Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra. (English) Zbl 1287.81063
Summary: We interpret the equivariant cohomology algebra \(H_{GL_n \times C^\ast}^\ast (T^\ast \mathcal F_{\lambda}; \mathbb C)\) of the cotangent bundle of a partial flag variety \(\mathcal F_{\lambda}\) parametrizing chains of subspaces \(0 = F_0 \subset F_1 \subset \cdots \subset F_N = \mathbb C^n\), \(\dim F_i/F_{i-1} = {\lambda}_i\), as the Yangian Bethe algebra \(\mathcal B^\infty(\frac{1}{D}\mathcal V_{\lambda}^{-})\) of the \(\mathfrak{gl}_{N^-}\) weight subspace \(\frac{1}{D}\mathcal V_{\lambda}^{-}\) of a \(Y(\mathfrak{gl}_N)\)-module \(\frac{1}{D}V^{-}\). Under this identification the dynamical connection of V. Tarasov and A. Varchenko [Acta Appl. Math. 73, No. 1–2, 141–154 (2002; Zbl 1013.17006)] turns into the quantum connection of A. Braverman et al. [Adv. Math. 227, No. 1, 421–458 (2011; Zbl 1226.14069)] and D. Maulik and A. Okounkov [“Quantum groups and quantum cohomology”, (2012), arXiv:1211.1287]. As a result of this identification we describe the algebra of quantum multiplication on \(H_{GL_n \times \mathbb C^\ast}^\ast (T^\ast \mathcal F_{\lambda}; \mathbb C)\) 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.
MSC:
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 53D50 | Geometric quantization |
| 81S10 | Geometry and quantization, symplectic methods |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
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