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Matrix models, complex geometry, and integrable systems: I

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Abstract

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We recall the general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of semiclassical integrable systems solved by constructing tau functions or prepotentials. We discuss the complex curves and tau functions of one-and two-matrix models in detail.

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References

  1. A. M. Polyakov, Gauge Fields and Strings (Contemporary Concepts in Physics, Vol. 3), Harwood, Chur, Switzerland (1987).

    Google Scholar 

  2. J. Polchinski, Phys. Rev. Lett., 75, 4724 (1995); hep-th/9510017 (1995); Progr. Theoret. Phys. (Suppl.), 123, 9 (1996); hep-th/9511157 (1995); Rev. Modern Phys., 68, 1245 (1996); hep-th/9607050 (1996); “TASI lectures on D-branes,” hep-th/9611050 (1996).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. J. Polchinski, String Theory, Vols. 1, 2, Cambridge Univ. Press, Cambridge (1998).

    Google Scholar 

  4. J. M. Maldacena, “TASI 2003 lectures on AdS/CFT,” hep-th/0309246 (2003).

  5. A. V. Marshakov, Phys. Usp., 45, 915 (2002); hep-th/0212114 (2002).

    Article  Google Scholar 

  6. G. ’t Hooft, Nucl. Phys. B, 72, 461 (1974).

    Article  ADS  Google Scholar 

  7. E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Comm. Math. Phys., 59, 35 (1978); F. David, Nucl. Phys. B, 257[FS14], 45 (1985); V. Kazakov, Phys. Lett. B, 150, 282 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  8. E. Witten, Comm. Math. Phys., 252, 189 (2004); hep-th/0312171 (2003); Adv. Theor. Math. Phys., 8, 779 (2004); hep-th/0403199 (2004); D. Polyakov, Phys. Lett. B, 611, 173 (2005); hep-th/0501219 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  9. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  10. S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, and A. Zabrodin, Nucl. Phys. B, 366, 569 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  11. T. Miwa, Proc. Japan Acad. A, 58, 9 (1982).

    MATH  MathSciNet  Google Scholar 

  12. S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 397, 339 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  13. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Phys. Lett. B, 275, 311 (1992); hep-th/9111037 (1991); Nucl. Phys. B, 380, 181 (1992); hep-th/9201013 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  14. M. Kontsevich, Funct. Anal. Appl., 25, 123 (1991); Comm. Math. Phys., 147, 1 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Fukuma, H. Kawai, and R. Nakayama, Internat. J. Mod. Phys. A, 6, 1385 (1991); Comm. Math. Phys., 143, 371 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  16. S. Kharchev and A. Marshakov, “Topological versus non-topological theories and p-q duality in c≤1 2d gravity models,” in: String Theory, Quantum Gravity, and the Unification of the Fundamental Interactions (Proc. Intl. Workshop on String Theory, Quantum Gravity, and the Unification of Fundamental Interactions, Rome, Italy, 1992, M. Bianchi, F. Fucito, E. Marinari, and A. Sagnotti, eds.), World Scientific, Singapore (1993), p. 331; hep-th/9210072 (1992); Internat. J. Mod. Phys. A, 10, 1219 (1995); hep-th/9303100 (1993).

    Google Scholar 

  17. A. A. Migdal, Phys. Rep., 102, 199 (1983).

    Article  ADS  Google Scholar 

  18. F. David, Phys. Lett. B, 302, 403 (1993); hep-th/9212106 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  19. G. Bonnet, F. David, and B. Eynard, J. Phys. A, 33, 6739 (2000); cond-mat/0003324 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  20. R. Dijkgraaf and C. Vafa, Nucl. Phys. B, 644, 3 (2002); hep-th/0206255 (2002); Nucl. Phys. B, 644, 21 (2002); hep-th/0207106 (2002); “A perturbative window into non-perturbative physics,” hep-th/0208048 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  21. G. Felder and R. Riser, Nucl. Phys. B, 691, 251 (2004); hep-th/0401191 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  22. I. Krichever, Comm. Pure. Appl. Math., 47, 437 (1992); hep-th/9205110 (1992).

    MathSciNet  Google Scholar 

  23. J. Fay, Theta-Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973).

    Google Scholar 

  24. B. de Wit and A. V. Marshakov, Theor. Math. Phys., 129, 1504 (2001); hep-th/0105289 (2001).

    Article  Google Scholar 

  25. L. Chekhov and A. Mironov, Phys. Lett. B, 552, 293 (2003); hep-th/0209085 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  26. L. Chekhov, A. Marshakov, A. Mironov, and D. Vasiliev, Phys. Lett. B, 562, 323 (2003); hep-th/0301071 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  27. V. A. Kazakov, A. Marshakov, J. A. Minahan, and K. Zarembo, JHEP, 0405, 024 (2004); hep-th/0402207 (2004); A. V. Marshakov, Theor. Math. Phys., 142, 222 (2005); hep-th/0406056 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  28. M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, Phys. Rev. Lett., 84, 5106 (2000); nlin.SI/0001007 (2000); I. K. Kostov, I. Krichever, M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, “ τ-Function for analytic curves,” in: Random Matrices and Their Applications (Math. Sci. Res. Inst. Publ., Vol. 40, P. Bleher and A. Its, eds.), Cambridge Univ. Press, Cambridge (2001), p. 285; hep-th/0005259 (2000); A. Zabrodin, “Matrix models and growth processes: From viscous flows to the quantum Hall effect,” hep-th/0412219 (2004).

    Article  ADS  Google Scholar 

  29. V. A. Kazakov and A. Marshakov, J. Phys. A, 36, 3107 (2003); hep-th/0211236 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  30. B. Eynard, JHEP, 0301, 051 (2003); hep-th/0210047 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  31. I. Krichever, A. Marshakov, and A. Zabrodin, Comm. Math. Phys., 259, 1 (2005); hep-th/0309010 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  32. A. V. Marshakov, “Matrix models, complex geometry, and integrable systems: II,” Theor. Math. Phys. (to appear); hep-th/0601214 (2006).

  33. A. Marshakov, P. Wiegmann, and A. Zabrodin, Comm. Math. Phys., 227, 131 (2002); hep-th/0109048 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  34. L. A. Takhtajan, Lett. Math. Phys., 56, 181 (2001); math.QA/0102164 (2001).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lebedev Physical Institute, RAS, Moscow, Russia

    A. V. Marshakov

  2. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Marshakov

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  1. A. V. Marshakov
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[This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.]

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 163–228, May, 2006.

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Marshakov, A.V. Matrix models, complex geometry, and integrable systems: I. Theor Math Phys 147, 583–636 (2006). https://doi.org/10.1007/s11232-006-0065-x

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