Cohomology of a flag variety as a Yangian Bethe algebra

Details Cohomology of a flag variety as a Yangian Bethe algebra Cohomology of a flag variety as a Yangian Bethe algebra

2012-04-23

arxiv.org/abs/1204.5138v1

Mathematics

Algebraic Geometry

Latex, 43 pages

Scientific paper

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...\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i, as the Yangian Bethe algebra of a gl_N-weight subspace of a Y(gl_N)-module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] if F_\lambda is the full flag variety. That allows us to construct the flat sections of the quantum connection in the form of hypergeometric integrals. For an arbitrary \lambda, we conjecture a description of the small quantum equivariant cohomology algebra of the cotangent bundle T^*F_\lambda as a suitable Yangian Bethe algebra.

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