Elliptic stable envelopes and 3d mirror symmetry

TL;DR

This thesis explores the limits of elliptic stable envelopes to connect elliptic cohomology with K-theory, formalizes symplectic duality, and extends quantum group actions, providing new computational tools and confirming conjectures in enumerative geometry.

Contribution

It introduces a method to derive K-theoretic stable envelopes from elliptic stable envelopes and formalizes symplectic duality, advancing the understanding of 3d mirror symmetry.

Findings

Derived K-theoretic stable envelopes for fixed point varieties.

Formalized symplectic duality as 3d mirror symmetry.

Extended quantum group actions to K-theory of dual varieties.

Abstract

In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety X^G of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. We formalize the notion of symplectic duality, also known as 3-dimensional mirror symmetry. We obtain a factorization theorem about the limit of elliptic stable envelopes to a wall, which generalizes the result of M.Aganagic and A.Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices etc., to actions on the K-theory of the symplectic dual variety. In the case of the Hilbert scheme of points in plane, our results imply the conjectures of E.Gorsky and A.Negut. We propose a new approach to K-theoretic quantum difference equations.