Skip to main content
Log in

Direct integration and non-perturbative effects in matrix models

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We show how direct integration can be used to solve the closed amplitudes of multicut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of nonholomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an onedimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg-Witten curve and the ring reduces to the nonholomorphic modular ring of the group Γ(2). On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and nonperturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multicut case requires a new class of nonperturbative sectors in the matrix model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aganagic, V. Bouchard and A. Klemm, Topological stringsand(almost) modularforms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. G. Akemann, Higher genus correlators for the Hermitian matrix model with multiple cuts, Nucl. Phys. B 482 (1996) 403 [hep-th/9606004] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  3. M. Alim and J.D. Lange, Polynomial structure of the(open) topologicalstringpartition function, JHEP 10 (2007) 045 [arXiv:0708.2886] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. M. Alim, J.D. Lange and P. Mayr, Global properties of topological string amplitudes and orbifold invariants, JHEP 03 (2010) 113 [arXiv:0809.4253] [SPIRES].

    Article  ADS  Google Scholar 

  5. L. Alvarez-Gaumé and J.L. Mañes, Supermatrix models, Mod. Phys. Lett. A6 (1991) 2039 [SPIRES].

    ADS  Google Scholar 

  6. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. D. Bessis, A new method in the combinatorics of the topological expansion, Commun. Math. Phys. 69 (1979) 147 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. D. Bessis, C. Itzykson and J.B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1 (1980) 109 [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Bonnet, F. David and B. Eynard, Breakdown of universality in multi-cut matrix models, J. Phys. A 33 (2000) 6739 [cond-mat/0003324] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [SPIRES].

    Article  MATH  ADS  Google Scholar 

  11. P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and physicists, SpringerVerlag, Germany (1954).

    MATH  Google Scholar 

  12. F. Cachazo, K.A. Intriligator and C. Vafa, A large-N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. F. Cachazo and C. Vafa, N=1 and N=2 geometry from fluxes, hep-th/0206017 [SPIRES].

  14. F. David, Phases of the large N matrix model and nonperturbative effects in 2-D gravity, Nucl. Phys. B 348 (1991) 507 [SPIRES].

    Article  ADS  Google Scholar 

  15. F. David, Nonperturbative effects in matrix models and vacua of two-dimensional gravity, Phys. Lett. B 302 (1993) 403 [hep-th/9212106] [SPIRES].

    ADS  Google Scholar 

  16. P. Desrosiers and B. Eynard, Supermatrix models, loop equations and duality, arXiv:0911.1762 [SPIRES].

  17. P. Di Francesco, P.H. Ginsparg and J. Zinn Justin, 2 -D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [SPIRES].

    Article  ADS  Google Scholar 

  18. R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  19. R. Dijkgraaf, A. Sinkovics and M. Temurhan, Matrix models and gravitational corrections, Adv. Theor. Math. Phys. 7 (2004) 1155 [hep-th/0211241] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  20. R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. R. Dijkgraaf and C. Vafa, N=1 supersymmetry, deconstruction and bosonic gauge theories, hep-th/0302011 [SPIRES].

  22. B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, JHEP 11 (2004) 031 [hep-th/0407261] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. B. Eynard, M. Mariño and N. Orantin, Holomorphic anomaly and matrix models, JHEP 06 (2007) 058 [hep-th/0702110] [SPIRES].

    Article  ADS  Google Scholar 

  24. B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].

  25. J.D. Fay, Theta functions on Riemann surfaces, Springer-Verlag, Germany (1973).

    MATH  Google Scholar 

  26. S. Garoufalidis, A. Its, A. Kapaev and M. Mariño, Asymptotics of the instantons of Painlevé I, arXiv:1002.3634 [SPIRES].

