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New Examples of Abelian D4D2D0 Indices
by Joseph McGovern
This is not the latest submitted version.
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: | Jan. 8, 2025, 9:56 a.m. |
| Submitted by: | Joseph McGovern |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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
Report
The present paper computes generalized DT invariants for certain compact Calabi-Yau threefolds, obtained as Z_p quotients of known examples, for which the ansatz is modified accordingly, and for some simple choices of charges (no D6-branes), under some assumptions on the mirrors as well as the usual assumptions on BMT inequality, and it reports on cases where such an ansatz does not work.
The input are GV/PT invariants, which the author computes and takes from the available literature, and results are checked against the expectation that generating sums are vector-valued modular forms.
I think the results are interesting, as they add some interesting data
points to an existing line of research, and this should be published
in Scipost.
Requested changes
I do not understand equation 3.16: if this is the relation between
reduced GW and reduced DT (i.e. PT), why is there any Macmahon, and why is 1\lambda multiplying the Kahler parameter?
Recommendation
Publish (meets expectations and criteria for this Journal)
Strengths
2-It is exceptionally well written and will provide a welcome entry point for people interested in learning about this complex subject.
Report
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%)