P-adic Gamma classes and overconvergent Frobenius structures for quantum connections
Shaoyun Bai, Daniel Pomerleano, Paul Seidel
TL;DR
The paper advances a p-adic perspective on the small quantum connection of monotone symplectic manifolds by formulating and proving a conjecture that the connection admits an overconvergent Frobenius structure with constant term governed by Morita’s $p$-adic Gamma class. The authors develop a multivariable toric framework and a mod $p$ mirror symmetry toolkit to construct a Dwork inverse Frobenius and show its overconvergence, explicitly tying the constant term to $ extGamma_p(TM)$. They prove the conjecture for smooth toric Fano varieties and Grassmannians, with supporting numerical evidence, and establish the single-variable specialization that yields a robust, overconvergent Frobenius across these classes. The approach blends mirror symmetry, $p$-adic analytic methods, and Dwork-type cohomological computations to connect arithmetic properties of the quantum connection with Gamma-class data, enabling evaluation at $(p-1)$-st roots of unity and yielding valuation patterns tied to the Betti numbers. This work thus provides a concrete arithmetic realization of Gamma-conjecture-type phenomena in the monotone setting and offers computational evidence for overconvergence in broader geometries.
Abstract
Consider the small quantum connection on a monotone symplectic manifold, with p-adic coefficients. We conjecture that this always admits an overconvergent Frobenius structure, whose constant term is given by a characteristic class associated to Morita's p-adic Gamma function. We prove this conjecture for toric Fano varieties and Grassmannians, and also supply additional experimental evidence.