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

TL;DR

This paper establishes a deep connection between the equivariant cohomology of cotangent bundles of flag varieties and Yangian Bethe algebras, revealing new algebraic structures and solutions related to quantum cohomology and mirror symmetry.

Contribution

It identifies the equivariant cohomology algebra with the Yangian Bethe algebra, providing explicit generators, relations, and hypergeometric solutions, thus linking quantum cohomology with integrable systems.

Findings

Equivariant cohomology algebra is isomorphic to Yangian Bethe algebra.

Describes quantum multiplication algebra as functions on fibers of a Wronski map.

Provides hypergeometric solutions to the quantum differential equation.

Abstract

We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag variety F_\lambda parametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i, as the Yangian Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_\lambda;\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.