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.
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REFERENCES
A. Polyakov, Gauge Fields and Strings, Harwood, Chur (1987); M. Green, J. Shwarz, and E. Witten, Superstring Theory, Vol. 1, 2, Cambridge Univ. Press, Cambridge (1987); A. Yu. Morozov, Sov. Phys. Usp., 35, 671 (1992); J. Polchinski, String Theory, Vol. 1, 2, Cambridge Univ. Press, Cambridge (1998); A. V. Marshakov, Phys. Usp., 45, 915 (2002); hep-th/0212114 (2002).
A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2589 (1995); hep-th/9405011 (1994); A. D. Mironov, A. Yu. Morozov, and L. Vinet, Theor. Math. Phys., 100, 890 (1995); S. M. Kharchev, A. D. Mironov, and A. Yu. Morozov, Theor. Math. Phys., 104, 866 (1995); A. D. Mironov, Theor. Math. Phys., 114, 127 (1998); A. Mironov and A. Morozov, Phys. Lett. B, 524, 217 (2002).
M. L. Mehta, Random Matrices, Acad. Press, New York (1991); E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Comm. Math. Phys., 59, 35 (1978); D. Bessis, Comm. Math. Phys., 69, 147 (1979); D. Bessis, C. Itzykson, and J.-B. Zuber, Adv. Appl. Math., 1, 109 (1980); C. Itzykson and J.-B. Zuber, J. Math. Phys., 21, 411 (1980).
A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565 (1991).
S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, and A. Zabrodin, Nucl. Phys. B, 366, 569 (1991).
A. Yu. Morozov, Phys. Usp., 37, 1 (1994); hep-th/9303139 (1993); “Matrix models as integrable systems,” hep-th/9502091 (1995).
A. Mironov, Internat. J. Mod. Phys. A, 9, 4355 (1994); Phys. Part. Nucl., 33, 537 (2002); I. K. Kostov, “Conformal field theory techniques in random matrix models,” hep-th/9907060 (1999).
A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 265, 99 (1991); S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, Nucl. Phys. B, 404, 717 (1993); A. D. Mironov and S. Z. Pakuliak, Theor. Math. Phys., 95, 604 (1993).
I. Kostov, Phys. Lett. B, 297, 74 (1992).
J. Alfaro and I. K. Kostov, “Generalized Hirota equations in models of 2D quantum gravity,” hep-th/9604011 (1996).
M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, Phys. Rev. Lett., 84, 5106 (2000); I. K. Kostov, I. Krichever, M.Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, “τ-function for analytic curves,” in: Random Matrices and Their Applications (MSRI Publs., Vol. 40, P. M. Bleher and A. R. Its, eds.), Cambridge Univ. Press, Cambridge (2001), p. 285; hep-th/0005259 (2000); A. Boyarsky, A. Marshakov, O. Ruchayskiy, P. Wiegmann, and A. Zabrodin, Phys. Lett. B, 515, 483 (2001); I. Krichever, A. Marshakov, and A. Zabrodin, “ Integrable structure of the Dirichlet boundary problem in multiply-connected domains,” hep-th/0309010 (2003).
M. L. Kontsevich, Funct. Anal. Appl., 25, No. 2, 123 (1991).
E. Witten, “On the Kontsevich model and other models of two-dimensional gravity,” in: Proc. 20th Intl. Conf. on Differential Geometric Methods in Theoretical Physics (S. Catt and A. Rocha, eds.), Vol. 1,2, World Scientific, River Edge, N. J. (1992), p. 176; A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 274, 280 (1992).
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Nucl. Phys. B, 380, 181 (1992); Phys. Lett. B, 275, 311 (1992).
S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 397, 339 (1993).
S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Modern Phys. Lett. A, 8, 1047 (1993).
L. Chekhov and Yu. Makeenko, Phys. Lett. B, 278, 271 (1992); L. Chekhov, “Matrix models and geometry of moduli spaces,” hep-th/9509001 (1995); S. Kharchev, “Kadomtsev-Petviashvili hierarchy and generalized Kontsevich model,” hep-th/9810091 (1998).
