Advertisement

Communications in Mathematical Physics

, Volume 316, Issue 1, pp 1–44 | Cite as

Characteristic Classes and Hitchin Systems. General Construction

  • A. Levin
  • M. Olshanetsky
  • A. Smirnov
  • A. ZotovEmail author
Article

Abstract

We consider topologically non-trivial Higgs G-bundles over Riemann surfaces Σ g with marked points and the corresponding Hitchin systems. We show that if G is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of the underlying bundles defined as elements of \({H^{2}(\Sigma_g, \mathcal{Z}(G))}\) , (\({\mathcal{Z}(G)}\) is a center of G). We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. We describe explicitly bundles in the genus one (g =  1) case. If Σ1 has one marked point and the bundles are trivial then the Hitchin systems coincide with Calogero-Moser (CM) systems. For the nontrivial bundles we call the corresponding systems the modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but less number of particles and greater number of spin variables. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, and define the phase spaces and the Poisson structure using dynamical r-matrices. The latter are completion of the classification list of Etingof-Varchenko corresponding to the trivial bundles. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of ’t Hooft matrices for sl(N). We find that the MCM systems contain the standard CM subsystems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the stable holomorphic bundles with non-trivial characteristic classes.

Keywords

Modulus Space Cartan Subalgebra Cartan Subgroup Coadjoint Orbit Higgs Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah M.: Vector bundles over an elliptic curve. Proc. London Math. Soc. 7, 414–452 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, V.: Mathematical Methods in Classical Mechanics. Berlin-Heidelberg-NewYork: Springer, 1978Google Scholar
  3. 3.
    Avan J., Talon M.: Classical R-matrix structure for the Calogero model. Phys. Lett. B 303, 33–37 (1993)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Babelon O., Viallet C-M.: Hamiltonian structures and Lax equations. Phys. Lett. B 237, 411 (1990)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Belavin A., Drinfeld V.: Solutions of the classical Yang - Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16(N 3), 159–180 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bernstein, J., Schwarzman, O.: Chevalley’s theorem for complex crystallographic Coxeter groups. (Russian) Funkt. Anal. i Prilo. 12(4), 79–80 (1978); Complex crystallographic Coxeter groups and affine root systems. J. Nonlinear Math. Phys. 13(2), 163–182 (2006)Google Scholar
  7. 7.
    Billey E., Avan J., Babelon O.: The r-matrix structure of the Euler-Calogero-Moser model. Phys. Lett. A 186, 114–118 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Bordner A., Corrigan E., Sasaki R.: Calogero-Moser models: I. A new formulation. Progr. Theor. Phys. 100, 1107–1129 (1998)MathSciNetADSGoogle Scholar
  9. 9.
    Bourbaki, N.: Lie Groups and Lie Algebras: Chapters 4–6. Berlin-Heidelberg-New York: Springer-Verlag, 2002Google Scholar
  10. 10.
    Braden H., Suzuki T.: R-matrices for Elliptic Calogero-Moser Models. Lett. Math. Phys. 30, 147–158 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Calogero, F.: Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971); Exactly solvable one-dimensional many-body problem. Lett. Nuovo Cim. 13, 411 (1975)Google Scholar
  12. 12.
    Dirac, P.: Lectures on quantum mechanics. Yeshiva Univ., NY: Academic Press, 1967Google Scholar
  13. 13.
    Braden H.W., Dolgushev V.A., Olshanetsky M.A., Zotov A.V.: Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions. J. Phys. A 36, 6979–7000 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Enriquez B., Rubtsov V.: Hitchin systems, higher Gaudin operators and R-matrices. Math. Res. Lett. 3, 343–357 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Etingof, P.: Lectures on Calogero-Moser systems. http://arxiv.org/abs/math/0606233v4 [math.QA], 2009
  16. 16.
    Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations, http://arxiv.org/abs/math/9908064v2 [math.QA], 2000
  17. 17.
