BGWM as second constituent of complex matrix model

Journal of High Energy Physics

Volumn 2009, Issue 12, 2009, Pages

  Alexandrov, A ^a,b   Mironov, A ^c   Morozov, A ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^c LEBEDEV PHYSICAL INSTITUTE   (Russian Federation)

Author keywords

Integrable Hierarchies; M Theory; Matrix Models; Topological Strings

Indexed keywords

EID: 76149111160     PISSN: 11266708     EISSN: 10298479     Source Type: Journal    
DOI: 10.1088/1126-6708/2009/12/053     Document Type: Article

References (243)

1. A.S. Alexandrov, A. Mironov and A. Morozov, 2007 Instantons and merons in matrix models, Physica D 235 126 [hep-th/0608228]
2. A.S. Alexandrov, A. Mironov and A. Morozov, 2007 M-theory of matrix models, Teor. Mat. Fiz. 150 179 [hep-th/0605171]
3. E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, 1978 Planar diagrams, Commun. Math. Phys. 59 35
4. D. Bessis, 1979 A new method in the combinatorics of the topological expansion, Commun. Math. Phys. 69 147
5. D. Bessis, C. Itzykson and J.-B. Zuber, 1980 Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1 109
6. M.-L. Mehta, 1981 A method of integration over matrix variables, Commun. Math. Phys. 79 327
7. nd edition, Academic Press, New York U.S.A. 1991
8. A.A. Migdal, 1983 Loop equations and 1/N expansion, Phys. Rept. 102 199
9. F. David, 1985 Planar diagrams, two-dimensional lattice gravity and surface models, Nucl. Phys. B 257 45
10. F. David, 1985 A model of random surfaces with nontrivial critical behavior, Nucl. Phys. B 257 543
11. J. Ambjørn, B. Durhuus and J. Fröhlich, 1985 Diseases of triangulated random surface models and possible cures, Nucl. Phys. B 257 433
12. V.A. Kazakov, I.K. Kostov and A.A. Migdal, 1985 Critical properties of randomly triangulated planar random surfaces, Phys. Lett. B 157 295
13. D.V. Boulatov, V.A. Kazakov, A.A. Migdal and I.K. Kostov, 1986 Possible types of critical behavior and the mean size of dynamically triangulated random surfaces, Phys. Lett. B 174 87
14. D.V. Boulatov, V.A. Kazakov, A.A. Migdal and I.K. Kostov, 1986 Analytical and numerical study of the model of dynamically triangulated random surfaces, Nucl. Phys. B 275 641
15. V.A. Kazakov, 1986 Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 140
16. V.A. Kazakov, 1989 The appearance of matter fields from quantum fluctuations of 2 D gravity, Mod. Phys. Lett. A 4 2125
17. L. Alvarez-Gaume, 1990 Random surfaces, statistical mechanics, and string theory, Lausanne lectures, Lausanne Switzerland
18. A. Levin and A. Morozov, 1990 On the foundations of the random lattice approach to quantum gravity, Phys. Lett. B 243 207
19. P.H. Ginsparg, Matrix models of 2D gravity, [hep-th/9112013]
20. P. Di Francesco and C. Itzykson, 1993 A generating function for fatgraphs Ann. Poincaré 59 117 [hep-th/9212108]
21. J.M. Daul, V.A. Kazakov and I.K. Kostov, 1993 Rational theories of 2D gravity from the two matrix model, Nucl. Phys. B 409 311 [hep-th/9303093]
22. M. Staudacher, 1993 Combinatorial solution of the two matrix model, Phys. Lett. B 305 332 [hep-th/9301038]
23. J. Ambjørn, L. Chekhov, C.F. Kristjansen and Y. Makeenko, 1993 Matrix model calculations beyond the spherical limit, Nucl. Phys. B 404 127 [Erratum ibid B 449 (1995) 681] [hep-th/9302014]
24. P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 1995 2D gravity and random matrices, Phys. Rept. 254 1 [hep-th/9306153]
