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TMF, 2012, Volume 171, Number 1, Pages 96–115 (Mi tmf6915)  

This article is cited in 19 scientific papers (total in 19 papers)

Resolvents and Seiberg–Witten representation for a Gaussian $\beta$-ensemble

A. D. Mironovab, A. Yu. Morozovb, A. V. Popolitovb, Sh. R. Shakirovbc

a Lebedev Physical Institute, RAS, Moscow, Russia
b Institute for Theoretical and Experimental Physics, Moscow, Russia
c Department of Mathematics, University of California, Berkeley, CA, USA

Abstract: The exact free energy of a matrix model always satisfies the Seiberg–Witten equations on a complex curve defined by singularities of the semiclassical resolvent. The role of the Seiberg–Witten differential is played by the exact one-point resolvent in this case. We show that these properties are preserved in the generalization of matrix models to $\beta$-ensembles. But because the integrability and Harer–Zagier topological recursion are still unavailable for $\beta$-ensembles, we must rely on the ordinary Alexandrov–Mironov–Morozov/Eynard–Orantin recursion to evaluate the first terms of the genus expansion. We restrict our consideration to the Gaussian model.

Keywords: matrix model, $\beta$-ensemble, integrability, Seiberg–Witten theory


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English version:
Theoretical and Mathematical Physics, 2012, 171:1, 505–522

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Received: 17.05.2011

Citation: A. D. Mironov, A. Yu. Morozov, A. V. Popolitov, Sh. R. Shakirov, “Resolvents and Seiberg–Witten representation for a Gaussian $\beta$-ensemble”, TMF, 171:1 (2012), 96–115; Theoret. and Math. Phys., 171:1 (2012), 505–522

Citation in format AMSBIB
\by A.~D.~Mironov, A.~Yu.~Morozov, A.~V.~Popolitov, Sh.~R.~Shakirov
\paper Resolvents and Seiberg--Witten representation for a~Gaussian $\beta$-ensemble
\jour TMF
\yr 2012
\vol 171
\issue 1
\pages 96--115
\jour Theoret. and Math. Phys.
\yr 2012
\vol 171
\issue 1
\pages 505--522

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    2. D. Krefl, J. Walcher, “ABCD of beta ensembles and topological strings”, J. High Energy Phys., 2012, no. 11, 111, 27 pp.  crossref  mathscinet  isi
    3. M. Aganagic, M. C. N. Cheng, R. Dijkgraaf, D. Krefl, C. Vafa, “Quantum geometry of refined topological strings”, J. High Energy Phys., 2012, no. 11, 019, 52 pp.  crossref  mathscinet  isi  elib
    4. M.-X. Huang, “Dijkgraaf-Vafa conjecture and $\beta$-deformed matrix models”, J. High Energy Phys., 2013, no. 7, 173  crossref  mathscinet  zmath  isi  elib
    5. L. Chekhov, B. Eynard, S. Ribault, “Seiberg-Witten equations and non-commutative spectral curves in Liouville theory”, J. Math. Phys., 54:2 (2013), 022306  crossref  mathscinet  zmath  adsnasa  isi  elib
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    7. N. S. Witte, P. J. Forrester, “Moments of the Gaussian $\beta$ ensembles and the large-$N$ expansion of the densities”, J. Math. Phys., 55:8 (2014), 083302  crossref  mathscinet  zmath  adsnasa  isi
    8. N. Nemkov, “S-duality as Fourier transform for arbitrary $\epsilon_1, \epsilon_2$”, J. Phys. A-Math. Theor., 47:10 (2014), 105401  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. N. S. Witte, P. J. Forrester, “Loop equation analysis of the circular $\beta$ ensembles”, J. High Energy Phys., 2015, no. 2, 173  crossref  mathscinet  zmath  isi
    10. I. Rumanov, “Classical integrability for $\beta$-ensembles and general Fokker-Planck equations”, J. Math. Phys., 56:1 (2015), 013508  crossref  mathscinet  zmath  adsnasa  isi
    11. H. Itoyama, R. Yoshioka, “Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models”, Prog. Theor. Exp. Phys., 2015, no. 11, 11B103  crossref  mathscinet  zmath  isi
    12. A. Mironov, A. Morozov, Y. Zenkevich, “_orig ding-iohara-miki symmetry of network matrix models”, Phys. Lett. B, 762 (2016), 196–208  crossref  mathscinet  zmath  isi  elib  scopus
    13. H. Awata, H. Kanno, T. Matsumoto, A. Mironov, A. Morozov, A. Morozov Yu. Ohkubo, Y. Zenkevich, “Explicit examples of DIM constraints for network matrix models”, J. High Energy Phys., 2016, no. 7, 103  crossref  mathscinet  zmath  isi  elib  scopus
    14. M. Manabe, P. Sulkowski, “Quantum curves and conformal field theory”, Phys. Rev. D, 95:12 (2017), 126003  crossref  isi
    15. F. Mezzadri, A. K. Reynolds, B. Winn, “Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles”, Nonlinearity, 30:3 (2017), 1034–1057  crossref  mathscinet  zmath  isi
    16. G. Bonelli, K. Maruyoshi, A. Tanzini, “Quantum Hitchin systems via $\beta$-deformed matrix models”, Commun. Math. Phys., 358:3 (2018), 1041–1064  crossref  mathscinet  zmath  isi
    17. Mironov A. Morozov A., “Sum Rules For Characters From Character-Preservation Property of Matrix Models”, J. High Energy Phys., 2018, no. 8, 163  crossref  zmath  isi  scopus
    18. Morozov A., Popolitov A., Shakirov Sh., “On (Q, T)-Deformation of Gaussian Matrix Model”, Phys. Lett. B, 784 (2018), 342–344  crossref  mathscinet  isi  scopus
    19. Morozov A., “On W-Representations of Beta- and Q, T-Deformed Matrix Models”, Phys. Lett. B, 792 (2019), 205–213  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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