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Anomalies of Generalized Symmetries from Solitonic Defects

by Lakshya Bhardwaj, Mathew Bullimore, Andrea E. V. Ferrari, Sakura Schafer-Nameki

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Submission summary

Authors (as registered SciPost users): Lakshya Bhardwaj · Andrea Ferrari
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-03-04
Date submitted: 2024-01-29 13:15
Submitted by: Ferrari, Andrea
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Author comments upon resubmission

We thank very much the referees for the helpful comments. We reply to these comments below:

Report 2: To the points on page 4 and 5: our formalism consistently implements the fact that endable defects are uncharged under certain global symmetries. It is true that there are relations between characteristic classes, but this redundancy does not affect our analysis. To the point on dynamical gauge field: we discuss the fate of solitonic defects and anomalies after gauging global symmetries in section 4

Report 1: 1. The definition is given at the beginning of the paper (introduction): solitonic defects are by definition defects that induce background fields for global symmetries. When a symmetry is gauged, by definition a solitonic defect that was only sourcing that symmetry may stop being a solitonic defect (if it does not happen to source background fields for another global symmetry obtained after gauging) 2. We corrected M_{d-1} to M_{d+1}, and changed the general class c_{d-1} to q_{d-1} not to confuse it with a Chern class. 3. The fact that representations of central extensions of a group are projective representations of the group has been well-known since the birth of quantum mechanics. The paper of Brennan et al. came after ours, and we not feel we need to emphasise this point further. 4. The referee may refer to dynamical vortex defects with some vacuum at infinity that breaks the gauge symmetry. The unit winding is in some sense a choice: given any vortex defect there is a co-character $\phi$ whose respective embedded U(1) has a unit vortex, and this is the one we choose to characterise the vortex by. We rephrased a little the discussion above 3.2 to make this clear 5. The notation \cG and \cF is used to refer to physical gauge and flavor groups, while G and F are reserved for mathematically useful groups related to these physical groups. We have tried to replace \cZ(\cG) with \cZ_\cG whenever confusing 6. This is an arbitrary example, the denominator is that one by definition. The discussion we provide is consistent (the order of the generator is correct). 7. This is a consequence of our definitions. We provide an example on that page. 8. We removed all double words (regular expression) 9. No comment

List of changes

Minor changes have been made according to the reply to the above reply to the referees. These include notational changes and correction of typos as per comment 2, Referee 2, and notational change as per comment 5, referee 2.

Published as SciPost Phys. 16, 087 (2024)

Reports on this Submission

Anonymous Report 1 on 2024-2-6 (Invited Report)


Dear Editor,

The authors addressed most of my comments and the draft looks improved.
As I mentioned in my previous report, the content of this work is solid and interesting and I recommend its publication.

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