On the exponential type conjecture

TL;DR

This paper proves that the small quantum t-connection on certain symplectic manifolds exhibits exponential type behavior and has quasi-unipotent monodromies, confirming a conjecture in the field.

Contribution

It establishes the exponential type property and quasi-unipotent monodromies of the quantum t-connection for a class of symplectic manifolds, solving a conjecture by Katzarkov-Kontsevich-Pantev and Galkin-Golyshev-Iritani.

Findings

Quantum t-connection is of exponential type.

Monodromies at t=0 are quasi-unipotent.

The proof uses reduction to positive characteristics.

Abstract

We prove that the small quantum t-connection on a closed monotone symplectic manifold is of exponential type and has quasi-unipotent regularized monodromies at t=0. This answers a conjecture of Katzarkov-Kontsevich-Pantev and Galkin-Golyshev-Iritani for those classes of symplectic manifolds. The proof follows a reduction to positive characteristics argument, and the main tools of the proof are Katz's local monodromy theorem in differential equations and quantum Steenrod operations in equivariant Gromov-Witten theory with mod p coefficients.