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2014 | 12 | 5 | 694-710

Tytuł artykułu

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

Treść / Zawartość

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Języki publikacji

EN

Abstrakty

EN
We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a $$\mathfrak{g}\mathfrak{l}_n$$ partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.

Wydawca

Czasopismo

Rocznik

Tom

12

Numer

5

Strony

694-710

Opis fizyczny

Daty

wydano
2014-05-01
online
2014-02-15

Twórcy

  • University of North Carolina at Chapel Hill

Bibliografia

  • [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
  • [2] Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys., 1992, 146(1), 1–60
  • [3] Givental A.B., Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices, 1996, 13, 613–663 http://dx.doi.org/10.1155/S1073792896000414
  • [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
  • [5] Givental A.B., Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J., 2001, 1(4), 551–568
  • [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
  • [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
  • [8] Maulik D., Okounkov A., Quantum groups and quantum cohomology, preprint available at http://arxiv.org/abs/1211.1287
  • [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 \(\mathfrak{g}\mathfrak{l}_{N + 1}\) 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/10.1142/9789814324373_0016
  • [11] Mukhin E., Tarasov V., Varchenko A., Three sides of the geometric Langlands correspondence for \(\mathfrak{g}\mathfrak{l}_N\) 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
  • [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
  • [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
  • [14] Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré-Birchoff-Witt expansions of the canonical \(U(\mathfrak{n}\_)\) -valued differential form, Ann. Comb., 2005, 9(1), 57–74 http://dx.doi.org/10.1007/s00026-005-0241-3
  • [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
  • [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
  • [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
  • [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
  • [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
  • [20] Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, SIGMA Symmetry Integrability Geom. Methods Appl., 2013, 9, #048
  • [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
  • [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

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Bibliografia

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