SciPost Submission Page
Engineering of Anyons on M5-Probes via Flux Quantization
by Hisham Sati, Urs Schreiber
Submission summary
| Authors (as registered SciPost users): | Urs Schreiber |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2501.17927v1 (pdf) |
| Date submitted: | Feb. 28, 2025, 11:14 a.m. |
| Submitted by: | Schreiber, Urs |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities ("geometric engineering" of anyons), which we motivate from fundamental open problems in the field of quantum computing. The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane's tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field. This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and "twistorial") form of unstable Cohomotopy ("Hypothesis H"). As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, results from algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates.
Current status:
Reports on this Submission
Strengths
2 - Complete and detailed list of references
3 - Informative discussion on motivation and connection to other topics
Weaknesses
1 - Hard to access for non-experts 2- Too schematic and condensed at times
Report
In these lecture notes, the authors review recent progress on topological aspects of M-theory with applications to and motivations from quantum computing. At a fundamental level, they focus on the idea (referred to as Hypothesis H, after one of the authors) that flux quantization in M-theory should occur within a specific unstable cohomotopy theory. From this, several known results in M-theory and string theory follows but also new phenomena are predicted, such as fractional M2-branes.
For these lectures, the emphasis is on the insights that Hypothesis H provides into key features of anyonic topological order, particularly as observed in fractional quantum Hall systems, through the study of M5-brane probes of certain orbifold singularities in 11-dimensional supergravity.
Section one discusses the motivation behind this work: to offer a new, fundamental description of anyon theory via geometric engineering of M-branes probing orbifold singularities. Section two reviews flux quantization on M5-branes as prescribed by Hypothesis H, with special attention to the case of orbifolds. Section three examines how the charge of soliton scattering can be related to Wilson loops in Chern-Simons theory and discusses topological observables. Section four presents the identification of solitonic anyons and, subsequently, anyonic defects, via geometric engineering on flux-quantized M5-branes. Section five summarises the main points discussed and highlights connections to further topics.
While the manuscript is meticulously written, it demands considerable effort from non-expert readers. Most claims appear plausible to me, although I do not have the expertise to verify all of them. If the lecture notes are intended for a broader audience, I would strongly recommend to include more pedagogical explanations. Section four, in particular, is both central and among the most challenging to understand. I would suggest considering the possibility of splitting it into two parts to enhance clarity.
Minor points:
- Page 8: why is the gravitino 1-form set to zero in the last equation (and similarly for other fluxes in the following page)?
- Page 12: why is there a hat" difference between $H^4(X^8, \mathbb{Z})$ and $H^7(\hat{X}^8, \mathbb{Z})$?
- Should one think of $S^7$ and $S^4$ of Hypothesis H as auxiliary or physical?
- Page 30: I perhaps spotted a typo,mangentic".
Recommendation
Ask for minor revision
Author: Urs Schreiber on 2025-05-29 [id 5531]
(in reply to Report 1 on 2025-04-27)Thanks for the thoughtful Report #1 by Anonymous (Referee 1), we do sincerely appreciate the time and work you invested in reading and evaluating our manuscript.
While waiting for further reports, we have meanwhile made requested adjustments (such as the splitting of section 4, which is a very sensible suggestion, thanks) in our local pdf manuscript (here). Also, we have added pointer there to the recent arXiv:2505.22144 where the statements from section 4 (and now also 5) are discussed and the proofs spelled out in more detail than would fit the lecture notes.