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Local Zeta Functions of Multiparameter Calabi-Yau Threefolds from the Picard-Fuchs Equations
by Philip Candelas, Xenia de la Ossa, Pyry Kuusela
Submission summary
| Authors (as registered SciPost users): | Pyry Kuusela · Xenia de la Ossa |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2405.08067v2 (pdf) |
| Code repository: | https://github.com/PyryKuusela/CY3Zeta |
| Date submitted: | April 2, 2025, noon |
| Submitted by: | Pyry Kuusela |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The deformation approach of arXiv:2104.07816 for computing zeta functions of one-parameter Calabi-Yau threefolds is generalised to cover also multiparameter manifolds. Consideration of the multiparameter case requires the development of an improved formalism. This allows us, among other things, to make progress on some issues left open in previous work, such as the treatment of apparent and conifold singularities and changes of coordinates. We also discuss the efficient numerical computation of the zeta functions. As examples, we compute the zeta functions of the two-parameter mirror octic, a non-symmetric split of the quintic threefold also with two parameters, and the $S_5$ symmetric five-parameter Hulek-Verrill manifolds. These examples allow us to exhibit the several new types of geometries for which our methods make practical computations possible. They also act as consistency checks, as our results reproduce and extend those of arXiv:hep-th/0409202 and arXiv:math/0304169. To make the methods developed here more approachable, a Mathematica package "CY3Zeta" for computing the zeta functions of Calabi-Yau threefolds, which is attached to this paper, is presented.
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Author: Pyry Kuusela on 2025-09-18 [id 5833]
(in reply to Report 1 on 2025-05-21)We thank the referee for a thorough reading of our manuscript and helpful comments. Below we briefly respond to the points raised by the referee.
1) We agree that the dependence of $\alpha_i$ on n was not clear. The $\alpha_i$ themselves are p-adic integers, which are independent of $n$, but the approximations displayed in table 2 depend on the chosen accuracy. We have now clarified the issue already in section 3.5, as well as above table 2, and in a new appendix B.1.
2) We have corrected the typo.
3-1) We agree that the basis chosen for the periods in appendix B is not the integral basis. Indeed, to make the comparison to the Calabi-Yau case as transparent as possible, we wish to use the Frobenius basis here. We have clarified this and changed the potentially misleading notation.
3-2) We have fixed the formula.
3-3) The apparent discrepancy comes down to the issue of twists, which we have now addressed in the main text briefly and in greater detail in new appendix B.1.
3-4) We have added the requested list, which we agree is helpful.
4) We thank the referee for pointing out the relevant reference, which we have now cited.