Teoreticheskaya i Matematicheskaya Fizika
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

 TMF, 2005, Volume 142, Number 3, Pages 419–488 (Mi tmf1792)

Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model

A. S. Alexandrovab, A. D. Mironovca, A. Yu. Morozova

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

Abstract: Although matrix model partition functions do not exhaust the entire set of $\tau$-functions relevant for string theory, they are elementary blocks for constructing many other $\tau$-functions and seem to capture the fundamental nature of quantum gravity an string theory properly. We propose taking matrix model partition functions as new special functions. This means that they should be investigated and represented in some standard form without reference to particular applications. At the same time, the tables and lists of properties should be sufficiently full to exclude unexpected peculiarities appearing in new applications. Accomplishing this task requires considerable effort, and this paper is only a first step in this direction. We restrict our consideration to the finite Hermitian one-matrix model an concentrate mostly on its phase and branch structure that arises when the partition function is considered as a $D$-module. We discuss the role of the CIV-DV prepotential (which generates a certain basis in the linear space of solutions of the Virasoro constraints, although an understanding of why and how this basis is distinguished is lacking) an evaluate several first multiloop correlators, which generalize the semicircular distribution to the case of multitrace and nonplanar correlators.

Keywords: matrix models, string theory, multiloop correlators

DOI: https://doi.org/10.4213/tmf1792

Full text: PDF file (606 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2005, 142:3, 349–411

Bibliographic databases:

Citation: A. S. Alexandrov, A. D. Mironov, A. Yu. Morozov, “Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model”, TMF, 142:3 (2005), 419–488; Theoret. and Math. Phys., 142:3 (2005), 349–411

Citation in format AMSBIB
\Bibitem{AleMirMor05} \by A.~S.~Alexandrov, A.~D.~Mironov, A.~Yu.~Morozov \paper Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model \jour TMF \yr 2005 \vol 142 \issue 3 \pages 419--488 \mathnet{http://mi.mathnet.ru/tmf1792} \crossref{https://doi.org/10.4213/tmf1792} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165901} \zmath{https://zbmath.org/?q=an:1178.81208} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2005TMP...142..349A} \elib{https://elibrary.ru/item.asp?id=9132034} \transl \jour Theoret. and Math. Phys. \yr 2005 \vol 142 \issue 3 \pages 349--411 \crossref{https://doi.org/10.1007/s11232-005-0031-z} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000228416900001} 

• http://mi.mathnet.ru/eng/tmf1792
• https://doi.org/10.4213/tmf1792
• http://mi.mathnet.ru/eng/tmf/v142/i3/p419

 SHARE:

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. Krotov D., Morozov A., “A solvable sector of AdS theory”, Journal of High Energy Physics, 2005, no. 10, 062
2. A. S. Alexandrov, A. D. Mironov, A. Yu. Morozov, “$M$-Theory of Matrix Models”, Theoret. and Math. Phys., 150:2 (2007), 153–164
3. Alexandrov A, Mironov A, Morozov A, “Instantons and merons in matrix models”, Physica D-Nonlinear Phenomena, 235:1–2 (2007), 126–167
4. Dolotin, V, “On the Shapes of Elementary Domains Or Why Mandelbrot Set Is Made From Almost Ideal Circles?”, International Journal of Modern Physics A, 23:22 (2008), 3613
5. Morozov A., Shakirov Sh., “On equivalence of two Hurwitz matrix models”, Modern Phys. Lett. A, 24:33 (2009), 2659–2666
6. Alexandrov, A, “Partition Functions of Matrix Models as the First Special Functions of String Theory II. Kontsevich Model”, International Journal of Modern Physics A, 24:27 (2009), 4939
7. Eynard, B, “Topological recursion in enumerative geometry and random matrices”, Journal of Physics A-Mathematical and Theoretical, 42:29 (2009), 293001
8. Mironov A., Morozov A., “Virasoro constraints for Kontsevich-Hurwitz partition function”, Journal of High Energy Physics, 2009, no. 2, 024
9. A. Yu. Morozov, “Unitary integrals and related matrix models”, Theoret. and Math. Phys., 162:1 (2010), 1–33
10. Mironov A., Morozov A., “On AGT relation in the case of U(3)”, Nuclear Phys. B, 825:1-2 (2010), 1–37
11. Mironov A., Morozov A., Shakirov Sh., “Matrix model conjecture for exact BS periods and Nekrasov functions”, J. High Energy Phys., 2010, no. 2, 030, 26 pp.
12. Mironov A., Morozov A., “Nekrasov functions and exact Bohr-Sommerfeld integrals”, J. High Energy Phys., 2010, no. 4, 040, 15 pp.
13. Mironov A., Morozov A., “Nekrasov functions from exact Bohr-Sommerfeld periods: the case of SU(N)”, J. Phys. A: Math. Theor., 43:19 (2010), 195401
14. A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory”, Theoret. and Math. Phys., 166:1 (2011), 1–22
15. Alexandrov A., “Matrix models for random partitions”, Nuclear Phys B, 851:3 (2011), 620–650
16. JETP Letters, 95:11 (2012), 586–593
17. A. Yu. Morozov, “Challenges of $\beta$-deformation”, Theoret. and Math. Phys., 173:1 (2012), 1417–1437
18. Andersen J.E., Chekhov L.O., Penner R.C., Reidys Ch.M., Sulkowski P., “Topological Recursion for Chord Diagrams, Rna Complexes, and Cells in Moduli Spaces”, Nucl. Phys. B, 866:3 (2013), 414–443
19. 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
20. Dubrovin B., Yang D., “Generating Series For Gue Correlators”, Lett. Math. Phys., 107:11 (2017), 1971–2012
21. Alexandrov A., “Cut-and-Join Description of Generalized Brezin-Gross-Witten Model”, Adv. Theor. Math. Phys., 22:6 (2018), 1347–1399
22. Morozov A., “On W-Representations of Beta- and Q, T-Deformed Matrix Models”, Phys. Lett. B, 792 (2019), 205–213
23. Dunin-Barkowski P., Popolitov A., Shadrin S., Sleptsov A., “Combinatorial Structure of Colored Homfly-Pt Polynomials For Torus Knots”, Commun. Number Theory Phys., 13:4 (2019), 763–826
•  Number of views: This page: 785 Full text: 245 References: 87 First page: 3