Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

Abstract

We interpret the equivariant cohomology algebra HGLn×C∗∗(T∗Fλ;C) of the cotangent bundle of a partial flag variety Fλ parametrizing chains of subspaces 0=F0⊂F1⊂⋯⊂FN=Cn, dimFi/Fi−1=λi, as the Yangian Bethe algebra B∞(1DVλ−) of the glN-weight subspace 1DVλ− of a Y(glN)-module 1DV−. Under this identification the dynamical connection of Tarasov and Varchenko (2002) [12] turns into the quantum connection of Braverman et al. (2010) [4] and Maulik and Okounkov (2012) [5]. As a result of this identification we describe the algebra of quantum multiplication on HGLn×C∗∗(T∗Fλ;C) as the algebra of functions on fibers of a discrete Wronski map. In particular this gives generators and relations of that algebra. This identification also gives us hypergeometric solutions of the associated quantum differential equation. That fact manifests the Landau–Ginzburg mirror symmetry for the cotangent bundle of the flag variety.