E. P. Wigner, Ann. Math., 53, 36 (1951).
F. J. Dyson, J. Math. Phys., 3, 140 (1962); D. Gross and E. Witten, Phys. Rev. D, 21, 446 (1980); T. Eguchi and H. Kawai, Phys. Rev. Lett., 48, 1063 (1982); D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables (CRM Monograph Series, Vol. 1), Amer. Math. Soc., Providence, R. I. (1992); P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, Phys. Rep., 254, 1 (1995).
F. David, Nucl. Phys. B, 257, 45 (1985); V. A. Kazakov, I. K. Kostov, and A. A. Migdal, Phys. Lett. B, 157, 295 (1985).
A. Givental, “Semisimple Frobenius structures at higher genus,”math.AG/0008067 (2000).
J. S. Song and Y. S. Song, J. Math. Phys., 45, 4539 (2004); hep-th/0103254 (2001); A. Alexandrov, J. Math. Phys., 44, 5268 (2003).
F. David, Phys. Lett. B, 302, 403 (1993); hep-th/9212106 (1992); G. Bonnet, F. David, and B. Eynard, J. Phys. A, 33, 6739 (2000); cond-mat/0003324 (2000); A. Klemm, M. Mariño, and S. Theisen, JHEP, 0303, 051 (2003).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products [in Russian], Nauka, Moscow (1971); English transl. 5th ed., Acad. Press, Boston, Mass. (1994).
R. Dijkgraaf and C. Vafa, Nucl. Phys. B, 644,3, 21 (2002); hep-th/0208048 (2002).
A. A. Migdal, Phys. Rep., 102, 199 (1983); J. Ambjørn, J. Jurkiewicz, and Yu. Makeenko, Phys. Lett. B, 251, 517 (1990).
E. Witten, Nucl. Phys. B, 460, 335 (1996).
I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B, 102, 27 (1981); 120, 166 (1983); Phys. Rev. D, 28, 2567 (1983); Nucl. Phys. B, 234, 106 (1984); J. Math. Phys., 26, 172 (1985).
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton, N. J. (1992); J. Gomis, J. París, and S. Samuel, Phys. Rep., 259, 1 (1995).
E. S. Fradkin and G. A. Vilkovisky, Phys. Lett. B, 55, 224 (1975); I. A. Batalin and G. A.Vilkovisky, Phys. Lett. B, 69, 309 (1977).
M. Henneaux, Phys. Rep., 126, 1 (1985).
E. Witten, Modern Phys. Lett. A, 5, 487 (1990).
A. S. Schwarz, Comm. Math. Phys., 155, 249 (1993).
J. Polchinski, Nucl. Phys. B, 231, 269 (1984); A. Mironov and A. Morozov, Phys. Lett. B, 490, 173 (2000).
N. Seiberg and E. Witten, Nucl. Phys. B, 426, 19 (1994).
F. David, Modern Phys. Lett. A, 5, 1019 (1990); A. Mironov and A. Morozov, Phys. Lett. B, 252, 47 (1990); J. Ambjτrn and Yu. Makeenko, Modern Phys. Lett. A, 5, 1753 (1990); H. Itoyama and Y. Matsuo, Phys. Lett. B, 255, 202 (1991).
M. Fukuma, H. Kawai, and R. Nakayama, Internat. J. Mod. Phys. A, 6, 1385 (1991); R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B, 348, 435 (1991).
Yu. Makeenko, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 356, 574 (1991).
H. Itoyama and A. Morozov, Internat. J. Mod. Phys. A, 18, 5889 (2003); hep-th/0301136 (2003).
J. Harer and D. Zagier, Invent. Math., 85, 457 (1986); S. K. Lando and A. K. Zvonkin, “Embedded graphs,” Preprint No. 63-01, Max-Plank-Institut für Mathematik, Bonn (2001).
C. Itzykson and J.-B. Zuber, Comm. Math. Phys., 134, 197 (1990).
E. Witten, Nucl. Phys. B, 340, 281 (1990); R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B, 352, 59 (1991); B. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620, M. Francaviglia and S. Greco, eds.), Springer, Berlin (1996), p. 120.
A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 389, 43 (1996); Modern Phys. Lett. A, 12, 773 (1997); Internat. J. Mod. Phys. A, 15, 1157 (2000).