25. B. Eynard, Large random matrices: eigenvalue distribution, [hep-th/9401165]
26. V.A. Kazakov, M. Staudacher and T. Wynter, 1996 Character expansion methods for matrix models of dually weighted graphs, Commun. Math. Phys. 177 451 [hep-th/9502132]
27. M. Adler, A. Morozov, T. Shiota and P. van Moerbeke, 1996 New matrix model solutions to the Kac-Schwarz problem, Nucl. Phys. (Proc. Suppl.) 49 201 [hep-th/9603066]
28. G. Akemann, 1996 Higher genus correlators for the Hermitian matrix model with multiple cuts, Nucl. Phys. B 482 403 [hep-th/9606004]
29. G. Akemann, P.H. Damgaard, U. Magnea and S. Nishigaki, 1997 Universality of random matrices in the microscopic limit and the Dirac operator spectrum, Nucl. Phys. B 487 721 [hep-th/9609174]
30. G. Akemann, P.H. Damgaard, U. Magnea and S. Nishigaki, 1998 Multicritical microscopic spectral correlators of Hermitian and complex matrices, Nucl. Phys. B 519 682 [hep-th/9712006]
31. T. Guhr, A. Müller-Groeling and H.A. Weidenmuller, 1998 Random matrix theories in quantum physics: common concepts, Phys. Rept. 299 189 [cond-mat/9707301]
32. H.W. Braden, A. Mironov and A. Morozov, 2001 QCD, Wick's theorem for KdV τ-functions and the string equation, Phys. Lett. B 514 293 [hep-th/0105169]
33. B. Eynard, 2000 Random matrices, Saclay Lecture Notes, France, http://www-spht.cea.fr/articles-k2/t01/014/publi.pdf
34. A. Zabrodin, 2003 New applications of non-Hermitian random matrices, Ann. Poincaré 4 S851 [cond-mat/0210331]
35. S.Y. Alexandrov, V.A. Kazakov and D. Kutasov, 2003 Non-perturbative effects in matrix models and D-branes J. High Energy Phys. JHEP09(2003)057 [hep-th/0306177]
36. P.J. Forrester, N.C. Snaith and J.J.M. Verbaarschot, 2003 Developments in random matrix theory, J. Phys. A 36 R1 [cond-mat/0303207]
37. P. Wiegmann and A. Zabrodin, Large-N expansion for normal and complex matrix ensembles, [hep-th/0309253]
38. M. Aganagic, R. Dijkgraaf, A. Klemm, M. Marĩo and C. Vafa, 2006 Topological strings and integrable hierarchies, Commun. Math. Phys. 261 451 [hep-th/0312085]
39. R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin and P. Wiegmann, 2005 Normal random matrix ensemble as a growth problem: evolution of the spectral curve, Nucl. Phys. B 704 407 [hep-th/0401165]
40. V.A. Kazakov and I.K. Kostov, Instantons in non-critical strings from the two-matrix model, [hep-th/0403152]
41. G. Akemann, Y.V. Fyodorov and G. Vernizzi, 2004 On matrix model partition functions for QCD with chemical potential, Nucl. Phys. B 694 59 [hep-th/0404063]
42. P. Di Francesco, 2 D quantum gravity, matrix models and graph combinatorics, [math-ph/0406013]
43. A. Morozov, Challenges of matrix models, [hep-th/0502010]
44. J. Harnad and A.Y. Orlov, 2007 Fermionic construction of τ functions and random processes, Physica D 235 168 [0704.1157]
45. A. Klemm and P. Sulkowski, 2009 Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 400 [0810.4944]
46. F.J. Dyson, 1962 Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 140
47. A. Morozov, 1992 String theory, what is it?, Sov. Phys. Usp. 35 671
48. A. Morozov, 1992 Usp. Fiz. Nauk 162 83
49. A. Morozov, 1994 Integrability and matrix models, Phys. Usp. 37 1 [hep-th/9303139]
50. Matrix models as integrable systems, [hep-th/9502091] - ref-separator.
51. A. Mironov, 1994 2D gravity and matrix models. 1. 2D gravity, Int. J. Mod. Phys. A 9 4355 [hep-th/9312212]
