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Random normal matrices, Bergman kernel and projective embeddings

Abstract

We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix model free energy and its relation to gravitational effective actions in two dimensions. We compute the leading terms of the free energy expansion in the pure bulk case, and make some observations on the structure of the expansion to all orders. As an application of these results, we propose an asymptotic formula for the Liouville action, restricted to the space of the Bergman metrics.

References

  1. O. Agam, E. Bettelheim, P. Wiegmann and A. Zabrodin, Viscous fingering and a shape of an electronic droplet in the quantum Hall regime, Phys. Rev. Lett. 88 (2002) 236801 [cond-mat/0111333] [INSPIRE].

    ADS  Article  Google Scholar 

  2. Y. Ameur, H. Hedenmalm and N. Makarov, Random normal matrices and Ward identities, arXiv:1109.5941.

  3. R. Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. 131 (2009) 1485 [arXiv:0710.4375].

    Article  MATH  MathSciNet  Google Scholar 

  4. R.J. Berman, Determinantal point processes and fermions on complex manifolds: bulk universality, arXiv:0811.3341.

  5. R.J. Berman, Determinantal point processes and fermions on complex manifolds: large deviations and bosonization, arXiv:0812.4224.

  6. R.J. Berman, S. Boucksom and D. Witt Nystrom, Fekete points and convergence towards equilibrium measures on complex manifolds, Acta Math. 207 (2011) 1 [arXiv:0907.2820].

    Article  MATH  MathSciNet  Google Scholar 

  7. R.J. Berman, Kähler-Einstein metrics emerging from free fermions and statistical mechanics, JHEP 10 (2011) 106 [arXiv:1009.2942] [INSPIRE].

    ADS  Article  Google Scholar 

  8. R.J. Berman, A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, arXiv:1011.3976.

  9. D. Catlin, The Bergman kernel and a theorem of Tian, in Analysis and geometry in several complex variables, Katata Japan (1997), Trends Math., Birkhäuser, Boston U.S.A. (1999), pg. 1.

  10. L.-L. Chau and Y. Yu, Unitary polynomials in normal matrix model and wave functions for the fractional quantum Hall effect, Phys. Lett. A 167 (1992) 452.

    Article  MathSciNet  Google Scholar 

  11. L.-L. Chau and O. Zaboronsky, On the structure of the correlation functions in the normal matrix model, Commun. Math. Phys. 196 (1998) 203 [hep-th/9711091] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  12. L. Chekhov and B. Eynard, Hermitean matrix model free energy: Feynman graph technique for all genera, JHEP 03 (2006) 014 [hep-th/0504116] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  13. X.X. Chen and G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002) 487 [math.DG/0010008].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  14. S.K. Donaldson, Scalar curvature and projective embeddings. I, J. Diff. Geom. 59 (2001) 479.

    MATH  MathSciNet  Google Scholar 

  15. S.K. Donaldson, Scalar curvature and projective embeddings. II, Quart. J. Math. 56 (2005) 345 [math.DG/0407534].

    Article  MATH  MathSciNet  Google Scholar 

  16. M.R. Douglas and S. Klevtsov, Bergman kernel from path integral, Commun. Math. Phys. 293 (2010) 205 [arXiv:0808.2451] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  17. P. Elbau and G. Felder, Density of eigenvalues of random normal matrices, Commun. Math. Phys. 259 (2005) 433 [math.QA/0406604].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. F. Ferrari, S. Klevtsov and S. Zelditch, Random geometry, quantum gravity and the Kähler potential, Phys. Lett. B 705 (2011) 375 [arXiv:1107.4022] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  19. F. Ferrari, S. Klevtsov and S. Zelditch, Random Kähler metrics, Nucl. Phys. B 869 (2013) 89 [arXiv:1107.4575] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  20. F. Ferrari, S. Klevtsov and S. Zelditch, Gravitational actions in two dimensions and the Mabuchi functional, Nucl. Phys. B 859 (2012) 341 [arXiv:1112.1352] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  21. F. Ferrari, S. Klevtsov and S. Zelditch, Simple matrix models for random Bergman metrics, J. Stat. Mech. 2012 (2012) P04012 [arXiv:1112.4382] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  22. J. Fine, Quantisation and the Hessian of Mabuchi energy, Duke Math. J. 161 (2012) 2753 [arXiv:1009.4543].

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Hedenmalm and A. Haimi, Asymptotic expansion of polyanalytic Bergman kernels, arXiv:1303.0720.

  24. H. Hedenmalm and N. Makarov, Quantum Hele-Shaw flow, math.PR/0411437.

  25. H. Hedenmalm and N. Makarov, Coulomb gas ensembles and Laplacian growth, Proc. London Math. Soc. 106 (2013) 859 [arXiv:1106.2971].

