Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Gaudin Hamiltonians generate the Bethe algebra of a tensor power of the vector representation of $ \mathfrak{gl}_N$

Authors: E. Mukhin, V. Tarasov and A. Varchenko
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 463-472
MSC (2010): Primary 82C23, 81Q12
Published electronically: March 18, 2011
MathSciNet review: 2729945
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Gaudin Hamiltonians $ H_1$, ..., $ H_n$ generate the Bethe algebra of the $ n$-fold tensor power of the vector representation of $ \mathfrak{gl}_N$. Surprisingly, the formula for the generators of the Bethe algebra in terms of the Gaudin Hamiltonians does not depend on $ N$. Moreover, this formula coincides with Wilson's formula for the stationary Baker-Akhiezer function on the adelic Grassmannian.

References [Enhancements On Off] (What's this?)

  • [B] H. M. Babujian, Off-shell Bethe ansatz equations and $ N$-point correlators in the $ {\rm SU}(2)$ WZNW theory, J. Phys. A 26 (1993), no. 23, 6981-6990. MR 1253889 (95a:82028)
  • [CT] A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, Preprint, 2006, 54 pp.; hep-th/0604128
  • [FFR] B. Feigin, E. Frenkel, and N. Reshetikhin, Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), no. 1, 27-62. MR 1309540 (96e:82012)
  • [FMTV] G. Felder, Y. Markov, V. Tarasov, and A. Varchenko, Differential equations compatible with KZ equations, Math. Phys. Anal. Geom. 3 (2000), no. 2, 139-177. MR 1797943 (2002e:17032)
  • [G1] M. Gaudin, Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), no. 10, 1089-1098. MR 0421442
  • [G2] -, La fonction d'onde de Bethe, Collection du Commissariat à l'Énergie Atomique: Sér. Sci., Masson, Paris, 1983. MR 0693905 (85h:82001)
  • [KS] P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, Lecture Notes in Phys., vol. 151, Springer, Berlin-New York, 1982, pp. 61-119. MR 0671263 (84m:81114)
  • [MTV1] E. Mukhin, V. Tarasov, and A. Varchenko, Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006, no. 8, P08002, 44 pp. MR 2249767 (2007h:82021)
  • [MTV2] -, A generalization of the Capelli identity, Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 383-398. MR 2641196.
  • [MTV3] -, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), no. 4, 909-940. MR 2525775
  • [MTV4] -, Generating operator of XXX or Gaudin transfer matrices has quasi-exponential kernel, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), 060, 31 pp. MR 2299861 (2008a:81095)
  • [MTV5] -, Spaces of quasi-exponentials and representations of $ \mathfrak{gl}_N$, J.Phys. A 41 (2008), 194017, 28 pp. MR 2452191 (2010g;17010)
  • [MTV6] -, Bethe algebra, Calogero-Moser space and Cherednik algebra, Preprint, 2009, 24 pp., arXiv:0906.5185
  • [RV] N. Reshetikhin and A. Varchenko, Quasiclassical asymptotics of solutions to the KZ equations, Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 293-322. MR 1358621 (96j:32025)
  • [SV] V. Schechtman and A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), no. 1, 139-194. MR 1123378 (93b:17067)
  • [T] D. Talalaev, Quantization of the Gaudin system, Preprint, 2004, 19 pp., hep-th/0404153
  • [W] H. Weyl, The classical groups. Their invariants and representations, Princeton Univ. Press, Princeton, NJ, 1939. MR 0000255 (1:5g)
  • [Wi] G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), 1-41. MR 1626461 (99f:58107)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 82C23, 81Q12

Retrieve articles in all journals with MSC (2010): 82C23, 81Q12

Additional Information

E. Mukhin
Affiliation: Department of Mathematical Sciences, Indiana University — Purdue University Indianapolis, 402 North Blackford St., Indianapolis, Indiana 46202-3216

V. Tarasov
Affiliation: Department of Mathematical Sciences, Indiana University — Purdue University Indianapolis, 402 North Blackford St., Indianapolis, Indiana 46202-3216
Address at time of publication: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

A. Varchenko
Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-325

Keywords: Gaudin model, Bethe algebra, Calogero–Moser space
Received by editor(s): November 15, 2009
Published electronically: March 18, 2011
Additional Notes: E. Mukhin was supported in part by NSF grant DMS-0900984
V. Tarasov was supported by NSF grant DMS-0901616
A. Varchenko was supported by NSF grant DMS-0555327
Dedicated: To Ludwig Dmitrievich Faddeev on his 75th birthday
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society