SciPost Submission Page
On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories
by Yuji Tachikawa, Hao Y. Zhang
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Hao Zhang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202407_00020v1 (pdf) |
| Date accepted: | 2024-08-21 |
| Date submitted: | 2024-07-12 06:24 |
| Submitted by: | Zhang, Hao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a $\bZ_3$-valued discrete topological term even after we turn off the $E_8\times E_8$ and $Spin(32)/\bZ_2$ gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. This result will be used to show further that the $Spin(16)\times Spin(16)$ non-tachyonic non-supersymmetric heterotic string has the same non-zero $\bZ_3$-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 $\bZ/3$-torsion element of $\pi_{-32}\TMF$ 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
Author comments upon resubmission
List of changes
Anonymous Report 2:
Minor suggestions:
1) We added a recent reference where these standard formulas were (re)derived carefully.
2) We improved the manuscript so that we only use $D^4$.
3) We expanded point 2 (c) to answer the referee's concern.
4) We switched to the terminology "weight".
We also corrected all the misprints kindly mentioned.
Anonymous Report 1:
As for the major comment about our treatment of $SO(16)\times SO(16)$, we added clarifying comments in the abstract and the introduction.
As for the major comment about how sure we (the authors) are of the crucial signs etc. in (2.20) to (2.22), we would like to emphasize that the anomaly polynomials $I_8$ of the Type I D5-brane and of the E-string theory have been known and used for decades, without leading to any problems. The $Y_8$ has been known even longer, since this gives the 10d Green-Schwarz term $B\wedge Y_8$, and again has been used extensively by many string theorists, without running into contradiction.
Therefore the authors are in the opinion that the left hand sides of these equations are well-established.
That the differences $I_8-Y_8$ have a factor of $Z_4$ as shown on the right hand sides gives another set of consistency checks.
In particular, randomly flipping some signs will ruin this factorization.
This makes us quite sure that no sign mistakes remain.
As for the minor points:
1) We tried to improve the language of the footnote 2.
2) 3) 4) We corrected the typos kindly mentioned.
5) We added a reference to a later section, as recommended by the referee.
6) We improved the sentence.
Published as SciPost Phys. 17, 077 (2024)
Reports on this Submission
Report
The authors have made appropriate modifications (although there seems to be a misprint in the newly added text below eq. (5.1): "oftertimes" instead of "oftentimes"). I recommend the manuscript for publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)