We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.
@article{bwmeta1.element.doi-10_2478_s11533-013-0376-8, author = {Vitaly Tarasov and Alexander Varchenko}, title = {Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety}, journal = {Open Mathematics}, volume = {12}, year = {2014}, pages = {694-710}, zbl = {1294.82016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0376-8} }
Vitaly Tarasov; Alexander Varchenko. Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety. Open Mathematics, Tome 12 (2014) pp. 694-710. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0376-8/
[1] Braverman A., Maulik D., Okounkov A., Quantum cohomology of the Springer resolution, Adv. Math., 2011, 227(1), 421–458 http://dx.doi.org/10.1016/j.aim.2011.01.021 | Zbl 1226.14069
[2] Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys., 1992, 146(1), 1–60 | Zbl 0760.17006
[3] Givental A.B., Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices, 1996, 13, 613–663 http://dx.doi.org/10.1155/S1073792896000414 | Zbl 0881.55006
[4] Givental A., Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 180, American Mathematical Society, Providence, 1997, 103–115 | Zbl 0895.32006
[5] Givental A.B., Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J., 2001, 1(4), 551–568 | Zbl 1008.53072
[6] Gorbounov V., Rimányi R., Tarasov V., Varchenko A., Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra, J. Geom. Phys., 2013, 74, 56–86 http://dx.doi.org/10.1016/j.geomphys.2013.07.006 | Zbl 1287.81063
[7] Markov Y., Varchenko A., Hypergeometric solutions of trigonometric KZ equations satisfy dynamical difference equations, Adv. Math., 2002, 166(1), 100–147 http://dx.doi.org/10.1006/aima.2001.2027 | Zbl 1018.32016
[8] Maulik D., Okounkov A., Quantum groups and quantum cohomology, preprint available at http://arxiv.org/abs/1211.1287 | Zbl 1226.14069
[9] Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp., 2006, 8, #P08002
[10] Mukhin E., Tarasov V., Varchenko A., Bethe algebra of the Gaudin model and algebra of functions on the critical set of the master function, In: New Trends in Quantum Integrable Systems, World Scientific, Hackensack, 2011, 307–324 http://dx.doi.org/10.1142/97898143243730016 | Zbl 1221.82038
[11] Mukhin E., Tarasov V., Varchenko A., Three sides of the geometric Langlands correspondence for Gaudin model and Bethe vector averaging maps, In: Arrangements of Hyperplanes-Sapporo 2009, Sapporo, August 1–13, 2009, Adv. Stud. Pure Math., 62, Mathematrical Society of Japan, Tokyo, 2012, 475–511 | Zbl 1260.82025
[12] Nakajima H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J., 1994, 76(2), 365–416 http://dx.doi.org/10.1215/S0012-7094-94-07613-8 | Zbl 0826.17026
[13] Nakajima H., Quiver varieties and Kac-Moody algebras, Duke Math. J., 1998, 91(3), 515–560 http://dx.doi.org/10.1215/S0012-7094-98-09120-7 | Zbl 0970.17017
[14] Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré-Birchoff-Witt expansions of the canonical -valued differential form, Ann. Comb., 2005, 9(1), 57–74 http://dx.doi.org/10.1007/s00026-005-0241-3 | Zbl 1088.33007
[15] Rimányi R., Tarasov V., Varchenko A., Partial flag varieties, stable envelopes and weight functions, Quantum Topol. (in press), preprint available at http://arxiv.org/abs/1212.6240 | Zbl 06452363
[16] Schechtman V.V., Varchenko A.N., Arrangements of hyperplanes and Lie algebra homology, Invent. Math., 1991, 106(1), 139–194 http://dx.doi.org/10.1007/BF01243909 | Zbl 0754.17024
[17] Tarasov V., Varchenko A., Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras, Invent. Math., 1997, 128(3), 501–588 http://dx.doi.org/10.1007/s002220050151 | Zbl 0877.33013
[18] Tarasov V., Varchenko A., Difference equations compatible with trigonometric KZ differential equations, Internat. Math. Res. Notices, 2000, 15, 801–829 http://dx.doi.org/10.1155/S1073792800000441 | Zbl 0971.39009
[19] Tarasov V., Varchenko A., Duality for Knizhnik-Zamolodchikov and dynamical equations, Acta Appl. Math., 2002, 73(1–2), 141–154 http://dx.doi.org/10.1023/A:1019787006990 | Zbl 1013.17006
[20] Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, SIGMA Symmetry Integrability Geom. Methods Appl., 2013, 9, #048 | Zbl 1288.82024
[21] Toledano Laredo V., The trigonometric Casimir connection of a simple Lie algebra, J. Algebra, 2011, 329, 286–327 http://dx.doi.org/10.1016/j.jalgebra.2010.05.025 | Zbl 1241.17012
[22] Varchenko A.N., Tarasov V.O., Jackson integral representations for solutions of the quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J., 1995, 6(2), 275–313