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

SIGMA 11 (2015), 091, 41 pages      arXiv:1505.07582      https://doi.org/10.3842/SIGMA.2015.091

Populations of Solutions to Cyclotomic Bethe Equations

Alexander Varchenko a and Charles A.S. Young b
a) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
b) School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK

Received June 17, 2014, in final form November 05, 2015; Published online November 14, 2015

Abstract
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain ''extended'' master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an ''extended'' non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.

Key words: Bethe equations; cyclotomic symmetry.

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References

  1. 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.
  2. Brochier A., A Kohno-Drinfeld theorem for the monodromy of cyclotomic KZ connections, Comm. Math. Phys. 311 (2012), 55-96, arXiv:1011.4285.
  3. Crampé N., Young C.A.S., Integrable models from twisted half-loop algebras, J. Phys. A: Math. Theor. 40 (2007), 5491-5509, math-ph/0609057.
  4. Enriquez B., Quasi-reflection algebras and cyclotomic associators, Selecta Math. (N.S.) 13 (2007), 391-463, math.QA/0408035.
  5. Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 27-62, hep-th/9402022.
  6. Frenkel E., Gaudin model and opers, in Infinite dimensional algebras and quantum integrable systems, Progr. Math., Vol. 237, Birkhäuser, Basel, 2005, 1-58, math.QA/0407524.
  7. Fuchs J., Schellekens B., Schweigert C., From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), 39-97, hep-th/9506135.
  8. Gaudin M., Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), 1089-1098.
  9. Gaudin M., La fonction d'onde de Bethe, Collection du Commissariat à l'Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
  10. Kac V.G., A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1977), 31-64.
  11. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  12. Kagan D., Young C.A.S., Conformal sigma models on supercoset targets, Nuclear Phys. B 745 (2006), 109-122, hep-th/0512250.
  13. Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), P08002, 44 pages, math.QA/0605015.
  14. Mukhin E., Varchenko A., Remarks on critical points of phase functions and norms of Bethe vectors, in Arrangements - Tokyo 1998, Adv. Stud. Pure Math., Vol. 27, Kinokuniya, Tokyo, 2000, 239-246, math.RT/9810087.
  15. Mukhin E., Varchenko A., Critical points of master functions and flag varieties, Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017.
  16. Mukhin E., Varchenko A., Miura opers and critical points of master functions, Cent. Eur. J. Math. 3 (2005), 155-182, math.QA/0312406.
  17. Mukhin E., Varchenko A., Quasi-polynomials and the Bethe ansatz, in Groups, Homotopy and Configuration Spaces, Geom. Topol. Monogr., Vol. 13, Geom. Topol. Publ., Coventry, 2008, 385-420, math.QA/0604048.
  18. 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, hep-th/9402126.
  19. Schechtman V.V., Varchenko A.N., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194.
  20. Scherbak I., Varchenko A., Critical points of functions, ${\mathfrak{sl}}_2$ representations, and Fuchsian differential equations with only univalued solutions, Mosc. Math. J. 3 (2003), 621-645, math.QA/0112269.
  21. Skrypnyk T., Integrable quantum spin chains, non-skew symmetric $r$-matrices and quasigraded Lie algebras, J. Geom. Phys. 57 (2006), 53-67.
  22. Skrypnyk T., ''$Z_2$-graded'' Gaudin models and analytical Bethe ansatz, Nuclear Phys. B 870 (2013), 495-529.
  23. Varagnolo M., Vasserot E., Cyclotomic double affine Hecke algebras and affine parabolic category ${\mathcal O}$, Adv. Math. 225 (2010), 1523-1588, arXiv:0810.5000.
  24. Varchenko A., Bethe ansatz for arrangements of hyperplanes and the Gaudin model, Mosc. Math. J. 6 (2006), 195-210, math.QA/0408001.
  25. Varchenko A., Quantum integrable model of an arrangement of hyperplanes, SIGMA 7 (2011), 032, 55 pages, arXiv:1001.4553.
  26. Vicedo B., Young C.A.S., Cyclotomic Gaudin models: construction and Bethe ansatz, arXiv:1409.6937.
  27. Vicedo B., Young C.A.S., Vertex Lie algebras and cyclotomic coinvariants, arXiv:1410.7664.
  28. Young C.A.S., Flat currents, ${\mathbb Z}_m$ gradings and coset space actions, Phys. Lett. B 632 (2006), 559-565.

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