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Mathematical Research Letters
Volume 29 (2022)
Number 3
Symplectic duality for $T^\ast Gr(k, n)$
Pages: 663 – 690
DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n3.a3
Author
Abstract
In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to quiver varieties. In this paper, we prove a formula which relates the vertex functions of $T^\ast Gr(k, n)$ and its symplectic dual. In the course of the proof, we study a family of $q$-difference operators which act diagonally on Macdonald polynomials. Our results may be interpreted from a combinatorial perspective as providing an evaluation formula for a $q$-Selberg type integral.
Received 14 September 2020
Accepted 1 June 2021
Published 30 November 2022