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One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
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 | |
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| Preprint Link: | https://arxiv.org/abs/2405.18141v3 (pdf) |
| Date accepted: | 2025-01-28 |
| Date submitted: | 2024-12-11 10:39 |
| Submitted by: | Furrer, Elias |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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 $G$ as long as the ground states are all bosonic. All computations are carried out in the effective $A$-model on $\Sigma_g$, whose $S^1$ ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and also 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 centre of the gauge group, and with a $\mathbb{Z}_r$ $\theta$-angle turned on. In the special cases with $N$ even, $\frac{N}{r}$ odd and $\frac{K}{r}$ even, we find a mixed 't Hooft anomaly between gravity and the $\mathbb{Z}_r^{(1)}$ one-form symmetry of the $SU(N)_K$ theory, and the infrared 3d TQFT after gauging is spin. In all cases, we count the Bethe states and the higher-genus states in terms of refinements of Jordan's totient function. This counting gives us the twisted indices if and only if the infrared 3d TQFT is bosonic. 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.
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Author comments upon resubmission
In the revised version, we have corrected an important point: When the infrared theory after the gauging of the 3d one-form symmetry is a spin TQFT, it becomes necessary to choose a spin structure on the base Riemann surface, and the index will depend on that choice. In such cases, the Bethe vacua of the gauged theory can become fermionic, which affects the supersymmetric index counting. Although the naive index still accurately counts the Bethe vacua or lines, respectively, it does not represent a supersymmetric index in the spin case. We have corrected the relevant statements in this version, and refer the study of fermionic vacua and the spin structure dependence to a future work.
List of changes
We have modified the discussion of the above correction throughout the paper, most importantly in sections 3.2 and 3.4.
Moreover, we have adjusted the title, abstract and introduction accordingly, and included several important references.
Published as SciPost Phys. 18, 066 (2025)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2025-1-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2405.18141v3, delivered 2025-01-08, doi: 10.21468/SciPost.Report.10460
Report
The authors have improved the manuscript. However, it will be helpful if the author can clarify that why the theory can be defined on a non-spin manifold in order for the gravity & 1-form mixed anomaly to be nontrivial.
(See e.g. https://arxiv.org/abs/1806.09592 https://arxiv.org/abs/1602.04251
https://arxiv.org/abs/1812.04716)
For example, for this to happen, the local operators need to be bosons, or the fermionic local operators need to carry charge. This can come from the A twist in the 1+1d theory.
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Author: Cyril Closset on 2025-01-09 [id 5098]
(in reply to Report 1 on 2025-01-08)Dear referee,
Thank you for the insightful comment on this and on the previous version of the paper.
We do agree that this anomaly is always trivial in a supersymmetric field theory, and our modified discussion in this section states this. See the last paragraph of section 3.2.
The root of the T^3 modular anomaly is the fact that the abelian anyon we want to condense might be fermionic (h=1/2) and in this case the expectation value of the topological line depends on the choice of spin structure. This is something we didn't fully appreciate when writing the paper, but that we are addressing in mode detail in a series of papers that will appear over the coming weeks.
We added comments to that effect in the introduction and throughout the paper. Correspondingly, we removed the statement that we computed the index "for all G" from the title and abstract, as our results in this paper only give a well-defined Witten index when all the Bethe vacua are bosonic. (This is the case for essentially all the literature on the Bethe-vacua approach to 3d partition functions in the last 10 years because people focussed on simply-connected gauge groups. We hope to give a full picture of the completely general case in that work to appear.)
To answer your specific query above: The 3d theory is defined on a (closed) 3-manifold M3, which is always spin. The partition function depends on the choice of spin structure, however, and the anomaly captures this possible dependence. The 4d anomaly functional can be non-trivial when we consider two distinct spin structures on M3, call it M3_\pm, and extend each to an 4-manifold M4_\pm with boundary M3_\pm, each with their distinct spin structure. Gluing the M4_+ to M4_- gives us a closed 4-manifold which is not spin, in general. The anomaly (a sign) arises when we change spin structures on M3 in this way. As we agreed, we can always trivialise the anomaly by considering the new abelian anyon obtained by fusing with a transparent fermionic line, something we mention (citing the literature, especially https://arxiv.org/abs/1812.04716) but will only explore in detail in a work to appear.
Does this clarify this issue, or did you have something else in mind? Thank you very much for your help in improving our manuscript.