Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Abstract

Let X be a holomorphic symplectic variety with a torus T action and a finite fixed point set of cardinality k. We assume that an elliptic stable envelope exists for X. Let AI, J=Stab(J)|I be the k×k matrix of restrictions of the elliptic stable envelopes of X to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of X. We say that two such varieties X and X′ are related by the 3d mirror symmetry if the fixed point sets of X and X′ have the same cardinality and can be identified so that the restriction matrix of X becomes equal to the restriction matrix of X′ after transposition and interchanging the equivariant and Kähler parameters of X, respectively, with the Kähler and equivariant parameters of X′. The first examples of pairs of 3d symmetric varieties were constructed in [Rimányi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle T*Gr(k,n) to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of Aₙ₋₁-type. In this paper, we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.