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


SIGMA 8 (2012), 100, 53 pages      arXiv:1110.4936      http://dx.doi.org/10.3842/SIGMA.2012.100

Geometry of Spectral Curves and All Order Dispersive Integrable System

Gaëtan Borot a and Bertrand Eynard b, c
a) Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
b) Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
c) Centre de Recherche Mathématiques de Montréal, Université de Montréal, P.O. Box 6128, Montréal (Québec) H3C 3J7, Canada

Received November 14, 2011, in final form December 11, 2012; Published online December 18, 2012

Abstract
We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between ''correlators'', the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Key words: topological recursion; Tau function; Sato formula; Hirota equations; Whitham equations.

pdf (742 kb)   tex (71 kb)

References

  1. Adler M., van Moerbeke P., Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J. 80 (1995), 863-911, solv-int/9706010.
  2. Akemann G., Pottier A., Ratios of characteristic polynomials in complex matrix models, J. Phys. A: Math. Gen. 37 (2004), L453-L459, math-ph/0404068.
  3. Albeverio S., Pastur L., Shcherbina M., On the 1/n expansion for some unitary invariant ensembles of random matrices, Comm. Math. Phys. 224 (2001), 271-305.
  4. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  5. Bergère M., Biorthogonal polynomials for potentials of two variables and external sources at the denominator, hep-th/0404126.
  6. Bergère M., Eynard B., Universal scaling limits of matrix models and (p,q) Liouville gravity, arXiv:0909.0854.
  7. Bertola M., Boutroux curves with external field: equilibrium measures without a minimization problem, Anal. Math. Phys. 1 (2011), 167-211, arXiv:0705.3062.
  8. Bertola M., Eynard B., Harnad J., Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem, Comm. Math. Phys. 243 (2003), 193-240, nlin.SI/0208002.
  9. Bertola M., Eynard B., Harnad J., Duality, biorthogonal polynomials and multi-matrix models, Comm. Math. Phys. 229 (2002), 73-120, nlin.SI/0108049.
  10. Bertola M., Eynard B., Harnad J., Partition functions for matrix models and isomonodromic tau functions, J. Phys. A: Math. Gen. 36 (2003), 3067-3083, nlin.SI/0204054.
  11. Bertola M., Eynard B., Harnad J., Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions, Comm. Math. Phys. 263 (2006), 401-437, nlin.SI/0410043.
  12. Bertola M., Eynard B., Harnad J., The duality of spectral curves that arises in two-matrix models, Theoret. Math. Phys. 134 (2003), 27-38, nlin.SI/0112006.
  13. Bertola M., Gekhtman M., Effective inverse spectral problem for rational Lax matrices and applications, Int. Math. Res. Not. 2007 (2007), no. 23, rnm103, 39 pages, arXiv:0705.0120.
  14. Bertola M., Marchal O., The partition function of the two-matrix model as an isomonodromic τ function, J. Math. Phys. 50 (2009), 013529, 17 pages, arXiv:0809.3367.
  15. Bertola M., Mo M.Y., Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights, Adv. Math. 220 (2009), 154-218, math-ph/0605043.
  16. Bleher P., Its A., Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), 185-266, math-ph/9907025.
  17. Bonnet G., David F., Eynard B., Breakdown of universality in multi-cut matrix models, J. Phys. A: Math. Gen. 33 (2000), 6739-6768, cond-mat/0003324.
  18. Borot G., Eynard B., All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, arXiv:1205.2261.
  19. Borot G., Guionnet A., Asymptotic expansion of β matrix models in the one-cut regime, Comm. Math. Phys., to appear, arXiv:1107.1167.
  20. Buchstaber V.M., Krichever I.M., Integrable equations, addition theorems and the Riemann-Schottky problem, Russian Math. Surveys 61 (2006), 19-78.
  21. Chekhov L., Eynard B., Marchal O., Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach, Theoret. Math. Phys. 166 (2011), 141-185, arXiv:1009.6007.
  22. Chekhov L., Eynard B., Marchal O., Topological expansion of the Bethe ansatz, and quantum algebraic geometry, arXiv:0911.1664.
  23. Chen Y., Ismail M.E.H., Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen. 30 (1997), 7817-7829.
  24. Deift P., Kriecherbauer T., McLaughlin K.D. T-R, Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
  25. Deift P., Venakides S., Zhou X., New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Not. 1997 (1997), no. 6, 285-299, arXiv:0705.0120.
  26. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), 295-368, math.AP/9201261.
  27. Dijkgraaf R., Vafa C., On geometry and matrix models, Nuclear Phys. B 644 (2002), 21-39, hep-th/0207106.
  28. Douglas M.R., Shenker S.H., Strings in less than one dimension, Nuclear Phys. B 335 (1990), 635-654.
  29. Dubrovin B.A., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  30. Dubrovin B.A., Integrable systems and classification of 2-dimensional topological field theories, in Integrable Systems (Luminy, 1991), Progr. Math., Vol. 115, Birkhäuser Boston, Boston, MA, 1993, 313-359, hep-th/9209040.
  31. Dubrovin B.A., Theta functions and non-linear equations, Russian Math. Surveys 36 (1981), no. 2, 11-92.
  32. Dubrovin B.A., Krichever I.M., Novikov S.P., Integrable systems. I, in Dynamical systems, IV, Encyclopaedia Math. Sci., Vol. 4, Springer, Berlin, 2001, 177-332.
  33. Dubrovin B.A., Zhang Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, math.DG/0108160.
  34. Ercolani N., McLaughlin K.D. T-R, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration, Int. Math. Res. Not. 2003 (2003), no. 14, 755-820, math-ph/0211022.
  35. Eynard B., Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence, J. High Energy Phys. 2009 (2009), no. 3, 003, 20 pages, arXiv:0802.1788.
  36. Eynard B., Topological expansion for the 1-Hermitian matrix model correlation functions, J. High Energy Phys. 2004 (2004), no. 11, 031, 35 pages, hep-th/0407261.
