Abstract:
Although matrix model partition functions do not exhaust the entire set of τ-functions relevant for string theory, they are elementary blocks for constructing many other τ-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.
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
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\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
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\jour Theoret. and Math. Phys.
\yr 2005
\vol 142
\issue 3
\pages 349--411
\crossref{https://doi.org/10.1007/s11232-005-0031-z}
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Linking options:
https://www.mathnet.ru/eng/tmf1792
https://doi.org/10.4213/tmf1792
https://www.mathnet.ru/eng/tmf/v142/i3/p419
This publication is cited in the following 30 articles:
Alexandrov A., “Kp Integrability of Triple Hodge Integrals. i. From Givental Group to Hierarchy Symmetries”, Commun. Number Theory Phys., 15:3 (2021), 615–650
Shakirov Sh., Sleptsov A., “Quantum Racah Matrices and 3-Strand Braids in Representation [3,3]”, J. Geom. Phys., 166 (2021), 104273
G. Carlet, J. van de Leur, H. Posthuma, S. Shadrin, “Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy”, Lett Math Phys, 111:3 (2021)
Morozov A., “On W-Representations of Beta- and Q, T-Deformed Matrix Models”, Phys. Lett. B, 792 (2019), 205–213
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
Alexandrov A., “Cut-and-Join Description of Generalized Brezin-Gross-Witten Model”, Adv. Theor. Math. Phys., 22:6 (2018), 1347–1399
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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
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
JETP Letters, 95:11 (2012), 586–593
A. Yu. Morozov, “Challenges of $\beta$-deformation”, Theoret. and Math. Phys., 173:1 (2012), 1417–1437
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
Alexandrov A., “Matrix models for random partitions”, Nuclear Phys B, 851:3 (2011), 620–650
A. Yu. Morozov, “Unitary integrals and related matrix models”, Theoret. and Math. Phys., 162:1 (2010), 1–33
Mironov A., Morozov A., “On AGT relation in the case of U(3)”, Nuclear Phys. B, 825:1-2 (2010), 1–37
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.
Mironov A., Morozov A., “Nekrasov functions and exact Bohr-Sommerfeld integrals”, J. High Energy Phys., 2010, no. 4, 040, 15 pp.
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
Morozov A., Shakirov Sh., “On equivalence of two Hurwitz matrix models”, Modern Phys. Lett. A, 24:33 (2009), 2659–2666
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
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