Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 032, 55 pages      arXiv:1001.4553      https://doi.org/10.3842/SIGMA.2011.032
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Quantum Integrable Model of an Arrangement of Hyperplanes

Alexander Varchenko
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received July 19, 2010, in final form March 19, 2011; Published online March 28, 2011

Abstract
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.

Key words: Gaudin model; arrangement of hyperplanes.

pdf (807 kb)   tex (53 kb)

References

  1. Arnold V.I., Gusein-Zade S.M., Varchenko A.M., Singularities of differentiable maps, Vol. II, Monodromy and asymptotics of integrals, Monographs in Mathematics, Vol. 83, Birkhäuser, Boston, MA, 1988.
  2. Babujian H.M., Off-shell Bethe ansatz equations and N-point correlators in the SU(2) WZNW theory, J. Phys. A: Math. Gen. 26 (1993), 6981-6990.
  3. Babujian H.M., Flume R., Off-shell Bethe ansatz equation for Gaudin magnets and solutions of Knizhnik-Zamolodchikov equations, Modern Phys. Lett. A 9 (1994), 2029-2039, hep-th/9310110.
  4. Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 29-62, hep-th/9402022.
  5. Feigin B., Frenkel E., Rybnikov L., Opers with irregular singularity and spectra of the shift of argument subalgebra, Duke Math. J. 155 (2010), 337-363, arXiv:0712.1183.
  6. Feigin B., Schechtman V., Varchenko A., On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 (1995), 219-247, hep-th/9407010.
  7. Gaudin M., Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), 1087-1098.
  8. Gaudin M., La fonction d'onde de Bethe, Collection du Commissariat l'Érgie Atomique, Série Scientifique, Masson, Paris, 1983.
  9. Griffiths Ph., Harris J., Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1994.
  10. Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), no. 8, P08002, 44 pages, math.QA/0605015.
  11. Mukhin E., Tarasov V., Varchenko A., The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), 863-881, math.AG/0512299.
  12. Mukhin E., Tarasov V., Varchenko A., Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), 909-940, arXiv:0711.4079.
  13. Mukhin E., Tarasov V., Varchenko A., Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of glN, Algebra i Analiz 22 (2010), no. 3, 177-190, arXiv:0904.2131.
  14. Mukhin E., Tarasov V., Varchenko A., Three sides of the geometric Langlands correspondence for glN Gaudin model and Bethe vector averaging maps, arXiv:0907.3266.
  15. Mukhin E., Tarasov V., Varchenko A., Bethe algebra of the glN+1 Gaudin model and algebra of functions on the critical set of the master function, arXiv:0910.4690.
  16. Mukhin E., Varchenko A., Norm of a bethe vector and the Hessian of the master function, Compos. Math. 141 (2005), 1012-1028, math.QA/0402349.
  17. Orlik P., Terao H., The number of critical points of a product of powers of linear functions, Invent. Math. 120 (1995), 1-14.
  18. Orlik P., Terao H., Arrangements and hypergeometric integrals, MSJ Memoirs, Vol. 9, Mathematical Society of Japan, Tokyo, 2001.
  19. Reshetikhin N., Varchenko A., Quasiclassical asymptotics of solutions to the KZ equations, in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, Vol. 4, Int. Press, Cambridge, MA, 1995, 293-322.
  20. Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré-Birkhoff-Witt expansions of the canonical U(n−)-valued differential form, Ann. Comb. 9 (2005), 57-74, math.CO/0407101.
  21. Silvotti R., On a conjecture of Varchenko, Invent. Math. 126 (1996), 235-248, alg-geom/9503016.
  22. Schechtman V., Terao H., Varchenko A., Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra 100 (1995), 93-102, hep-th/9411083.
  23. Schechtman V., Varchenko A., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194.
  24. Talalaev D., Quantization of the Gaudin system, hep-th/0404153.
  25. Varchenko A., Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics, Vol. 21, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  26. Varchenko A., Critical points of the product of powers of linear functions and families of bases of singular vectors, Compos. Math. 97 (1995), 385-401, hep-th/9312119.
  27. Varchenko A., Bethe ansatz for arrangements of hyperplanes and the Gaudin model, Mosc. Math. J. 6 (2006), 195-210, 223-224, math.QA/0408001.
  28. Yuzvinsky S., Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), 5339-5354.
  29. Zaslavsky T., Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. 154.

Previous article   Next article   Contents of Volume 7 (2011)