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Tr. Mat. Inst. Steklova, 2005, Volume 251, Pages 265–306 (Mi tm54)  

This article is cited in 27 scientific papers (total in 27 papers)

Complex Geometry of Matrix Models

L. O. Chekhova, A. V. Marshakovb, A. D. Mironovb, D. Vasilievcd

a Steklov Mathematical Institute, Russian Academy of Sciences
b P. N. Lebedev Physical Institute, Russian Academy of Sciences
c Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
d Moscow Institute of Physics and Technology

Abstract: The paper contains some new results and a review of recent achievements concerning multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by semiclassical, or generalized Whitham, hierarchies and are directly related to the superpotentials of four-dimensional ${\mathcal N}=1$ SUSY gauge theories. We study the derivatives of tau-functions for these solutions associated with families of Riemann surfaces (with possible double points) and find that they satisfy the Witten–Dijkgraaf–Verlinde–Verlinde equations. We also find the free energy in the subleading order in the matrix size and prove that it satisfies certain determinant relations.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 251, 254–292

Bibliographic databases:

Document Type: Article
UDC: 530.1
Received in August 2005

Citation: L. O. Chekhov, A. V. Marshakov, A. D. Mironov, D. Vasiliev, “Complex Geometry of Matrix Models”, Nonlinear dynamics, Collected papers, Tr. Mat. Inst. Steklova, 251, Nauka, MAIK Nauka/Inteperiodika, M., 2005, 265–306; Proc. Steklov Inst. Math., 251 (2005), 254–292

Citation in format AMSBIB
\Bibitem{CheMarMir05}
\by L.~O.~Chekhov, A.~V.~Marshakov, A.~D.~Mironov, D.~Vasiliev
\paper Complex Geometry of Matrix Models
\inbook Nonlinear dynamics
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2005
\vol 251
\pages 265--306
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm54}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2234386}
\zmath{https://zbmath.org/?q=an:1138.81493}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2005
\vol 251
\pages 254--292


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Marshakov, “Matrix models, complex geometry, and integrable systems: II$^*$”, Theoret. and Math. Phys., 147:3 (2006), 777–820  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Chekhov L.O., “Solving matrix models in the $1/N$-expansion”, Russian Math. Surveys, 61:3 (2006), 483–543  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Chekhov L., “Matrix models with hard walls: geometry and solutions”, J. Phys. A, 39:28 (2006), 8857–8893  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. A. S. Alexandrov, A. D. Mironov, A. Yu. Morozov, “$M$-Theory of Matrix Models”, Theoret. and Math. Phys., 150:2 (2007), 153–164  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Alexandrov A., Mironov A., Morozov A., “Instantons and merons in matrix models”, Phys. D, 235:1-2 (2007), 126–167  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Alexandrov A., Mironov A., Morozov A., “BGWM as second constituent of complex matrix model”, Journal of High Energy Physics, 2009, no. 12, 053  crossref  mathscinet  isi  scopus
    7. Alexandrov A., Mironov A., Morozov A., Putrov P., “Partition functions of matrix models as the first special functions of string theory. II. Kontsevich model”, Internat. J. Modern Phys. A, 24:27 (2009), 4939–4998  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. A. Yu. Morozov, “Unitary integrals and related matrix models”, Theoret. and Math. Phys., 162:1 (2010), 1–33  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Mironov A., Morozov A., “Nekrasov functions from exact Bohr–Sommerfeld periods: the case of SU(N)”, J. Phys. A, 43:19 (2010), 195401  crossref  adsnasa  isi  elib  scopus
    10. Mironov A., Morozov A., Shakirov Sh., “Matrix model conjecture for exact BS periods and Nekrasov functions”, Journal of High Energy Physics, 2010, no. 2, 030  crossref  mathscinet  isi  scopus
    11. Mironov A., Morozov A., “On AGT relation in the case of U(3)”, Nuclear Phys. B, 825:1-2 (2010), 1–37  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    12. Morozov A., Shakirov Sh., “The matrix model version of AGT conjecture and CIV-DV prepotential”, Journal of High Energy Physics, 2010, no. 8, 066  crossref  mathscinet  zmath  isi  scopus
    13. Mironov A., Morozov A., Shakirov Sh., “Conformal blocks as Dotsenko-Fateev integral discriminants”, Internat. J. Modern Phys. A, 25:16 (2010), 3173–3207  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Marshakov A., Mironov A., Morozov A., “On AGT relations with surface operator insertion and a stationary limit of beta-ensembles”, J. Geom. Phys., 61:7 (2011), 1203–1222  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Mironov A., Morozov A., Shakirov Sh., “A direct proof of AGT conjecture at beta=1”, Journal of High Energy Physics, 2011, no. 2, 067  crossref  mathscinet  zmath  isi  scopus
    16. Mironov A., Morozov A., Morozov A., “Conformal blocks and generalized Selberg integrals”, Nuclear Phys. B, 843:2 (2011), 534–557  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. A. V. Marshakov, “Gauge theories as matrix models”, Theoret. and Math. Phys., 169:3 (2011), 1704–1723  mathnet  crossref  crossref  mathscinet  isi
    18. A. D. Mironov, A. Yu. Morozov, A. V. Popolitov, Sh. R. Shakirov, “Resolvents and Seiberg–Witten representation for a Gaussian $\beta$-ensemble”, Theoret. and Math. Phys., 171:1 (2012), 505–522  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    19. Mironov A., Morozov A., Shakirov Sh., “Towards a Proof of AGT Conjecture by Methods of Matrix Models”, Internat J Modern Phys A, 27:1 (2012), 1230001  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    20. JETP Letters, 95:11 (2012), 586–593  mathnet  crossref  isi  elib  elib
    21. A. Yu. Morozov, “Challenges of $\beta$-deformation”, Theoret. and Math. Phys., 173:1 (2012), 1417–1437  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    22. A. V. Popolitov, “Relation between Nekrasov functions and Bohr–Sommerfeld periods in the pure $SU(N)$ case”, Theoret. and Math. Phys., 178:2 (2014), 239–252  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Morozov A.A., “The properties of conformal blocks, the AGT hypothesis, and knot polynomials”, Phys. Part. Nuclei, 47:5 (2016), 775–837  crossref  mathscinet  isi  elib  scopus
    24. Andrei Mironov, Alexei Morozov, “Check-Operators and Quantum Spectral Curves”, SIGMA, 13 (2017), 047, 17 pp.  mathnet  crossref
    25. Itoyama H., Mironov A., Morozov A., “Ward Identities and Combinatorics of Rainbow Tensor Models”, J. High Energy Phys., 2017, no. 6, 115  crossref  mathscinet  zmath  isi  scopus
    26. Itoyama H., Mironov A., Morozov A., “Rainbow Tensor Model With Enhanced Symmetry and Extreme Melonic Dominance”, Phys. Lett. B, 771 (2017), 180–188  crossref  zmath  isi  scopus
    27. Mironov A., Morozov A., “On the Complete Perturbative Solution of One-Matrix Models”, Phys. Lett. B, 771 (2017), 503–507  crossref  zmath  isi  scopus
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