Enumerative geometry via elliptic stable envelope

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.