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/2412.01149v3 (pdf) |
| Date submitted: | Dec. 1, 2025, 6:04 a.m. |
| Submitted by: | Joseph McGovern |
| 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 Rødland'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.
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Author comments upon resubmission
I have implemented all requested changes, and corrected several typos, as I detail in the more extensive changelog. I also added some very minor clarifying remarks.
In particular, concerning Referee 2's point on the old equation (3.16): I have corrected this as it was indeed mistaken.
Please note that this does not affect any of the conclusions of the paper, as the Plethystic relation (current equation 3.18) remains correct and was used for all computations.
Further to this, I have deleted the old equation (3.17), which does not play a role in my computations, to avoid confusion.
Concerning Referee 1's comment on singular curves: The arguments in the appendix B only require some curve C to not be singular, so I have made this comment and the conclusions stand unchanged.
In particular this gives the claimed relation between the Kahler parameters of the cover and quotient.
To secure the (2.7) relating the GV invariants, I included a new argument based on the GLSM sphere partition function. I was unable to come up with a more elegant geometrical argument for (2.7).
Many thanks,
Joseph.
List of changes
page 1: Paragraph 1, spaces were added to "D4, D2, D0,".
page 2: Penultimate paragraph, typo with a comma fixed.
page 3: Paragraph 1, "or the related Pfaffian threefold" added.
page 4: fixed a typo "Determinantal".
page 6: The discussion following (2.6) was reworked, with a new GLSM argument.
page 8: Typo fixed in (3.1), q instead of q^t. Similar for following paragraph.
page 8: Final paragraph, typo fixed, \log(\varphi) replaced with \varpi_{0}\log\varphi.
page 10: Paragraph after (3.10): I added an explanation of how to obtain this normalisation k_c .
page 10: Second paragraph after (3.10), removed typo "or".
page 12: (3.16) was corrected.
page 12: (3.17) was deleted.
page 14: Fixed a typo in (3.25).
page 18: Added a full-stop to (3.44).
page 19: Fixed a typo below (3.49), " Q' < Q "
page 20: A new large footnote was added, explaining how to obtain (3.51).
page 20: A remark was added below (3.51) on \chi.
page 20: A typo in (3.53) and (3.54) was fixed, I added the argument to the LHS.
page 21: Word "case" added to the paragraphs above (3.55).
page 23: "Then" added to first paragraph.
page 30: First paragraph, \Delta is defined. Second paragraph, the relation between \varphi and y is given. An x was changed to a \varphi.
page 33: Fixed a typo "are not".
page 35: First paragraph, typo fixed, "The".
page 38: First paragraph, added "four of which" to clarify examples.
page 47: Added a comma in (B.26).
page 48: Made a tilde in (B.31) smaller.
page 49: Third paragraph, added a sentence on GCDs of Wall data.