### Abstract

Original language | English |
---|---|

Pages (from-to) | 56-86 |

Number of pages | 31 |

Journal | Journal of Geometry and Physics |

Volume | 74 |

Early online date | 26 Jul 2013 |

DOIs | |

Publication status | Published - Dec 2013 |

### Fingerprint

### Keywords

- Yangian
- Bethe algebra
- quantum cohomology
- quantum connection
- discrete Wronski map

### Cite this

*Journal of Geometry and Physics*,

*74*, 56-86. https://doi.org/10.1016/j.geomphys.2013.07.006

**Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra.** / Gorbunov, V. ; Rimanyi, R. ; Tarasov, V. ; Varchenko, A. .

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 74, pp. 56-86. https://doi.org/10.1016/j.geomphys.2013.07.006

}

TY - JOUR

T1 - Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

AU - Gorbunov, V.

AU - Rimanyi, R.

AU - Tarasov, V.

AU - Varchenko, A.

PY - 2013/12

Y1 - 2013/12

N2 - We interpret the equivariant cohomology algebra View the MathML source of the cotangent bundle of a partial flag variety View the MathML source parametrizing chains of subspaces 0=F0⊂F1⊂⋯⊂FN=Cn, dimFi/Fi−1=λi, as the Yangian Bethe algebra View the MathML source of the glN-weight subspace View the MathML source of a Y(glN)-module View the MathML source. 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 View the MathML source 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.

AB - We interpret the equivariant cohomology algebra View the MathML source of the cotangent bundle of a partial flag variety View the MathML source parametrizing chains of subspaces 0=F0⊂F1⊂⋯⊂FN=Cn, dimFi/Fi−1=λi, as the Yangian Bethe algebra View the MathML source of the glN-weight subspace View the MathML source of a Y(glN)-module View the MathML source. 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 View the MathML source 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.

KW - Yangian

KW - Bethe algebra

KW - quantum cohomology

KW - quantum connection

KW - discrete Wronski map

U2 - 10.1016/j.geomphys.2013.07.006

DO - 10.1016/j.geomphys.2013.07.006

M3 - Article

VL - 74

SP - 56

EP - 86

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -