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Gauge theories as matrix models

Abstract

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.

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Correspondence to A. V. Marshakov.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 169, No. 3, pp. 391–412, December, 2011.

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Marshakov, A.V. Gauge theories as matrix models. Theor Math Phys 169, 1704–1723 (2011). https://doi.org/10.1007/s11232-011-0146-3

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Keywords

  • supersymmetric gauge theory
  • matrix model
  • two-dimensional conformal field theory
  • highest-weight module of the Virasoro algebra