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Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants
by Philip Candelas, Xenia de la Ossa, Joseph McGovern
Submission summary
| Authors (as registered SciPost users): | Joseph McGovern |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2410.07107v2 (pdf) |
| Date submitted: | 2025-01-07 15:17 |
| 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 revisit the series solutions of the attractor equations of 4d N=2 supergravities obtained by Calabi--Yau compactifications previously studied in Candelas, Kuusela, and McGovern (2021). While only convergent for a restricted set of black hole charges, we find that they are summable with Pad\'e resummation providing a suitable method. By specialising these solutions to rank-two attractors, we obtain many conjectural identities of the type discovered in CKM. These equate ratios of weight-four special L-values with an infinite series whose summands are formed out of genus-0 Gromov--Witten invariants. We also present two new rank-two attractors which belong to moduli spaces each interesting in their own right. Each moduli space possesses two points of maximal unipotent monodromy. One has already been studied by Hosono and Takagi, and we discuss issues stemming from the associated L-function having nonzero rank.
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 reports on (numerical) evidence of a conjectural remarkable identity between L-function values on the one hand and suitably evaluated Mellin transforms of modular forms on the other hand, which in enumerative geometry corresponds to a relationship between GW invariants and point counts over finite fields of Calabi-Yau manifolds.
2. The paper reports on newly discovered attractor points of in families of Calabi-Yau threefolds.
3. To carry out the numerical checks of the conjectured identities and to reach higher numerical precision, the authors use Padé's approximation as an innovative tool in this context. This technique may prove useful in similar contexts in the future as well.
Weaknesses
1. From a conceptual point of view, the paper does not offer deep new insights. Nevertheless, the results can be useful for future investigations as any new example is relevant in this demanding research field.
2. As the authors state themselves, there are some weaknesses in the precession of the numerical evidence.
Report
I believe that the submitted manuscript is an interesting research article, which offers solid and important results. While there are some numerical weaknesses the paper offers convincing numerical evidence for the above mentioned conjectured number theoretic identity. Therefore, I recommend publication of the paper as it stands.
Requested changes
No requested changes.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)