Enumerative geometry via elliptic stable envelope

TL;DR

This paper constructs $q$-difference equations from elliptic stable envelopes of varieties, linking curve counting, quantum difference equations, and equivariant elliptic cohomology, thus connecting enumerative geometry with integrable systems.

Contribution

It introduces a natural method to derive $q$-difference equations from elliptic stable envelopes, providing a bridge between curve counting and elliptic cohomology.

Findings

Constructed $q$-difference equations from elliptic stable envelopes.

These equations match quantum difference equations in examples.

Solutions encode generating functions for curve counting in $X$.

Abstract

Assume $X$ is a variety for which the elliptic stable envelope exists. In this note we construct natural $q$-difference equations from the elliptic stable envelope of $X$. In examples, these equations coincide with the quantum difference equations, which give a natural $q$-deformation of the Dubrovin connection of $X$. Solutions of the quantum difference equations provide generating functions counting curves in $X$. In this way, our construction connects curve counting and equivariant elliptic cohomology. This is an overview paper based on the author's talk at the workshop The 16th MSJ-SI: Elliptic Integrable Systems, Representation Theory and Hypergeometric Functions, Tokyo 2023.