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MTC[M3,G]: 3d Topological Order Labeled by Seifert Manifolds
by Federico Bonetti, Sakura Schäfer-Nameki, Jingxiang Wu
Submission summary
| Authors (as registered SciPost users): | Federico Bonetti |
| Submission information | |
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| Preprint Link: | scipost_202511_00072v1 (pdf) |
| Date submitted: | Nov. 27, 2025, 12:20 p.m. |
| Submitted by: | Federico Bonetti |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of $G$ a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of $G$. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank $r\leq 5$ in terms of Seifert manifolds and a choice of Lie group $G$.
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:
Reports on this Submission
Strengths
2-They describe a combinatorial process for constructing ribbon fusion categories from the data associated with a Seifert surface, namely the sequence of pairs (p_i,q_i).
3-They illustrate that their methods recover the modular data for all cases up to rank 5, and the majority of rank 6, identifying the data with an existing enumeration.
4-There are some mathematical conjectures that the community might appreciate.
Weaknesses
2-Since all rank at most 5 categories can be realized by means of type A constructions, it is not surprising that this recovers all of those. Indeed, the missing cases in rank 6 are constructed from Spin-odd type B. At least the graded Deligne construction should still work there--perhaps only the characters of Seifert manifolds are missing.
Report
Requested changes
1-The tables have both Spin(even) and E7, but the construction is not explicitly described.
2-footnote 1: the first *known* such ambiguity...
3-the graded Deligne product has appeared in papers of Nikshych and Green (and perhaps before) under the name "fiber product." It could be good to add a reference.
4-page 6: it is not stated what the parameter b is. It is assumed to be 0, so it is not incredibly important, but should be defined.
5-page 9 top line: tt->it.
6-Only one example is provided. It would be nice to see an example where the final category has no (faithful) grading, such as a (2,1),(3,1),(5,-) case. I believe the point is that (2,1) makes it so that everything sits in the trivial component. This was confusing to me until I worked it out myself.
7-Quite a few of the references are only to the arxiv, but must be published by now. Making the references consistent would be good.
Recommendation
Ask for minor revision