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One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices for any $G$

by Cyril Closset, Elias Furrer, Osama Khlaif

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Submission summary

Authors (as registered SciPost users): Cyril Closset · Elias Furrer · Osama Khlaif
Submission information
Preprint Link: https://arxiv.org/abs/2405.18141v2  (pdf)
Date submitted: June 12, 2024, 2:49 p.m.
Submitted by: Elias Furrer
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter gauge theories with a one-form symmetry in the $A$-model formalism on $\Sigma_g\times S^1$. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface $\Sigma_g$ for any real compact gauge group $G$. All computations are carried out in the effective $A$-model on $\Sigma_g$, whose ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and how its 't Hooft anomaly constrains the vacuum structure. In the special case of the $SU(N)_K$ $\mathcal{N}=2$ Chern-Simons theory, we obtain results for the $(SU(N)/\mathbb{Z}_r)^{\theta}_K$ $\mathcal{N}=2$ Chern-Simons theories, for all non-anomalous $\mathbb{Z}_r \subseteq \mathbb{Z}_N$ subgroups of the center of the gauge group, and with the associated $\mathbb{Z}_r$ $\theta$-angle turned on, reproducing and extending various results in the literature. In particular, we find an interesting mixed 't Hooft anomaly between gravity and the $\mathbb{Z}_r$ one-form symmetry of the $SU(N)_K$ theory (for $N$ even, $\frac{N}{r}$ odd and $\frac{K}{r}$ even). This plays a key role in our derivation of the Witten index, which we explicitly compute for any $N$, $K$ and $r$ in terms of refinements of Jordan's totient function. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted

Reports on this Submission

Report #3 by Anonymous (Referee 3) on 2024-8-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2405.18141v2, delivered 2024-08-28, doi: 10.21468/SciPost.Report.9666

Strengths

1.) The paper discusses 3d QFTs with ${\cal N}=2$ SUSY and a $U(1)_R$ symmetry. This class of theories can be usefully analyzed by compactifying on a circle (retaining all the KK modes) and studying the resulting 2d (2,2) theory, more precisely its topological A-twist, which provides a natural setting to study many BPS observables of the 3d theory. The main goal of this paper is to generalize previous results (including by Closset and collaborators) on this formalism to 3d gauge theories with more general gauge groups, e.g. simple but non-simply-connected ones, which can be obtained from simply connected ones by gauging a discrete 1-form global symmetry in 3d. The paper is thus situated at the interface between SUSY gauge theories and generalized global symmetries.

2.) The main focus is on pure SUSY Chern-Simons theories, though theories with matter are briefly discussed as well. Much of the paper is dedicated to working out some non-trivial examples in great detail, tracking their symmetries and anomalies to the 2d description, and gauging anomaly-free symmetries to obtain new theories.

3.) A nice application of this formalism is the calculation of observables such as the Witten index (which has fascinating number-theoretic properties) and $S^3$ partition functions.

Weaknesses

As already pointed out by the first referee, the paper highlights a certain mixed anomaly between a 1-form symmetry and spacetime, with 4d anomaly inflow action $\sim \int w_2 \cup B_2$, where $B_2$ is the 1-form symmetry background field, and $w_2$ is the 2nd Stiefel-Whittney class of the spacetime four-manifold; but this anomaly is trivial on spin manifolds, which is the setting for generic SUSY theories. This needs to be addressed and clarified by the authors. It is possible that something slightly stronger can be said in pure SUSY Chern-Simons gauge theory (with or without a Yang-Mills term), by noting that the only fermions are the gauginos, with unit $R$-charge, so that the $U(1)_R$ background gauge field is a Spin$_c$ connection, in terms of which a version of the anomaly may survive.

Report

Modulo the minor issues mentioned above, which should be revised by the authors, I recommend the paper for publication.

Recommendation

Ask for minor revision

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report #2 by Anonymous (Referee 2) on 2024-8-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2405.18141v2, delivered 2024-08-26, doi: 10.21468/SciPost.Report.9654

Strengths

This paper provides an analysis of line operators in three-dimensional gauge theories with one-form symmetries, following up especially C. Closset's extensive work on three-dimensional gauge theories. It is (1) reasonably general and thorough, and certainly (2) this is a timely topic.

Weaknesses

Aside from some minor issues mentioned by the other referee, it looks fine to me, at least on a short reading.

Report

I think the journal's acceptance criteria are met. Modulo minor issues mentioned by the other referee, I recommend it for publication.

Recommendation

Publish (easily meets expectations and criteria for this Journal; among top 50%)

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

Report #1 by Anonymous (Referee 1) on 2024-8-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2405.18141v2, delivered 2024-08-23, doi: 10.21468/SciPost.Report.9639

Report

The manuscript investigates one-form symmetry in 3d N=2 supersymmetric Chern-Simons theory on a circle with topological twist in 2d. The manuscript studies the action of one-form symmetry on the Hilbert space, and computes the 't Hooft anomaly of the one-form symmetry and use it to constrain the vacuum structure.

Before I can recommend the manuscript for publication, here are some minor comments

  • In the discussion of the mixed anomaly between gravity and one-form symmetry, this comes from the generator of the one-form symmetry being an emergent fermion. However, such anomaly is only nontrivial in a bosonic theory. Since the theory is supersymmetric, there is also physical fermion, so the theory is fermionic, and the anomaly becomes trivial (e.g. one can tensor the generator with physical fermion to obtain a boson). Equivalently, the second Stiefel-Whitney class is trivial on spin 4-manifolds.

  • there is no theta angle in 3d, it is the holonomy of a 0-form symmetry along S1, maybe the author can clarify it?

Recommendation

Ask for minor revision

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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