3d Mirror Symmetry and Elliptic Stable Envelopes

TL;DR

This paper explores the relationship between 3d mirror symmetry of quiver varieties and elliptic stable envelopes, providing explicit formulas and demonstrating a duality in their restriction matrices.

Contribution

It introduces formulas for elliptic stable envelopes on mirror pairs of quiver varieties and establishes a duality relation via the Mother function in equivariant elliptic cohomology.

Findings

Formulas for elliptic stable envelopes on both sides of the mirror pair.

Existence of a Mother function linking the stable envelopes.

Restriction matrices are equal after transposition and parameter identification.

Abstract

We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters on one side with the K\"ahler parameters on the dual side.