooft anomalies of generalized global symmetries can be understood in terms of the properties of solitonic defects, which generically are non-topological defects. The defining property of such defects is that they act as sources for background fields of generalized symmetries. 't Hooft anomalies arise when solitonic defects are charged under these generalized symmetries. We illustrate this idea for several kinds of anomalies in various spacetime dimensions. A systematic exploration is performed in 3d for 0-form, 1-form, and 2-group symmetries, whose 't Hooft anomalies are related to two special types of solitonic defects, namely vortex line defects and monopole operators. This analysis is supplemented with detailed computations of such anomalies in a large class of 3d gauge theories. Central to this computation is the determination of the gauge and 0-form charges of a variety of monopole operators: these involve standard gauge monopole operators, but also fractional gauge monopole operators, as well as monopole operators for 0-form symmetries. The charges of these monopole operators mainly receive contributions from Chern-Simons terms and fermions in the matter content. Along the way, we interpret the vanishing of the global gauge and ABJ anomalies, which are anomalies not captured by local anomaly polynomials, as the requirement that gauge monopole operators and mixed monopole operators for 0-form and gauge symmetries have non-fractional integer charges.

Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices

“…Following [89], the above result implies that the mixed gauge/zero-form monopole operator such that the flux for the U(1) gauge group is 1 2q and that for the Cartan subalgebra of SO(3) f is 1/2 carries charge 1 2q under the U(1) w symmetry and charge 0 (mod 2) under the Cartan subalgebra of su(2) f . This implies the mixed anomaly between the U(1) w symmetry and the SO(3) f symmetry for all q characterised by the anomaly theory 4…”

“…This theory has a two-group symmetry between the Z 2 one-form symmetry, arising from gauging the magnetic symmetry, and the flavour symmetry [89,Section 7.4]. One can indeed see this as follows.…”

“…where w f,w 2 denotes the second Stiefel-Whitney class that is an obstruction to lifting the SO(3) f,w bundles to the SU(2) f,w bundles, and the integration is taken over a spin four-manifold whose boundary is the three-manifold in which the 3d theory in question lives. In particular, as proposed in [89,Section 7.2] this mixed anomaly can be detected by examining the mixed gauge/zero-form monopole operators with fractional magnetic flux for both the gauge group and the Cartan subalgebra of the flavour symmetry group. Our interpretation in terms of the index is to investigate such a mixed anomaly by setting the magnetic flux of the U(1) gauge group to be ±1/2 and that of the the Cartan subalgebra of SO(3) f to be 1/2.…”

A Panorama Of Physical Mathematics c. 2022

“…Since then there has been widespread use of such generalized symmetries. Combined with the theory of anomalies striking applications have been made to determine dynamics of nontrivial theories [168][169][170][171][172][173][174][175]. Motivated, in part, by these developments there has been some exploration into the notion of "two-group symmetries" [176,177] and these works clearly point the way to potentially interesting future developments.…”

“…In the past few years they have been combined with the new ideas based on invertible topological field theories and generalized symmetries to discover new constraints on nontrivial RG flows in quantum field theories, see e.g. [167][168][169][170][171][172][173][174][175].…”