# The Fate of Discrete 1-Form Symmetries in 6d

### Submission summary

 As Contributors: Ling Lin Arxiv Link: https://arxiv.org/abs/2008.09117v2 (pdf) Date submitted: 2020-09-21 18:09 Submitted by: Lin, Ling Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### 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 tensor multiplets. We study their interplay with large gauge transformations for dynamical tensor fields. In a non-trivial background for the global 1-form symmetry, this leads to an ambiguity of the effective field theory partition function. This anomaly is eliminated by the inclusion of BPS strings. However, the non-trivial 1-form background can induce fractional string charges which are not compatible with Dirac quantization, and hence the symmetry is absent. The anomalous term therefore serves as a tool to detect whether the discrete 1-form symmetries are realized, which we demonstrate in explicit examples originating from string compactifications. We also corroborate this by finding that a non-trivial ambiguity is related to states which explicitly break the global 1-form symmetry, which appear as generally massive excitations of the 6d BPS strings. For 6d theories consistently coupled to gravity, this ambiguity of the partition function hints at the presence of a symmetry breaking tower of states. When the ambiguity is absent, the F-theory realization of the theories points to the gauging of the 1-form symmetries via the presence of non-trivial Mordell--Weil torsion.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2008.09117v2 on 21 September 2020

## Reports on this Submission

### Report

This is an interesting paper in the context of higher-dimensional supersymmetric theories, which is currently a very active area.

This work presents some steps of an analysis of the higher form symmetries of six-dimensional theories from the perspective of the low energy effective action, and it points out an intriguing effect: there is a potential mixed anomaly between discrete global 1-form symmetry groups and the higher gauge transformation for anti-self dual tensors.

The referee recommends accepting this work for publication, provided the questions presented in the section "requested changes" below are addressed satisfactorily.

### Requested changes

These are some questions/suggestions for the authors.

1.)
In 6d ordinary gauge anomaly cancellation strongly constrains the matter content of the theory.

The matter introduced to cancel the anomaly is typically charged with respect to the centre symmetry thus breaking it completely or to certain subgroups. For this reason only few examples of six-dimensional theories end up having interesting discrete 1-form centre symmetries.

As an example of this fact for 6d (1,0) theories the only gauge groups that can appear without matter are SU(3), SO(8), F4, E6, E7 and E8. These gauge groups are discussed in section 3 and are free of the dangerous fractionalization.

Similarly, this happens for the other non-higgsable models analyzed in section 3.

From the examples presented it seems that the constraints from 6d gauge anomaly cancellation are such that the effect observed around equations (3.9) and (3.10) is generic in the context of models with conventional matter (e.g. bi-fundamentals of various kinds): is there evidence in favor or against this idea? For instance, can the authors provide a more detailed analysis of the example in equation (3.18)?

2.)
In the same spirit of the above question: is there an alternative argument to argue that gauging the 0-form global symmetries of various types of conformal matter is not breaking the centre symmetry of the corresponding 0-form gauge groups? Same question also for gauging various subgroups of the E8 global symmetries of the E-string . One possible consistency check is given by the circle reduction of the corresponding models.

3.)
The theories considered in Section 5 are gravitational, hence should not have any global symmetry.

Gauging a 1-form centre symmetry does not necessarily produce a theory that does not have a higher symmetry, rather it typically give rise to a parent magnetic higher form symmetry.

For instance gauging the centre symmetry of a 4d SU(N) gauge theory one obtains PSU(N) which has a magnetic $\mathbb{Z}_N$ 1-form symmetry.

In 6d by a similar token gauging an electric 1-form symmetry might give rise to a theory with a magnetic 3-form symmetry.

If the theories discussed in section 5 are obtained by gauging 1-form symmetries, is there a way to argue that there are no other emergent magnetic 3-form higher symmetries from the gauging as required by a consistent gravitational model?

4.)
What is the matter content of the theories discussed in equation (5.17) of section 5.2? The intersection theory in equations (5.19) and (5.21) is reminiscent of the computations about the coefficients for the ordinary anomaly polynomial: can this remark be used to answer the referee's question 1.)?

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