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

Richárd Rimányi, University of North Carolina at Chapel Hill, USA, Andrey Smirnov, Alexander Varchenko, Zijun Zhou, University of North Carolina at Chapel Hill, USA, University of North Carolina at Chapel Hill, USA, Stanford University, USA
2019 Symmetry, Integrability and Geometry: Methods and Applications  
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 $A_{I,J}= \operatorname{Stab}(J)|_{I}$ 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 K\"ahler parameters and equivariant parameters of $X$. We say that two such varieties $X$
more » ... ch 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 K\"ahler parameters of $X$, respectively, with the K\"ahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rim\'anyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ 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.
doi:10.3842/sigma.2019.093 fatcat:2fcecq32cvfk7bxdttz4skdzau