Arithmetic geometry of quantum connections on Calabi-Yau 3-folds

2026/01/04 by Shaoyun Bai, Jae Hee Lee, Daniel Pomerleano · 1 voice
Mathematics · #math.SG #math.AG

paper · pdf

arxiv published 2026/01/04 · arxiv updated 2026/03/24

Abstract

Fix a prime p > 3. Working over 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.

Citations

• The Getzler-Gauss-Manin connection and Kontsevich-Soibelman operations on the periodic cyclic homology
• The cyclic open--closed map and variations of Hodge structures
• Quantum Adams operations in quasimap K-theory
• P-adic Gamma classes and overconvergent Frobenius structures for quantum connections
• THH(Z) and the image of J
• 3D mirror symmetry in positive characteristic
• On the exponential type conjecture
• Cohomological splitting over rationally connected bases
• Frobenius intertwiners for q-difference equations
• Cyclotomic Structures in Symplectic Topology
• Quantum Steenrod operations of symplectic resolutions
• The quantum connection, Fourier-Laplace transform, and families of A-infinity-categories
• Universally counting curves in Calabi--Yau threefolds
• Constructing the relative Fukaya category
• A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants
• Local Zeta Functions From Calabi-Yau Differential Equations
• The Gopakumar-Vafa finiteness conjecture
• On the periodic topological cyclic homology of DG categories in characteristic p
• Topological cyclic homology
• The Gauss-Manin connection on the periodic cyclic homology
• Connections on equivariant Hamiltonian Floer cohomology
• The Gopakumar-Vafa formula for symplectic manifolds
• Frobenius map and the p-adic Gamma function
• Frobenius map for quintic threefolds
• Integrality of instanton numbers and p-adic B-model
• Fourier transforms and p-adic "Weil II"
• Fourier transforms and p-adic ‘Weil II’
• Geometry and analytic theory of Frobenius manifolds
• The Hodge filtration on nonabelian cohomology
• Mirror Symmetry and Algebraic Geometry
• Nilpotent connections and the monodromy theorem : applications of a result of Turrittin

Discussions

• Shaoyun Bai, Jae Hee Lee: The $p$-curvature of quantum connections for Calabi-Yau threefolds https://arxiv.org/abs/2601.01654 https://arxiv.org/pdf/2601.01654 https://arxiv.org/html/2601.01654 [bsky, 0 points, 0 comments]

Related

• The quantum connection and its mod p reduction
• P-adic Gamma classes and overconvergent Frobenius structures for quantum connections
• P-adic splittings of the quantum connection
• Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties
• Quantum Steenrod operations of symplectic resolutions