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 $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}=\mathrm{Stab}(J){|}_I$ be the $k\times k$ 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 $X^{\prime }$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X^{\prime }$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X^{\prime }$ after transposition and interchanging the equivariant and Kähler parameters of $X$, respectively, with the Kähler and equivariant parameters of $X^{\prime }$. 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 $T^{\ast }\mathrm{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-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.