Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety

Abstract

Abstract:We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety. MSC:82B23, 17B80, 55N91

Trigonometric weight functions asK-theoretic stable envelope maps for the cotangent bundle of a flag variety

“…This conjecture is the K -theoretic analog of the main theorem in [1] that describes the quantum multiplication in the equivariant cohomology of Nakajima varieties. The case of the quantum multiplication in the equivariant cohomologies of the varieties T * F λ was considered also in [23,14,24]. Lemmas 13.12 and 13.14 mean that modulo Conjecture 13.15, the quantum equivariant K -theory algebra QK T (T * F λ ) ⊗ C(h) is isomorphic to the Bethe algebra ρ λ (B q ) and is isomorphic to the algebra K q λ ⊗ C(h).…”

Combinatorial Formulae for Nested Bethe Vectors

“…Let A 1 , A 2 be two copies of the same algebra A. Then for any a, b ∈ A we have a (1) = a ⊗ 1, b (2) = 1 ⊗ b, (a ⊗ b) (12) = a ⊗ b and (a ⊗ b) (21) = b ⊗ a.…”

“…It is published with no intention to give any review of the subject or reflect the state of the art. Let us only mention a few papers making progress in particularly close problems [5,6,4,14,1,2] and those exploring recently the results of the paper [16,21,13].…”

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

“…[TV2]. These hypergeometric integrals provide flat sections of the quantum connection defined by (3.20), see [TV3]. This presentation of flat sections of the quantum connection as hypergeometric integrals is in the spirit of the mirror symmetry, see Candelas et al [COGP], Givental [G1, G2], and [BCK, BCKS, GKLO, I, JK].…”