General information
Latest issue
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS


Personal entry:
Save password
Forgotten password?

TMF, 2011, Volume 166, Number 2, Pages 163–215 (Mi tmf6603)  

This article is cited in 29 scientific papers (total in 29 papers)

Topological expansion of the $\beta$-ensemble model and quantum algebraic geometry in the sectorwise approach

L. O. Chekhovabc, B. Eynardd, O. Marchald

a Institute for Theoretical and Experimental Physics, Moscow, Russia
b Steklov Mathematical Institute, Moscow, Russia
c Laboratoire Poncelet, Moscow, Russia
d Institite de Physique Th\'eorique, Centre des Etudes Atomiques, Gif-sur-Yvette, France

Abstract: We construct the solution of the loop equations of the $\beta$-ensemble model in a form analogous to the solution in the case of the Hermitian matrices $\beta=1$. The solution for $\beta=1$ is expressed in terms of the algebraic spectral curve given by $y^2=U(x)$. The spectral curve for arbitrary $\beta$ converts into the Schrödinger equation $((\hbar\partial)^2-U(x))\psi(x)=0$, where $\hbar\propto (\sqrt\beta-1/\sqrt\beta )/N$. The basic ingredients of the method based on the algebraic solution retain their meaning, but we use an alternative approach to construct a solution of the loop equations in which the resolvents are given separately in each sector. Although this approach turns out to be more involved technically, it allows consistently defining the $\mathcal B$-cycle structure for constructing the quantum algebraic curve (a D-module of the form $y^2-U(x)$, where $[y,x]=\hbar$) and explicitly writing the correlation functions and the corresponding symplectic invariants $\mathcal F_h$ or the terms of the free energy in an $1/N^2$-expansion at arbitrary $\hbar$. The set of “flat”; coordinates includes the potential times $t_k$ and the occupation numbers $\widetilde{\epsilon}_\alpha$. We define and investigate the properties of the $\mathcal A$- and $\mathcal B$-cycles, forms of the first, second, and third kinds, and the Riemann bilinear identities. These identities allow finding the singular part of $\mathcal F_0$, which depends only on $\widetilde{\epsilon}_\alpha$.

Keywords: Schrödinger equation, Bergman kernel, correlation function, Riemann identity, flat coordinates, Riccati equation


Full text: PDF file (962 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2011, 166:2, 141–185

Bibliographic databases:

Received: 18.08.2010
Revised: 13.09.2010

Citation: L. O. Chekhov, B. Eynard, O. Marchal, “Topological expansion of the $\beta$-ensemble model and quantum algebraic geometry in the sectorwise approach”, TMF, 166:2 (2011), 163–215; Theoret. and Math. Phys., 166:2 (2011), 141–185

Citation in format AMSBIB
\by L.~O.~Chekhov, B.~Eynard, O.~Marchal
\paper Topological expansion of the~$\beta$-ensemble model and quantum algebraic geometry in the~sectorwise approach
\jour TMF
\yr 2011
\vol 166
\issue 2
\pages 163--215
\jour Theoret. and Math. Phys.
\yr 2011
\vol 166
\issue 2
\pages 141--185

