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Higher central charges and topological boundaries in 2+1-dimensional TQFTs
by Justin Kaidi, Zohar Komargodski, Kantaro Ohmori, Sahand Seifnashri, Shu-Heng Shao
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Sahand Seifnashri |
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| Preprint Link: | https://arxiv.org/abs/2107.13091v2 (pdf) |
| Date accepted: | 2022-08-16 |
| Date submitted: | 2022-04-05 02:36 |
| Submitted by: | Seifnashri, Sahand |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge $c_-$ has to vanish. In this paper, we consider conditions associated with "higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.
Published as SciPost Phys. 13, 067 (2022)
Reports on this Submission
Anonymous Report 2 on 2022-8-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.13091v2, delivered 2022-08-07, doi: 10.21468/SciPost.Report.5507
Weaknesses
No significant weakness was found.
Report
This work discusses the obstructions to the existence of topological gapped boundary conditions in 2+1d TQFTs. In particular, the vanishing of chiral central charge $c_-$ is not sufficient for its existence. For Abelian TQFTs, sufficient and necessary conditions for the existence of gapped boundary conditions were found, in terms of the vanishing of certain higher central charges. For non-Abelian TQFTs, necessary (but not sufficient) conditions were found as well.
Although equivalent conditions (i.e. Lagrangian subgroups/sub-algebras) have been discussed in the literature, the higher central charges in this work require much less computation and hence is more practically accessible. Moreover, the derivation in this work also sheds new insights on the generalized global symmetries and their gauging.
This work is original, and introduces a new concept -- higher central charge, which proves to be useful. The paper is self-contained and extremely well-written. I strongly recommend its publication.
Anonymous Report 1 on 2022-8-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.13091v2, delivered 2022-08-05, doi: 10.21468/SciPost.Report.5502
Strengths
1-Nice combination of physics and mathematics. The paper uses a clear physical approach to find criteria for the existence of gapped boundaries for certain TFTs in 2+1 dimensions that are equivalent to and sometimes simpler than those appearing in the more mathematical literature. More generally, the results give nice insight into this problem.
2-The paper is systematically and coherently written with occasional examples that nicely illustrate what is going on.
Weaknesses
Not really any significant weaknesses.
Report
While my report is very late and I think the paper has already been accepted, I certainly agree it should be published in this journal. It is an original take with computationally and physically interesting results on the well known problem of understanding which TFTs in 2+1 dimensions admit gapped boundaries.