    Article  MATH  MathSciNet  Google Scholar 

  26. D. Karabali and V. Nair, Quantum Hall effect in higher dimensions, Nucl. Phys. B 641 (2002) 533 [hep-th/0203264] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  27. D. Karabali and V. Nair, The effective action for edge states in higher dimensional quantum Hall systems, Nucl. Phys. B 679 (2004) 427 [hep-th/0307281] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  28. D. Karabali and V. Nair, Edge states for quantum Hall droplets in higher dimensions and a generalized WZW model, Nucl. Phys. B 697 (2004) 513 [hep-th/0403111] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  29. V.A. Kazakov, M. Staudacher and T. Wynter, Exact solution of discrete two-dimensional R 2 gravity, Nucl. Phys. B 471 (1996) 309 [hep-th/9601069] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  30. S. Klevtsov, Bergman kernel from the lowest Landau level, Nucl. Phys. (Proc. Suppl.) B 192-193 (2011) 154 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. Z. Lu, On the lower order terms of the asymptotic expansion of Zelditch, Amer. J. Math. 122 (2000) 235 [math.DG/9811126].

    Article  MATH  MathSciNet  Google Scholar 

  32. X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progr. Math. 254, Birkhäuser Boston U.S.A. (2006).

  33. X. Ma and G. Marinescu, Berezin-Toeplitz quantization on Kähler manifolds, J. Reine Angew. Math. 662 (2012) 1 [arXiv:1009.4405].

    MATH  MathSciNet  Google Scholar 

  34. A. Marshakov, P. Wiegmann and A. Zabrodin, Integrable structure of the Dirichlet boundary problem in two-dimensions, Commun. Math. Phys. 227 (2002) 131 [hep-th/0109048] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  35. D.H. Phong and J. Sturm, Lectures on stability and constant scalar curvature, Curr. Devel. Math. 2007 (2009) 101, Int. Press, Somerville U.S.A. (2009) [arXiv:0801.4179].

  36. R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin and P. Wiegmann, Normal random matrix ensemble as a growth problem: evolution of the spectral curve, Nucl. Phys. B 704 (2005) 407 [hep-th/0401165] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  37. G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geom. 32 (1990) 99.

    MATH  Google Scholar 

  38. H. Xu, A closed formula for the asymptotic expansion of the Bergman kernel, Commun. Math. Phys. 314 (2012) 555 [arXiv:1103.3060].

    ADS  Article  MATH  Google Scholar 

  39. P. Wiegmann and A. Zabrodin, Conformal maps and dispersionless integrable hierarchies, Commun. Math. Phys. 213 (2000) 523 [hep-th/9909147] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  40. P. Wiegmann and A. Zabrodin, Large-N expansion for normal and complex matrix ensembles, in Proc. of Les Houches Spring School, (2003) [hep-th/0309253] [INSPIRE].

  41. A. Zabrodin, Matrix models and growth processes: from viscous flows to the quantum Hall effect, in Applications of random matrices in physics, Springer U.S.A. (2006), pg. 261 [hep-th/0412219] [INSPIRE].

  42. P. Wiegmann and A. Zabrodin, Large-N expansion of the 2D Dyson gas, J. Phys. A 39 (2006) 8933 [hep-th/0601009] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  43. O. Zeitouni and S. Zelditch, Large deviations of empirical zero point measures on Riemann surfaces, I: g = 0, Int. Math. Res. Notices 2010 (2010) 3939 [arXiv:0904.4271].

    Google Scholar 

  44. S. Zelditch, Szegö kernels and a theorem of Tian, Int. Math. Res. Notices 1998 (1998) 317 [math-ph/0002009].

    Article  MATH  MathSciNet  Google Scholar 

  45. S. Zelditch, Large deviations of empirical measures of zeros on Riemann surfaces, Int. Math. Res. Notices 2013 (2013) 592 [arXiv:1101.0417].

    MathSciNet  Google Scholar 

  46. S.-C. Zhang and J.-P. Hu, A four-dimensional generalization of the quantum Hall effect, Science 294 (2001) 823 [cond-mat/0110572] [INSPIRE].

    ADS  Article  Google Scholar 

  47. J.-P. Hu and S.-C. Zhang, Collective excitations at the boundary of a 4D quantum Hall droplet, Phys. Rev. B 66 (2002) 125301 [cond-mat/0112432] [INSPIRE].

    ADS  Article  Google Scholar 

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Correspondence to Semyon Klevtsov.

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ArXiv ePrint: 1309.7333

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Klevtsov, S. Random normal matrices, Bergman kernel and projective embeddings. J. High Energ. Phys. 2014, 133 (2014). https://doi.org/10.1007/JHEP01(2014)133

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Keywords

  • Matrix Models
  • 2D Gravity
  • Differential and Algebraic Geometry