A. S. Losev, JETP Letters, 65, 386 (1997); K. Ito and S.-K. Yang, Phys. Lett. B, 433, 56 (1998); G. Bertoldi and M. Matone, Phys. Rev. D, 57, 6483 (1998); A. Morozov, Phys. Lett. B, 427, 93 (1998); A. Mironov and A. Morozov, Phys. Lett. B, 424, 48 (1998); H. W. Braden, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 448, 195 (1999); A. Veselov, Phys. Lett. A, 261, 297 (1999); J. M. Isidro, Nucl. Phys. B, 539, 379 (1999); A. Mironov, “WDVV equations and Seiberg-Witten theory,” hep-th/9903088 (1999); A. V. Marshakov, Theor. Math. Phys., 132, 895 (2002).
F. Cachazo, K. Intriligator, and C. Vafa, Nucl. Phys. B, 603, 3 (2001); F. Cachazo and C. Vafa, “N = 1 and N = 2 geometry from fluxes,” hep-th/0206017 (2002).
N. Dorey, T. J. Hollowood, S. Prem Kumar, and A. Sinkovics, JHEP, 0211,039, 040 (2002); 0212, 003 (2002); F. Ferrara, Nucl. Phys. B, 648, 161 (2003); Phys. Rev. D, 67, 085013 (2003); D. Berenstein, Phys. Lett. B, 552, 255 (2003); R. Dijkgraaf, S. Gukov, V. Kazakov, and C. Vafa, Phys. Rev. D, 68, 045007 (2003); A. Gorsky, Phys. Lett. B, 554, 185 (2003); R. Dijkgraaf, M. T. Grisaru, C. S. Lam, C. Vafa, and D. Zanon, Phys. Lett. B, 573, 138 (2003); B. Feng, “Seiberg duality in matrix model,” hep-th/0211202 (2002); B. Feng, Nucl. Phys. B, 661, 113 (2003); hep-th/0212010; F. Cachazo, M. R. Douglas, N. Seiberg, and E. Witten, JHEP, 0212, 071 (2002); F. Cachazo, N. Seiberg, and E. Witten, JHEP, 0302, 042 (2003); 0304, 018 (2003); A. Dymarsky and V. Pestun, Phys. Rev. D, 67, 125001 (2003); R. Boels, Jan de Boer, R. Duivenvoorden, and J. Wijnhout, JHEP, 0403, 010 (2004); hep-th/0305189 (2003).
G. Bonelli, Nucl. Phys. B, 649, 130 (2003); hep-th/0209225 (2002); H. Fujiand Y. Ookouchi, JHEP, 0212, 067 (2002); hep-th/0210148 (2002); R. Argurio, V. L. Campos, G. Ferretti, and R. Heise, Phys. Rev. D, 67, 065005 (2003); hep-th/0210291 (2002); Phys. Lett. B, 553, 332 (2003); hep-th/0211249 (2002); J. McGreevy, JHEP, 0301, 047 (2003); hep-th/0211009 (2002); H. Suzuki, JHEP, 0303, 005, 036 (2003); hep-th/0211052, hep-th/0212121 (2002); I. Bena and R. Roiban, Phys. Lett. B, 555, 117 (2003); hep-th/0211075 (2002); Y. Demasure and R. A. Janik, Phys. Lett. B, 553, 105 (2003); hep-th/0211082 (2002); R. Gopakumar, JHEP, 0305, 033 (2003); hep-th/0211100 (2002); I. Bena, R. Roiban, and R. Tatar, Nucl. Phys. B, 679, 168 (2004); hep-th/0211271 (2002); Y. Tachikawa, Phys. Lett. B, 573, 235 (2003); hep-th/0211189 (2002); Progr. Theor. Phys., 110, 841 (2003); hep-th/0211274 (2002); Y. Ookouchi, JHEP, 0401, 014 (2004); hep-th/0211287 (2002); S. K. Ashok, R. Corrado, N. Halmagyi, K. D. Kennaway, and C. Romelsberger, Phys. Rev. D, 67, 086004 (2003); hep-th/0211291 (2002); K. Ohta, JHEP, 0302, 057 (2003); hep-th/0212025 (2002); R. A. Janik and N. A. Obers, Phys. Lett. B, 553, 309 (2003); hep-th/0212069 (2002); S. Seki, Nucl. Phys. B, 661, 257 (2003); hep-th/0212079 (2002); C. Hofman, JHEP, 0310, 022 (2003); hep-th/0212095 (2002); C. H. Ahn and S. Nam, Phys. Lett. B, 562, 141 (2003); hep-th/0212231 (2002); C. H. Ahn, Phys. Lett. B, 560, 116 (2003); hep-th/0301011 (2003); S. Aoyama and T. Masuda, JHEP, 0403, 072 (2004); hep-th/0309232 (2003).
