Skip to main content

2-D Quantum and Topological Gravities, Matrix Models, and Integrable Differential Systems

  • Chapter
The Painlevé Property

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

These lecture notes are partly based on the review [1] on 2-D quantum gravity and random matrix models. The subdivision into two parts corresponds to two very different approaches to the quantization of 2-D gravity, on the one hand through a discretization of the problem (Part A), on the other hand through a mathematical formulation thereof, as intersection theory on the moduli space of Riemann surfaces (Part B).

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, 2-D quantum gravity and random matrix models, Phys. Rep. 254 (1995), 1–131.

    Article  ADS  Google Scholar 

  2. A. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B103 (1981), 207–210; Quantum geometry of fermionic strings, 211–213.

    MathSciNet  ADS  Google Scholar 

  3. V. Knizhnik, A. Polyakov, and A. Zamolodchikov, Fractal structure of 2-D quantum gravity, Mod. Phys. Lett. A3 (1988) 819; F. David, Conformai field theories coupled to 2-D gravity in the conformai gauge, Mod. Phys. Lett. A3 (1988), 1651; J. Distler and H. Kawai, Conformai field theory and 2-D quantum gravity or who’s afraid of Joseph Liouville? Nucl. Phys. B321 (1989), 509.

    Google Scholar 

  4. L. Onsager, Crystal statistics. 1. A two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944) 117–149.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformai Field Theory (Springer, New York, 1996).

    Google Scholar 

  6. C. Itzykson, H. Saleur, and J.-B. Zuber (eds.), Conformai Invariance and Applications to Statistical Mechanics, (World Scientific, 1988).

    Google Scholar 

  7. E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978), 35–51.

    Article  ADS  MATH  Google Scholar 

  8. D. Bessis, A new method in the combinatorics of the topological expansion, Comm. Math. Phys. 69 (1979), 147–163.

    Article  MathSciNet  ADS  Google Scholar 

  9. D. Bessis, C. Itzykson, and J.-B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), 109–157.

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Itzykson and J.-B. Zuber, The planar approximation. 2., J. Math. Phys. 21 (1980), 411; Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. S. Chadha, G. Mahoux, and M.-L. Mehta, A method of integration over matrix variables. 2., J. Phys. A14 (1981), 579.

    MathSciNet  ADS  Google Scholar 

  12. M. Staudacher, The Yang-Lee edge singularity on a dynamical planar random surface, Nucl. Phys. B336 (1990), 349.

    Article  MathSciNet  ADS  Google Scholar 

  13. V. Kazakov, Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A119 (1986), 140–144.

    MathSciNet  ADS  Google Scholar 

  14. C. Cernkovic, P. Ginsparg, and G. Moore, The Ising model, the Yang-Lee edge singularity, and 2-D quantum gravity, Phys. Lett. B237 (1990), 196.

    ADS  Google Scholar 

  15. E. Brézin and V. Kazakov, Exactly solvable field theories of closed strings, Phys. Lett. B236 (1990), 144–150; M. Douglas and S. Shenker, Strings in less than one dimension, Nucl. Phys. B335 (1990), 635; D. Gross and A. Migdal, Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127; A nonperturbative treatment of two-dimensional quantum gravity, Nucl. Phys. B340 (1990), 333-365.

    ADS  Google Scholar 

  16. M. Douglas, Strings in less than one dimension and generalized KP hierarchies, Phys. Lett. B238 (1990), 176.

    ADS  Google Scholar 

  17. I. Gelfand and L. Dikii, Fractional powers of operators, and Hamiltonian systems, Funct. Anal. Appl. 10:4 (1976), 13 and The resolvent, and Hamiltonian systems, Funct. Anal. Appl. 11:2 (1977), 93.

    MathSciNet  Google Scholar 

  18. A. Cappelli, C. Itzykson, and J.-B. Zuber, The ADE classification of minimal and A(1) conformai invariant theories, Comm. Math. Phys. 113 (1987), 1–26.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. P. Di Francesco and D. Kutasov, Unitary minimal models coupled to gravity, Nucl. Phys. B342 (1990), 589 and Integrable models of 2-D quantum gravity, in Random Surfaces and Quantum Gravity, NATO ASI Series B262 (1991), 35-51.

    Article  ADS  Google Scholar 

  20. V. Drinfcld and V. Sokolov, Lie algebras and equations of Korteweg-De Vries type, J. Sov. Math. 30 (1985) 1975; Sov. Math. Dokl. 23 No.3 (1981), 457.

    Google Scholar 

  21. E. Witten, On the structure of the topological phase of two-dimensional gravity, Nucl. Phys. B340 (1990), 281–332 and Two-dimensional gravity and and intersection theory on moduli space, Surv. in Diff. Geom. 1 (1991), 243-310.

    Article  MathSciNet  ADS  Google Scholar 

  22. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1–23.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. C. Itzykson and J.-B. Zuber, Combinatorics of the modular group. 2. The Kontsevich integrals, Int. J. Mod. Phys. A7 (1992), 5661–5705.

    MathSciNet  ADS  Google Scholar 

  24. P. Di Francesco, C. Itzykson, and J.-B. Zuber, Polynomial averages in the Kontsevich model, Comm. Math. Phys. 151 (1993), 193–219.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum coho-mology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. T. Eguchi and S.-K. Yang, The topological CP1 model and the large N matrix integral, Mod. Phys. Lett. A9 (1994), 2893–2902.

    MathSciNet  ADS  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Francesco, P.D. (1999). 2-D Quantum and Topological Gravities, Matrix Models, and Integrable Differential Systems. In: Conte, R. (eds) The Painlevé Property. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1532-5_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1532-5_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98888-7

  • Online ISBN: 978-1-4612-1532-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics