Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

Abstract We consider a pair of quiver varieties $(X;X^{\prime})$ 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^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.