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355, 466 (1995); E. Martinec and N. Warner, Nucl. Phys. B, 459, 97 (1996); R. Donagi and E. Witten, Nucl. Phys. B, 460, 299 (1996); A. Gorsky, A. Mironov, A. Marshakov, and A. Morozov, Nucl. Phys. B, 527, 690 (1998); H. Itoyama and A. Morozov, Nucl. Phys. B, 477, 855 (1996); 491, 529 (1997); “Integrability and Seiberg-Witten theory,” hep-th/9601168 (1996); E. D’Hoker and D. H. Phong, “Lectures on supersymmetric Yang-Mills theory and integrable systems,” hep-th/9912271 (1999); A. Marshakov, Seiberg-Witten Theory an Integrable Systems, World Scientific, Singapore (1999); H. W. Braden and I. M. Krichever (eds.), Integrability: The Seiberg-Witten an Whitham Equations, Gordon and Breach, Amsterdam (2000); A.Gorsky and A. Mironov, “Integrable many-body systems and gauge theories,” hep-th/0011197 (2000).
L. Chekhov and A. Mironov, Phys. Lett. B, 552, 293 (2003); V. Kazakov and A. Marshakov, J. Phys. A, 36, 3107 (2003).
H. Itoyama and A. Morozov, Nucl. Phys. B, 657, 53 (2003); Phys. Lett. B, 555, 287 (2003).
H. Itoyama and A. Morozov, Progr. Theor. Phys., 109, 433 (2003).
L. Chekhov, A. Marshakov, A. Mironov, and D. Vasiliev, Phys. Lett. B, 562, 323 (2003); A. Mironov, Fortschr. Phys., 51, 781 (2003).
J. Ambjτrn, L. Chekhov, C. F. Kristjansen, and Yu. Makeenko, Nucl. Phys. B, 404, 127 (1993); J. Ambjτrn, L. Chekhov, and Yu. Makeenk, Phys. Lett. B, 282, 341 (1992); G. Akemann, Nucl. Phys. B, 482, 403 (1996).
G.’t Hooft, Nucl. Phys. B, 72, 461 (1974).
G. Veneziano, Nucl. Phys. B, 117, 519 (1976); D. De Wit and G.’t Hooft, Phys. Lett., 69, 61 (1977); E. Witten, “The 1/N expansion in atomic and particle physics,” in: Recent Developments in Gauge Theories (G.’t Hooft et al., eds.), Plenum, New York (1980), p. 403; S. R. Wadia, Phys. Rev. D, 24, 970 (1981); A. Mironov, A.Morozov, and G. Semenoff, Internat. J. Mod. Phys. A, 11, 5031 (1996); B. Eynard, JHEP, 0311, 018 (2003); hep-th/0309036 (2003).
E. Brézin and V. A. Kazakov, Phys. Lett. B, 236, 144 (1990); D. Gross and A. A. Migdal, Phys. Rev. Lett., 64, 127 (1990); Nucl. Phys. B, 340, 333 (1990); M. Douglas and S. Shenker, Nucl. Phys. B, 335, 635 (1990).
D. Berenstein, J. Maldacena, and H. Nastase, JHEP, 0204, 013 (2002); N. R. Constable, D. Z. Freedman, M. Headrick, and S. Minwalla, JHEP, 0210, 068 (2002); D. J. Gross, A. Mikhailov, and R. Roiban, Ann. Phys., 301, 31 (2002); hep-th/0205066 (2002); JHEP, 0305, 025 (2003); N. Beisert, C. Kristjansen, J. Plefka, G. W. Semenoff, and M. Staudacher, Nucl. Phys. B, 643, 3 (2002); 650, 125 (2003).
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Translated from Teoreticheskaya i Matematicheskaya Fizika,Vol. 142, No. 3, pp. 419–488, March, 2005.
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Alexandrov, A.S., Mironov, A.D. & Morozov, A.Y. Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model. Theor Math Phys 142, 349–411 (2005). https://doi.org/10.1007/s11232-005-0031-z
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DOI: https://doi.org/10.1007/s11232-005-0031-z