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Marshakov, “Gauge theories as matrix models”, Theoret. and Math. Phys., 169:3 (2011), 1704–1723  mathnet  crossref  crossref  mathscinet  isi
    2. Bonelli G., Maruyoshi K., Tanzini A., Yagi F., “Generalized matrix models and AGT correspondence at all genera”, J. High Energy Phys., 2011, no. 7, 055  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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
    4. A. Yu. Morozov, “Challenges of $\beta$-deformation”, Theoret. and Math. Phys., 173:1 (2012), 1417–1437  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    5. Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.  mathnet  crossref  mathscinet
    6. Bergère M., Eynard B., Marchal O., Prats-Ferrer A., “Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model”, J. High Energy Phys., 2012, no. 3, 098  crossref  mathscinet  zmath  isi  elib  scopus
    7. Nishinaka T., Rim Ch., “$\beta$-deformed matrix model and Nekrasov partition function”, J. High Energy Phys., 2012, no. 2, 114  crossref  mathscinet  zmath  isi  elib  scopus
    8. Orantin N., Veliz-Osorio A., “Non-homogenous disks in the chain of matrices”, J. High Energy Phys., 2012, no. 1, 080  crossref  mathscinet  zmath  isi  scopus
    9. Aganagic M., Cheng M.C.N., Dijkgraaf R., Krefl D., Vafa C., “Quantum geometry of refined topological strings”, J. High Energy Phys., 2012, no. 11, 019  crossref  mathscinet  isi  elib  scopus
    10. Nishinaka T., Rim Ch., “Matrix models for irregular conformal blocks and Argyres-Douglas theories”, J. High Energy Phys., 2012, no. 10, 138  crossref  mathscinet  isi  elib  scopus
    11. Bourgine J.-E., “Large $N$ limit of $\beta$-ensembles and deformed Seiberg-Witten relations”, J. High Energy Phys., 2012, no. 8, 046  crossref  mathscinet  isi  scopus
    12. Huang M.-x., “Dijkgraaf-Vafa Conjecture and Beta-Deformed Matrix Models”, J. High Energy Phys., 2013, no. 7, 173  crossref  mathscinet  zmath  isi  elib  scopus
    13. Bourgine J.-E., “Large N Techniques for Nekrasov Partition Functions and AGT Conjecture”, J. High Energy Phys., 2013, no. 5, 047  crossref  isi  elib  scopus
    14. Baek J.-H., “Genus One Correction to Seiberg-Witten Prepotential From Beta-Deformed Matrix Model”, J. High Energy Phys., 2013, no. 4, 120  crossref  mathscinet  zmath  isi  scopus
    15. Rumanov I., “Hard Edge For Beta-Ensembles and Painlevé III”, Int. Math. Res. Notices, 2014, no. 23, 6576–6617  crossref  mathscinet  zmath  isi  scopus
    16. Marchal O., “Elements of Proof For Conjectures of Witte and Forrester About the Combinatorial Structure of Gaussian Beta Ensembles”, J. High Energy Phys., 2014, no. 9, 003  crossref  isi  scopus
    17. Desrosiers P., Liu D.-Zh., “Asymptotics For Products of Characteristic Polynomials in Classical Beta-Ensembles”, Constr. Approx., 39:2 (2014), 273–322  crossref  mathscinet  zmath  isi  scopus
    18. Majumdar S.N., Schehr G., “TOP Eigenvalue of a Random Matrix: Large Deviations and Third Order Phase Transition”, J. Stat. Mech.-Theory Exp., 2014, P01012  crossref  mathscinet  isi  scopus
    19. Witte N.S., Forrester P.J., “Loop Equation Analysis of the Circular Beta Ensembles”, J. High Energy Phys., 2015, no. 2, 173  crossref  mathscinet  zmath  isi  scopus
    20. Safnuk B., “Topological recursion for open intersection numbers”, Commun. Number Theory Phys., 10:4 (2016), 833–857  crossref  mathscinet  zmath  isi  scopus
    21. Chan Ch.-Ts., Irie H., Niedner B., Yeh Ch.-H., “Wronskians, dualities and FZZT-Cardy branes”, Nucl. Phys. B, 910 (2016), 55–177  crossref  mathscinet  zmath  isi  elib  scopus
    22. Choi S.K., Rim Ch., Zhang H., “Irregular conformal block, spectral curve and flow equations”, J. High Energy Phys., 2016, no. 3, 118  crossref  mathscinet  zmath  isi  scopus
    23. Chaiho Rim, “Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach”, SIGMA, 13 (2017), 012, 23 pp.  mathnet  crossref
    24. Manabe M., Sulkowski P., “Quantum Curves and Conformal Field Theory”, Phys. Rev. D, 95:12 (2017), 126003  crossref  isi  scopus
    25. 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
    26. Mezzadri F., Reynolds A.K., Winn B., “Moments of the Eigenvalue Densities and of the Secular Coefficients of Beta-Ensembles”, Nonlinearity, 30:3 (2017), 1034–1057  crossref  mathscinet  zmath  isi  scopus
    27. Bonelli G., Maruyoshi K., Tanzini A., “Quantum Hitchin Systems Via -Deformed Matrix Models”, Commun. Math. Phys., 358:3 (2018), 1041–1064  crossref  mathscinet  zmath  isi  scopus
    28. Cordova C., Heidenreich B., Popolitov A., Shakirov Sh., “Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models”, Commun. Math. Phys., 361:3 (2018), 1235–1274  crossref  mathscinet  zmath  isi  scopus
    29. Ciosmak P., Hadasz L., Manabe M., Sulkowski P., “Singular Vector Structure of Quantum Curves”, Topological Recursion and Its Influence in Analysis, Geometry, and Topology, Proceedings of Symposia in Pure Mathematics, 100, eds. Liu C., Mulase M., Amer Mathematical Soc, 2018, 119–149  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:652
    Full text:123
    First page:16

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020