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Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
by Guillermo Arias-Tamargo, Chris Hull, Maxwell L. Velásquez Cotini Hutt
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Submission summary
| Authors (as registered SciPost users): | Guillermo Arias-Tamargo · Maxwell L. Velásquez Cotini Hutt |
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| Preprint Link: | https://arxiv.org/abs/2503.20865v2 (pdf) |
| Date submitted: | May 16, 2025, 2:42 p.m. |
| Submitted by: | Guillermo Arias-Tamargo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this work we discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. This requires that the target space has an isometry with compact orbits that acts without fixed points. Our approach allows us to include target spaces without non-trivial 1-cycles, does not require the NLSM to be conformal, and when it is conformal it does not need to be rational; moreover, it highlights the microscopic origin of the topological terms that are responsible for the non-invertibility of the defect. An interesting class of examples are Wess-Zumino-Witten models, which are self-dual under a discrete gauging of a subgroup of the isometry symmetry and so host a topological defect line with Tambara-Yamagami fusion. Along the way, we discuss how the usual 0-form symmetries match across T-dual models in target spaces without 1-cycles, and how global obstructions can prevent locally conserved currents from giving rise to topological operators.
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- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
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Reports on this Submission
Report
The authors show that, given specific choices of the radius and WZ level, nonlinear sigma models with a global U(1) isometry can host interesting duality-type symmetries, which in general are not Verlinde lines.
The paper is very well written and the results clear and crisp. The authors have already published interesting follow-ups and I expect several other applications to appear in the future.
I thus happily recommend this paper for publication in SciPost.
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Report #1 by Anonymous (Referee 1) on 2025-7-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.20865v2, delivered 2025-07-17, doi: 10.21468/SciPost.Report.11589
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Weaknesses
Report
This paper provides a clear an interesting extension of the construction of T-duality defects, extensively discussed in the case of the 2d compact boson, to the case of more generic non linear sigma models. The construction still relies on the method of half-space gauging which sends the theory to its dual version upon a topological manipulation. The paper is clear and nicely written; I therefore recommend it for publication, though I would like the authors to answer/comment to the following: 1. At the end of pag. 4, the authors comment on the fact that, in order to perform the half space gauging, the T-duality defect must be placed on homologically trivial cycles. Even if I agree that one cannot perform a gauging on half space when the cycle is non-trivial, the statement, as written, is a little bit confusing since the T-duality defect can be placed on any cycle of the world sheet (contractible or not). See e.g. reference [36] for explicit constructions of partition function of the compact boson on a torus with the duality defect placed on non-trivial cycles. 2. Regarding the last comment in Section 2.2 on page 9, I would like to point out that, by coupling to the flat gauge field, one can obtain either of the Lagrangians with radius R/p or p/(2\pi R) depending on which gauge field is integrated out first. 3. Should the last term in eq. 4.20 be \Phi^_+ instead of \Phi^_- ? 4. In the second paragraph on page 59, it is claimed that on a worldsheet without non-trivial 1-cycles, the T-duality defect is trivial, it reduces to T-duality and is invertible. I do not agree with this statement. Even though in this case the flat gauge field is trivial in empty space, one can still consider inserting this defect together with other local operators in the theory. In such cases, one will detect the non-invertibility of the defect, as it will project out operators charged under the symmetry being gauged. 5. At the beginning of page 52, the authors attempt to interpret the condition for the existence of the duality defect in non-conformal theories as a symmetry enhancement at some energy scale. I do not fully understand this argument. All derivations in the paper are carried out using the Lagrangian description of the sigma model. Since these theories are asymptotically free in two dimensions, I would expect that, if the parameters of the Lagrangian are fixed as discussed in Section 4.5, the theory exhibits the non-invertible symmetry already in the UV, making it an exact symmetry of the model. As a side comment, I believe one could verify this by computing the one-loop beta function for the coupling and showing that quantum corrections do not spoil the duality conditions. 6. One curiosity I would like to ask is: do 2d SCFTs related by mirror symmetry fit into this framework? Specifically, can 2d mirror symmetry be used to construct duality defects in the same spirit as those discussed in this paper?
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