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Random normal matrices, Bergman kernel and projective embeddings

Journal of High Energy Physics volume 2014, Article number: 133 (2014) Cite this article

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.

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  1. Mathematisches Institut and Institut für Theoretische Physik, Universität zu Köln, Weyertal 86-90, 50931, Köln, Germany

    Semyon Klevtsov

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  1. Semyon Klevtsov
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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