    Etingof P., Varchenko A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. 192, 77–120 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Fairlie D., Fletcher P., Zachos C.: Infinite Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras. J. Math. Phys. 31, 1088–1094 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Feher L.: Poisson-Lie dynamical r-matrices from Dirac reduction. Czech. J. Phys. 54, 1265–1274 (2004)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Felder, G.: Conformal field theory and integrable systems associated with elliptic curves, Proc. of the ICM 94, Basel: Birkhaeuser, 1994, pp. 1247–1255Google Scholar
  21. 21.
    Felder G., Gawedzki K., Kupiainen A.: Spectra Of Wess-Zumino-Witten Models With Arbitrary Simple Groups. Commun. Math. Phys. 117, 127–158 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Friedman, R., Morgan, J.: Holomorphic principal bundles over elliptic curves, http://arxiv.org/abs/math/9811130v1 [math.AG], 1998 R. Friedman, J. Morgan, Holomorphic Principal Bundles Over Elliptic Curves II: The Parabolic Construction, http://arxiv.org/abs/math/0006174v2 [math.AG], 2001
  23. 23.
    Friedman R., Morgan J., Witten E.: Principal G-bundles over elliptic curves. Math. Res. Lett. 5, 97–118 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gibbons J., Hermsen T.: A generalization of the Calogero-Moser systems. Physica 11D, 337–348 (1984)MathSciNetADSGoogle Scholar
  25. 25.
    Gordeev N., Popov V.: Automorphism groups of finite dimensional simple algebras. Ann. of Math. 158, 1041–1065 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gorsky, A., Nekrasov, N.: Elliptic Calogero-Moser system from two dimensional current algebra. http://arxiv.org/abs/hep-th/9401021v1, 1994
  27. 27.
    Hitchin N.: Stable bundles and integrable systems. Duke Math. Jour. 54, 91–114 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    D’Hoker E., Phong D.H.: Calogero-Moser Lax pairs with spectral parameter for general Lie algebras. Nuclear Phys. B 530, 537–610 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Hurtubise J., Markman E.: Calogero-Moser systems and Hitchin systems. Commun. Math. Phys. 223, 533–552 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Kac V.: Automorphisms of finite order of semisimple Lie algebras. Funct.Anal. and Applic. 3, 94–96 (1969)Google Scholar
  31. 31.
    Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. http://arxiv.org/abs/hep-th/0604151v3, 2007
  32. 32.
    Kazdan D., Kostant B., Sternberg S.: Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math. 31, 481–507 (1978)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Khesin B., Levin A., Olshanetsky M.: Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus. Comm. Math. Phys. 250, 581–612 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    : Elliptic solutions of the KP equation and integrable systems of particles. Funct. Anal. Applic. 14(N4), 45–54 (1980)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Levin A., Olshanetsky M., Zotov A.: Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version. Comm. Math. Phys. 236, 93–133 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Levin A., Olshanetsky M.: Isomonodromic deformations and Hitchin Systems. Amer. Math. Soc. Transl. (2) 191, 223–262 (1999)MathSciNetGoogle Scholar
  37. 37.
    Levin A., Olshanetsky M., Zotov A.: Painlevé VI, rigid tops and reflection equation. Commun. Math. Phys. 268, 67–103 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Chernyakov Yu., Levin A., Olshanetsky M., Zotov A.: Elliptic Schlesinger system and Painlevé VI. J. Phys. A 39, 12083–120102 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Zotov A.V.: 1+1 Gaudin Model. SIGMA 7, 067 (2011)MathSciNetGoogle Scholar
  40. 40.
    Levin A., Olshanetsky M., Zotov A.: Monopoles and modifications of bundles over elliptic curves. SIGMA 5, 065 (2009)MathSciNetGoogle Scholar
  41. 41.
    Levin, A.M., Olshanetsky, M.A., Smirnov, A.V., Zotov, A.V.: Characteristic Classes of SL (\({N, \mathbb{C}}\))-Bundles and Quantum Dynamical Elliptic R-Matrices, http://arxiv.org/abs/1208.5750v1 [math.ph], 2012
  42. 42.
