Flat endomorphisms for mod $p$ equivariant quantum connections from quantum Steenrod operations
Jae Hee Lee
TL;DR
The paper develops a covariant-endomorphism framework for the mod $p$ equivariant quantum connection using quantum Steenrod operations, with a detailed realization on $X=T^*\mathbb{P}^1$. Central to the approach is a local Calabi--Yau reduction that converts curve-counting into computable equivariant Chern-integral counts, enabling full determination of the covariantly constant endomorphism in characteristic $p$. The main contribution is the explicit computation of the $S^1$-equivariant quantum Steenrod operations for the fiber class on $T^*\mathbb{P}^1$, including a proof of uniqueness of the flat endomorphism and a precise link to arithmetic flat sections in Varchenko's framework. The results extend covariant constancy beyond the previously known regime, illuminate the role of multiple covers, and connect symplectic geometry with mod $p$ Gauss--Manin systems and arithmetic holomorphic-curve counts, with potential implications for mod $p$ mirror symmetry and arithmetic flat sections.
Abstract
We present a method for constructing covariantly constant endomorphisms for the mod $p$ equivariant quantum connection, using the quantum Steenrod power operations of Fukaya and Wilkins. The example of the cotangent bundle of the projective line is fully computed, and we discuss the relationship with the mod $p$ solutions of trigonometric KZ equation recently constructed by Varchenko. As a byproduct, we compute the first examples of quantum Steenrod operations that are not a priori determined by ordinary Gromov--Witten theory and classical Steenrod operations, which may be of independent interest.