SciPost Submission Page
New Examples of Abelian D4D2D0 Indices
by Joseph McGovern
Submission summary
| Authors (as registered SciPost users): | Joseph McGovern |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2412.01149v2 (pdf) |
| Date submitted: | 2025-01-08 09:56 |
| Submitted by: | McGovern, Joseph |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We apply the methods of \cite{Alexandrov:2023zjb} to compute generating series of D4D2D0 indices with a single unit of D4 charge for several compact Calabi-Yau threefolds, assuming modularity of these indices. Our examples include a $\mathbb{Z}_{7}$ quotient of R{\o}dland's pfaffian threefold, a $\mathbb{Z}_{5}$ quotient of Hosono-Takagi's double quintic symmetroid threefold, the $\mathbb{Z}_{3}$ quotient of the bicubic intersection in $\mathbb{P}^{5}$, and the $\mathbb{Z}_{5}$ quotient of the quintic hypersurface in $\mathbb{P}^{4}$. For these examples we compute GV invariants to the highest genus that available boundary conditions make possible, and for the case of the quintic quotient alone this is sufficiently many GV invariants for us to make one nontrivial test of the modularity of these indices. As discovered in \cite {Alexandrov:2023zjb}, the assumption of modularity allows us to compute terms in the topological string genus expansion beyond those obtainable with previously understood boundary data. We also consider five multiparameter examples with $h^{1,1}>1$, for which only a single index needs to be computed for modularity to fix the rest. We propose a modification of the formula in \cite{Alexandrov:2022pgd} that incorporates torsion to solve these models. Our new examples are only tractable because they have sufficiently small triple intersection and second Chern numbers, which happens because all of our examples are suitable quotient manifolds. In an appendix we discuss some aspects of quotient threefolds and their Wall data.
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
1-The paper provides new examples of generating series of MSW-invariants and extends previous results to non-simply connected Calabi-Yau.
2-It is exceptionally well written and will provide a welcome entry point for people interested in learning about this complex subject.
Report
The paper builds on recent results that combine the direct integration of the holomorphic anomaly equations satisfied by topological string free energies, the MNOP relation and wall crossing between D6-D2-D0 and D4-D2-D0 bound states in order to fix the modular generating series of rank one D4-D2-D0 invariants (MSW-invariants). Once the generating series has been determined, the results can be used in turn to obtain boundary conditions on the holomorphic ambiguity and to obtain higher genus results for the topological string.
The work extends the previous results by studying geometries with non-trivial fundamental groups, including quotients of complete intersections in projective space but also of non-complete intersections as well as several multi-parameter models. Considering quotient spaces allows the author to obtain geometries with small intersection numbers which reduces the number of coefficients in the modular ansatz that need to be determined.
Few examples of generating series of MSW-invariants are known and the results are a valuable addition to the literature. The author also corrects a mistake from the previous literature in the case of a Z5-quotient of the quintic.
The paper is well written. It manages to cover the necessary technical background in way that is complete yet concise, such that the exposition is self-contained while maintaining readability. It also reviews the relevant topology in a useful appendix. Slightly tangential to the main subject of the paper, this appendix also discusses an interesting example of non-homotopic Calabi-Yau threefolds that share the same Wall data and clarifies a previously open question about two geometries that had appeared in the study of non-Abelian gauged linear sigma models.
I fully recommend the paper for publication.
Requested changes
1-Do the arguments above (2.7) and (B.12) also hold if the curve is singular? If not, this poses a potential loophole that should be addressed.
2-In the third equation in (3.26) it should read k^jk^kp^0
3-Below (3.50) it should read Q'<Q
4-Perhaps one or two more sentences could be added on how one obtains (3.52) from what has been introduced before
Cosmetic remarks:
5-In the third row of the right column of (B.26), a comma is missing.
6-In the first equality of (B.31) the \widetilde is too wide
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)