Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 093, 22 pages      arXiv:1906.00134      https://doi.org/10.3842/SIGMA.2019.093
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

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

Richárd Rimányi a, Andrey Smirnov ab, Alexander Varchenko ac and Zijun Zhou d
a) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
b) Institute for Problems of Information Transmission, Bolshoy Karetny 19, Moscow 127994, Russia
c) Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Leninskiye Gory 1, 119991 Moscow GSP-1, Russia
d) Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA 94305, USA

Received July 08, 2019, in final form November 18, 2019; Published online November 28, 2019

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 $A_{I,J}= \operatorname{Stab}(J)|_{I}$ 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 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 $T^*\operatorname{Gr}(k,n)$ 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.

Key words: equivariant elliptic cohomology; elliptic stable envelope; 3d mirror symmetry.

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