52. A. Mironov, 2002 Matrix models of two-dimensional gravity Phys. Part. Nucl. 33 537
53. A. Mironov, 2002 Fiz. Elem. Chast. Atom. Yadra 33 1051
54. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, 1995 Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 466 [hep-th/9505035]
55. A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, 1998 RG equations from Whitham hierarchy, Nucl. Phys. B 527 690 [hep-th/9802007]
56. A. Marshakov, A. Mironov and A. Morozov, 1996 WDVV-like equations in N=2 SUSY Yang-Mills theory, Phys. Lett. B 389 43 [hep-th/9607109]
57. A. Marshakov, A. Mironov and A. Morozov, 2000 More evidence for the WDVV equations in N=2 SUSY Yang-Mills theories, Int. J. Mod. Phys. A 15 1157 [hep-th/9701123]
58. H.W. Braden and I. Krichever eds., 2000 Integrability: the Seiberg-Witten and Whitham equations, Gordon and Breach, U.S.A.
59. A. Gorsky and A. Mironov, Integrable many-body systems and gauge theories, [hep-th/0011197]
60. A. Hurwitz, Uber Riemann'sche Flachen mit gegebenen Verzweigungpunkten (in German), Math. Ann. 39 (1891) 1
61. Uber die Anzal der Riemann'sche Flachen mit gegebenen Verzweigungpunkten (in German), Math. Ann. 55 1902 53
62. G. Frobenius, Uber gruppencharakter (in German), Sitzberg. Koniglich Preuss. Akad. Wiss. Berlin, Germany (1896), pg. 985
63. R. Dijkgraaf, 1995 Mirror symmetry and elliptic curves, in The moduli spaces of curves Prog. Math. 129 149, Brikhäuser, U.S.A.
64. R. Vakil, 1997 Enumerative geometry of curves via degeneration methods, Harvard Ph.D. thesis, U.S.A.
65. I. Goulden and D. Jackson, 1997 Transitive factorisations into transpositions and holomorphic mappings on the sphere Proc. Am. Math. Soc. 125 51 [math.CO/9903094]
66. S. Lando and D. Zvonkine, 1999 On multiplicitites of the Lyashko-Looijenga mapping on the discriminant strata, Funct. Anal. Appl. 33 178
67. Counting ramified coverings and intersection theory on spaces of rational functions. I, [math.AG/0303218] - ref-separator.11
68. S. Natanzon and V. Turaev, 1999 A compactification of Hurwitz space, Topology 38 889
69. D. Goulden, D.M. Jackson and A. Vainshtein, 2000 The number of ramifified coverings of the sphere by torus and surfaces of higher genera, Ann. Comb. 4 27, Brikhäuser, U.S.A.
70. A. Okounkov, 2000 Toda equations for Hurwitz numbers Math. Res. Lett. 7 447
71. A. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians, [math.AG/0108100]
72. T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein, 2001 Hurvitz naumbers and intersections on moduli spaces of curves, Invent. Math. 146 297
73. S. Lando, 2002 Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves, Russ. Math. Surveys 57 463
74. A. Alexeevski and S. Natanzon, 2006 Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves, Selecta Math. 12 307 [math.GT/0202164]
75. S. Natanzon, 2006 The normality of some elliptic functional-differential operators of second order, Russ. Math. Surveys 61 185
76. Disk single Hurwitz numbers, [0804.0242] - ref-separator.
77. A. Alexeevski and S. Natanzon, 2008 Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebra of finite groups Am. Math. Soc. Transl. 224 1 [0709.3601]
78. A. Alexeevski and S. Natanzon, 2008 Algebra of two-fold graphs and Hurwitz numbers for seamed surfaces Izvestiya RAN. Seriya fizicheskaya 12 3
79. J. Zhou, Hodge integrals, Hurwitz numbers and symmetric groups, [math.AG/0308024]