  37. Eynard B., Mariño M., A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011), 1181-1202, arXiv:0810.4273.
  38. Eynard B., Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452, math-ph/0702045.
  39. Eynard B., Orantin N., Topological expansion of mixed correlations in the Hermitian 2-matrix model and x-y symmetry of the Fg algebraic invariants, J. Phys. A: Math. Theor. 41 (2008), 015203, 28 pages, arXiv:0705.0958.
  40. Farkas H.M., Kra I., Riemann surfaces, Graduate Texts in Mathematics, Vol. 71, 2nd ed., Springer-Verlag, New York, 1992.
  41. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin, 1973.
  42. Flaschka H., Forest M.G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33 (1980), 739-784.
  43. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, American Mathematical Society, Providence, RI, 2006.
  44. Fyodorov Y.V., Strahov E., An exact formula for general spectral correlation function of random Hermitian matrices, J. Phys. A: Math. Gen. 36 (2003), 3203-3213, math-ph/0204051.
  45. Gurevich A., Pitaevskii L., Nonstationary structure of a collisionless shock wave, Soviet Phys. JEPT 38 (1974), 291-297.
  46. Harnad J., Tracy C.A., Widom H., Hamiltonian structure of equations appearing in random matrices, in Low-Dimensional Topology and Quantum Field Theory (Cambridge, 1992), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 315, Editor H. Osborn, Plenum, New York, 1993, 231-245, hep-th/9301051.
  47. Hirota R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192-1194.
  48. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  49. Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A., Differential equations for quantum correlation functions, Internat. J. Modern Phys. B 4 (1990), 1003-1037.
  50. Its A.R., Kitaev A.V., Fokas A.S., An isomonodromic approach in the theory of two-dimensional quantum gravity, Russian Math. Surveys 45 (1990), no. 6, 155-157.
  51. Its A.R., Matveev V.B., Hill's operator with finitely many gaps, Funct. Anal. Appl. 9 (1975), 65-66.
  52. Its A.R., Matveev V.B., Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation, Theoret. Math. Phys. 23 (1975), 343-355.
  53. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  54. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981), 26-46.
  55. Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
  56. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Phys. D 2 (1981), 306-352.
  57. Kharchev S., Marshakov A., Mironov A., Morozov A., Zabrodin A., Unification of all string models with c<1, Phys. Lett. B 275 (1992), 311-314, hep-th/9111037.
  58. Kokotov A., Korotkin D., Invariant Wirtinger projective connection and tau-functions on spaces of branched coverings, in Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, Vol. 37, Amer. Math. Soc., Providence, RI, 2004, 91-97, math-ph/0310008.
  59. Korotkin D., Matrix Riemann-Hilbert problems related to branched coverings of CP1, in Factorization and Integrable Systems (Faro, 2000), Oper. Theory Adv. Appl., Vol. 141, Birkhäuser, Basel, 2003, 103-129, math-ph/0106009.
  60. Korotkin D., Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann. 329 (2004), 335-364, math-ph/0306061.
  61. Kostov I., Bilinear functional equations in 2d quantum gravity, hep-th/9602117.
  62. Krichever I.M., Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977), 12-26.
  63. Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys 32 (1977), no. 6, 185-213.
  64. Krichever I.M., Perturbation theory in periodic problems for two-dimensional integrable systems, Sov. Sci. Rev. Sect. C 9 (1992), no. 2, 1-103.
  65. Krichever I.M., The τ-function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994), 437-475, hep-th/9205110.
  66. Lax P., Levermore C., The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), 253-290.
  67. Lax P., Levermore C., The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math. 36 (1983), 571-593.
  68. Lax P., Levermore C., The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math. 36 (1983), 809-830.
  69. Mehta M.L., Random matrices, Pure and Applied Mathematics (Amsterdam), Vol. 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004.
  70. Moore G., Geometry of the string equations, Comm. Math. Phys. 133 (1990), 261-304.
  71. Mumford D., Tata lectures on theta. I, Progress in Mathematics, Vol. 28, Birkhäuser Boston Inc., Boston, MA, 1983.
  72. Mumford D., Tata lectures on theta. II, Progress in Mathematics, Vol. 43, Birkhäuser Boston Inc., Boston, MA, 1984.
  73. Mumford D., Tata lectures on theta. III, Progress in Mathematics, Vol. 97, Birkhäuser Boston Inc., Boston, MA, 1991.
  74. Rauch H.E., Weierstrass points, branch points, and moduli of Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 543-560.
  75. Sato M., Sato Y., Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., Vol. 81, North-Holland, Amsterdam, 1983, 259-271.
  76. Segal G.B., Loop groups, in Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., Vol. 1111, Springer, Berlin, 1985, 155-168.
  77. Shabat A.B., Zakharov V.E., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 11 (1977), 226-235.
  78. van Moerbeke P., The spectrum of random matrices and integrable systems, in in Group21, Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras, Vol. II, World Scientific, Singapore, 1997, 835-852, solv-int/9706009.
  79. Venakides S., The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38 (1985), 883-909.
  80. Venakides S., The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990), 335-361.
  81. Whitham G.B., Linear and nonlinear waves, Pure and Applied Mathematics, Wiley Interscience, New York, 1974.

Previous article  Next article   Contents of Volume 8 (2012)