    Levin, A.M., Olshanetsky, M.A., Smirnov, A.V., Zotov, A.V.: Hecke transformations of conformal blocks in WZW theory. I. KZB equations for non-trivial bundles. http://arxiv.org/abs/1207.4386v1 [math.ph], 2012
  43. 43.
    Levin A., Zotov A.: Integrable systems of interacting elliptic tops. Theor. Math. Phys. 146(1), 55–64 (2006)MathSciNetGoogle Scholar
  44. 44.
    Looijenga E.: Root systems and elliptic curves. Invent. Math. 38, 17–32 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Li L.-C., Xu P.: Integrable spin Calogero-Moser systems. Commun. Math. Phys. 231, 257–286 (2002)ADSzbMATHCrossRefGoogle Scholar
  46. 46.
    Markman E.: Spectral curves and integrable systems. Comp. Math. 93, 255–290 (1994)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Moser J.: Three integrable systems connected with isospectral deformations. Adv. Math. 16, 1–23 (1975)ADSCrossRefGoogle Scholar
  48. 48.
    Levin, A.M., Olshanetsky, M.A., Smirnov, A.V., Zotov, A.V.: Characteristic Classes and Integrable Systems. General Construction, http://arxiv.org/abs/1006.0702v4 [math.ph], 2010
  49. 49.
    Levin, A.M., Olshanetsky, M.A., Smirnov, A.V., Zotov, A.V.: Characteristic Classes and Integrable Systems for Simple Lie Groups, http://arxiv.org/abs/1007.4127v2 [math.ph], 2010
  50. 50.
    Levin, A., Olshanetsky, M., Smirnov, A., Zotov, A.: Calogero-Moser systems for simple Lie groups and characteristic classes of bundles. J. Geom. Phys. 62, 1810–1850 (2012)Google Scholar
  51. 51.
    Mumford, D.: Tata Lectures on Theta I, II, Boston, MA: Birkhäuser Boston, 1983, 1984Google Scholar
  52. 52.
    Narasimhan M.S., Seshadri C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82, 540–64 (1965)Google Scholar
  53. 53.
    Nekrasov N.: Holomorphic Bundles and Many-Body Systems. Commun. Math. Phys. 180, 587–604 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  54. 54.
    Olshanetsky M., Perelomov A.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981)MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    Olshanetsky M., Perelomov A.: Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric space of zero curvature. Lett. Nuovo Cim. 16, 333–339 (1976)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Onishchik, A., Vinberg, E.: Seminar on Lie groups and algebraic groups, Moscow (1988), (in Russian) English transl. Berlin-Heidelberg-New York: Springer-Verlag, 1990Google Scholar
  57. 57.
    Reyman, A., Semenov-Tian-Schansky, M.: Lie algebras and Lax equations with spectral parameter on elliptic curve, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 150 (1986), Voprosy Kvant. Teor. Polya i Statist. Fiz. 6, 104–118, 221; translation in J. Soviet Math. 46(1), 1631–1640 (1989)Google Scholar
  58. 58.
    Ramanathan A.: Stable principal bundles on a compact Riemann surface. Math. Ann. 213, 129–152 (1998)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Schiffmann O.: On classification of dynamical r-matrices. Math. Res. Lett. 5, 13–30 (1998)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Schweigert C.: On moduli spaces of flat connections with non-simply connected structure group. Nucl. Phys. B 492, 743–755 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  61. 61.
    Simpson C.: Harmonic bundles on Noncompact Curves. J. Am. Math. Soc. 3, 713–770 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Sklyanin, E.: Dynamical r-matrices for the Elliptic Calogero-Moser Model. Alg. Anal. 6, 227–237 (1994); St.Petersburg Math. J. 6, 397–406 (1995)Google Scholar
  63. 63.
    Weyl, A.: Elliptic functions according to Eisenstein and Kronecker. Berlin-Heidelberg-New York: Springer-Verlag, 1976Google Scholar
  64. 64.
    Wojciechowski S.: An integrable marriage of the Euler equations with the Calogero-Moser systems. Phys. Lett. A 111, 101–103 (1985)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • A. Levin
    • 1
    • 2
    • 3
  • M. Olshanetsky
    • 1
    • 3
  • A. Smirnov
    • 1
    • 4
  • A. Zotov
    • 1
    Email author
  1. 1.Institute of Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Laboratory of Algebraic GeometryGU-HSEMoscowRussia
  3. 3.Max Planck Institute for MathematicsBonnGermany
  4. 4.Math. DeptColumbia UniversityNew YorkUSA

Personalised recommendations