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

SIGMA 8 (2012), 072, 11 pages      arXiv:1201.3990

KZ Characteristic Variety as the Zero Set of Classical Calogero-Moser Hamiltonians

Evgeny Mukhin a, Vitaly Tarasov a, b and Alexander Varchenko c
a) Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
b) St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia
c) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received June 09, 2012, in final form October 04, 2012; Published online October 16, 2012

We discuss a relation between the characteristic variety of the KZ equations and the zero set of the classical Calogero-Moser Hamiltonians.

Key words: Gaudin Hamiltonians; Calogero-Moser system; Wronski map.

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  1. 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.
  2. Cherednik I., Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Adv. Math. 106 (1994), 65-95.
  3. Cherednik I., Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, in Quantum Many-Body Problems and Representation Theory, MSJ Mem., Vol. 1, Math. Soc. Japan, Tokyo, 1998, 1-96.
  4. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, math.AG/0011114.
  5. Felder G., Markov Y., Tarasov V., Varchenko A., Differential equations compatible with KZ equations, Math. Phys. Anal. Geom. 3 (2000), 139-177, math.QA/0001184.
  6. Felder G., Veselov A.P., Polynomial solutions of the Knizhnik-Zamolodchikov equations and Schur-Weyl duality, Int. Math. Res. Not. 2007 (2007), no. 15, 21 pages, math.RT/0610383.
  7. Felder G., Veselov A.P., Shift operators for the quantum Calogero-Sutherland problems via Knizhnik-Zamolodchikov equation, Comm. Math. Phys. 160 (1994), 259-273.
  8. Finkelberg M., Ginzburg V., Calogero-Moser space and Kostka polynomials, Adv. Math. 172 (2002), 137-150, math.RT/0110190.
  9. Gaudin M., La fonction d'onde de Bethe, Collection du Commissariat à l'Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
  10. Givental A., Kim B., Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), 609-641, hep-th/9312096.
  11. Gordon I., Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), 321-336, math.RT/0202301.
  12. Jurco B., Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys. 30 (1989), 1289-1293.
  13. Kazhdan D., Kostant B., Sternberg S., Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-507.
  14. Matsuo A., Integrable connections related to zonal spherical functions, Invent. Math. 110 (1992), 95-121.
  15. Mukhin E., Tarasov V., Varchenko A., Bethe algebra, Calogero-Moser space and Cherednik algebra, arXiv:0906.5185.
  16. Mukhin E., Tarasov V., Varchenko A., Gaudin Hamiltonians generate the Bethe algebra of a tensor power of the vector representation of glN, St. Petersburg Math. J. 22 (2011), 463-472, arXiv:0904.2131.
  17. Mukhin E., Tarasov V., Varchenko A., Generating operator of XXX or Gaudin transfer matrices has quasi-exponential kernel, SIGMA 3 (2007), 060, 31 pages, math.QA/0703893.
  18. 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.
  19. 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.
  20. Reshetikhin N., Varchenko A., Quasiclassical asymptotics of solutions to the KZ equations, in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, Vol. IV, Int. Press, Cambridge, MA, 1995, 293-322, hep-th/9402126.
  21. Schechtman V.V., Varchenko A.N., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194.
  22. Schechtman V.V., Varchenko A.N., Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990), 279-283.
  23. Wilson G., Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), 1-41.

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