  27. T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [SPIRES].

    Article  ADS  Google Scholar 

  28. B. Haghighat and A. Klemm, Topological strings on grassmannian Calabi-Yau manifolds, JHEP 01 (2009) 029 [arXiv:0802.2908] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. B. Haghighat and A. Klemm, Solving the topological string on K3 fibrations, JHEP 01 (2010) 009 [arXiv:0908.0336] [SPIRES].

    Article  ADS  Google Scholar 

  30. B. Haghighat, A. Klemm and M. Rauch, Integrability of the holomorphic anomaly equations, JHEP 10 (2008) 097 [arXiv:0809.1674] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. M.x. Huang and A. Klemm, Holomorphic anomaly in gauge theories and matrix models, JHEP 09 (2007) 054 [hep-th/0605195] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  32. M.x. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [SPIRES].

    Article  ADS  Google Scholar 

  33. M.x. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  34. S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  35. A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [SPIRES].

    Article  ADS  Google Scholar 

  36. A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. H. Klingen, Introductory lectures on Siegel modular forms, Cambridge University Press, Cambridge U.K. (1990).

    Book  MATH  Google Scholar 

  38. J.C. Le Guillou and J. Zinn-Justin, Large order behavior of perturbation theory, NorthHolland, Amsterdam The Netherlands (1990).

    Google Scholar 

  39. M. Mariño, Les Houches lectures on matrix models and topological strings, hep-th/0410165 [SPIRES].

  40. M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [SPIRES].

    Article  ADS  Google Scholar 

  41. M. Mariño, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, arXiv:0711.1954 [SPIRES].

  42. M. Mariño, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009) 052301 [arXiv:0809.2619] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  43. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].

    MathSciNet  Google Scholar 

  44. T. Okuda and T. Takayanagi, Ghost D-branes, JHEP 03 (2006) 062 [hep-th/0601024] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  45. H. Ooguri and C. Vafa, Gravity induced C-deformation, Adv. Theor. Math. Phys. 7 (2004) 405 [hep-th/0303063] [SPIRES].

    MathSciNet  Google Scholar 

  46. H. Ooguri and C. Vafa, The C-deformation of gluino and non-planar diagrams, Adv. Theor. Math. Phys. 7 (2003) 53 [hep-th/0302109] [SPIRES].

    MathSciNet  Google Scholar 

  47. S. Pasquetti and R. Schiappa, Borel and stokes nonperturbative phenomena in topological string theory and c=1 matrix models, Annales Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [SPIRES].

    Article  ADS  Google Scholar 

  48. T.M. Seara and D. Sauzin, Ressumació de Borel i teoria de la ressurgència, Butl. Soc. Catalana Mat. 18 (2003) 131.

    MathSciNet  Google Scholar 

  49. N. Seiberg and E. Witten, Monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  50. S.H. Shenker, The strength of nonperturbative effects in string theory, in Random surfaces and quantum gravity, O. Álvarez, E. Marinari and P. Windey eds., Plenum Press, New York U.S.A. (1992).

  51. P. Sulkowski, Matrix models for 2* theories, Phys. Rev. D 80 (2009) 086006 [arXiv:0904.3064] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  52. C. Vafa, Brane/anti-brane systems and U(NM) supergroup, hep-th/0101218 [SPIRES].

  53. S. Yamaguchi and S.T. Yau, Topological string partition functions as polynomials, JHEP 07 (2004) 047 [hep-th/0406078] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. S.A. Yost, Supermatrix models, Int. J. Mod. Phys. A7 (1992) 6105 [hep-th/9111033] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  55. D. Zagier, Elliptic modular forms and their applications, in The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway, J.H. Bruinier, G. Van Der Geer and G. Harder, Springer, Heidelberg Germany (2008).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marcos Mariño.

Additional information

ArXiv ePrint: 1002.3846

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klemm, A., Mariño, M. & Rauch, M. Direct integration and non-perturbative effects in matrix models. J. High Energ. Phys. 2010, 4 (2010). https://doi.org/10.1007/JHEP10(2010)004

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP10(2010)004

Keywords

Navigation