Arithmetic geometry of quantum connections on Calabi-Yau $3$-folds
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arXiv
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| Natura: | Preprint |
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2026
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| author | Bai, Shaoyun Lee, Jae Hee Pomerleano, Daniel |
| author_facet | Bai, Shaoyun Lee, Jae Hee Pomerleano, Daniel |
| contents | Fix a prime $p > 3$. Working over $\mathbb{Z}_p$, we show that the quantum connection of any closed Calabi-Yau threefold gives rise to a Fontaine-Laffaile module when restricted to the even degree and torsion-free part of $p$-adic quantum cohomology, whose associated Frobenius endomorphism has leading order term prescribed by the $p$-adic Gamma class. After reducing mod $p$, the divided Frobenius endomorphism defines an analogue of the inverse Cartier operator on mod $p$ quantum cohomology. We establish an $A$-model analogue of a classical result due to Katz: the conjugation of the $p$-curvature of the mod $p$ quantum connection by the inverse Cartier operator is equal to the Frobenius pullback of the quantum product, the $A$-model counterpart of the Kodaira-Spencer class. Moreover, we identify the quantum Steenrod operation with the $p$-curvature of the mod $p$ quantum connection in this setting for any prime $p$.
We propose several conjectures concerning how these arithmetic structures may extend to quantum connections on more general semi-positive symplectic manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01654 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic geometry of quantum connections on Calabi-Yau $3$-folds Bai, Shaoyun Lee, Jae Hee Pomerleano, Daniel Symplectic Geometry Algebraic Geometry Fix a prime $p > 3$. Working over $\mathbb{Z}_p$, we show that the quantum connection of any closed Calabi-Yau threefold gives rise to a Fontaine-Laffaile module when restricted to the even degree and torsion-free part of $p$-adic quantum cohomology, whose associated Frobenius endomorphism has leading order term prescribed by the $p$-adic Gamma class. After reducing mod $p$, the divided Frobenius endomorphism defines an analogue of the inverse Cartier operator on mod $p$ quantum cohomology. We establish an $A$-model analogue of a classical result due to Katz: the conjugation of the $p$-curvature of the mod $p$ quantum connection by the inverse Cartier operator is equal to the Frobenius pullback of the quantum product, the $A$-model counterpart of the Kodaira-Spencer class. Moreover, we identify the quantum Steenrod operation with the $p$-curvature of the mod $p$ quantum connection in this setting for any prime $p$. We propose several conjectures concerning how these arithmetic structures may extend to quantum connections on more general semi-positive symplectic manifolds. |
| title | Arithmetic geometry of quantum connections on Calabi-Yau $3$-folds |
| topic | Symplectic Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2601.01654 |