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On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories
by Yuji Tachikawa, Hao Y. Zhang
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Submission summary
| Authors (as registered SciPost users): | Hao Zhang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2403.08861v2 (pdf) |
| Date submitted: | 2024-05-08 07:35 |
| Submitted by: | Zhang, Hao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a $\mathbb{Z}_3$-valued discrete topological term even after we turn off the $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$ gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. We also find that the $Spin(16)\times Spin(16)$ non-tachyonic non-supersymmetric heterotic string has the same non-zero $\mathbb{Z}_3$-valued discrete topological term. We will also explain the relation of our findings to the theory of topological modular forms. The paper is written as a string theory paper, except for an appendix translating the content in mathematical terms. We will explain there that our finding identifies a representative of the $\mathbb{Z}/3$-torsion element of $\pi_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.
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
Report #2 by Anonymous (Referee 1) on 2024-6-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2403.08861v2, delivered 2024-06-24, doi: 10.21468/SciPost.Report.9294
Report
The authors show that the 10d effective actions of $E_8\times E_8$ and $SO(32)$ heterotic string theories, with gauge fields turned off, have different $\mathbb{Z}_3$-valued topological terms. The difference can be detected by putting the theories on the spacetime which is the $Sp(2)$ group manifold with a non-trivial 3-form flux. To determine the difference, the authors relate 10d topological terms to the global anomalies of the NS5-brane worldvolume theories via Green-Schwarz mechanism. These global anomalies are in turn related to the perturbative anomalies (known in part from previous works) via an Elitzur-Nair-like relation.
The authors also comment on possible alternative approaches to determine the difference.
I believe that the results and the techniques used in the manuscript are quite interesting for theoretical physicists working on string theory, global anomalies, and related topics and possibly also for algebraic topologists. I find the manuscript well-written. I would like to recommend it for publication.
Requested changes
I have only very minor suggestions:
1) I think it would be better to add some brief explanation or references about the origin of the formulas (2.1)-(2.6).
2) The convention for ball/disk is not very consistent, e.g. on page 4 it is a disk $D^4$, while on page 15 it is a ball $B^4$.
3) In point 2(c) I think it would be better to mention that $t$ and $J_8$ will fixed later on. Otherwise, I found that part a bit confusing (until I reached the end of (3.2)).
4) After (5.1) the authors write "... set of modular forms of degree $d$." I think "weight" is a much more commonly used term in the context of usual modular forms (instead of "degree", which is more appropriate in the cohomology theory context) .
I've also noticed some misprints:
6) middle of page 5: I think "be" is missing in "... can only detected ..."
7) missing closing parenthesis in the second line after eq. (4.4)
8) sentence on page 25: "... solve s difficult ..."
Recommendation
Ask for minor revision