80. A. Okounkov and R. Pandharipande, 2006 Gromov-Witten theory, Hurwitz theory, and completed cycles Ann. Math. 163 517 [math.AG/0204305]
81. T. Graber and R. Vakil, 2003 Hodge integrals and Hirwitz numbers via virtual localization Compositio Math. 135 25
82. M. Kazarian and S. Lando, 2004 Towards the intersection theory of Hurwitz spaces Izv. Ross. Akad. Nauk Ser. Mat. 68 82 [math.AG/0410388]
83. An algebro-geometric proof of Witten's conjecture, [math.AG/0601760] - ref-separator.
84. M. Kazarian, KP hierarchy for Hodge integrals, based on the talk at the Moscow Workshop on Combinatorics of moduli spaces, Hurwitz numbers and cluster algebras, Moscow Russia June 2008, [0809.3263]
85. V. Bouchard and M. Marĩo, 2008 Hurwitz numbers, matrix models and enumerative geometry, in From Hodge Theory to Integrability and tQFT: tt*-geometry, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, U.S.A. [0709.1458]
86. A. Mironov and A. Morozov, 2009 Virasoro constraints for Kontsevich-Hurwitz partition function J. High Energy Phys. JHEP02(2009)024 [0807.2843]
87. A. Mironov, A. Morozov and S. Natanzon, Complete set of cut-and-join operators in Hurwitz-Kontsevich theory, [0904.4227]
88. The standard tau-functions of KP-Toda families are considered in many places, see for example E. Date, M. Jimbo, M. Kashiwara and T. Miwa, 1983 Transformation groups for soliton equations, RIMS Symp. Non-linear integrable systems - classical theory and quantum theory, World Scientific, Singapore
89. V. Kac, 1985 Infinite-dimensional Lie algebras, section 14, Cambridge University Press, Cambridge U.K.
90. V. Kac and M. Wakimoto, 1989 Exceptional hierarchies of soliton equations, Proceedings of Symposia in Pure Mathematics 49 191
91. For a concept of generalized τ-functions see A. Mironov, A. Morozov and L. Vinet, 1995 On a c-number quantum τ function, Theor. Math. Phys. 100 890 [hep-th/9312213]
92. For a concept of generalized τ-functions see A. Mironov, A. Morozov and L. Vinet, 1994 Teor. Mat. Fiz. 100 119
93. q(2), Int. J. Mod. Phys. A 10 2589 [hep-th/9405011]
94. S. Kharchev, A. Mironov and A. Morozov, 1995 Non-standard KP evolution and quantum τ-function, Theor. Math. Phys. 104 866 [q-alg/9501013]
95. A. Mironov, Quantum deformations of τ functions, bilinear identities and representation theory, [hep-th/9409190]
96. A. Mironov, 1998 τ-function within group theory approach and its quantization, Theor. Math. Phys. 114 127 [q-alg/9711006]
97. A. Mironov, 1998 Teor. Mat. Fiz. 114 163
98. E. Witten, 1990 On the structure of the topological phase of two-dimensional gravity, Nucl. Phys. B 340 281
99. M. Fukuma, H. Kawai and R. Nakayama, 1991 Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Int. J. Mod. Phys. A 6 1385
100. R. Dijkgraaf, E. Verlinde and H. Verlinde, 1991 Topological strings in D < 1, Nucl. Phys. B 352 59
101. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, 1991 Matrix models of 2D gravity and Toda theory, Nucl. Phys. B 357 565
102. A. Gerasimov et al., 1991 Matrix models as integrable systems: From universality to geometrodynamical principle of string theory, Mod. Phys. Lett. A 6 3079
103. A. Mironov and A. Morozov, 1990 On the origin of Virasoro constraints in matrix models: Lagrangian approach, Phys. Lett. B 252 47
104. F. David, 1990 Loop equations and nonperturbative effects in two-dimensional quantum gravity, Mod. Phys. Lett. A 5 1019
105. H. Itoyama and Y. Matsuo, 1991 Noncritical Virasoro algebra of d < 1 matrix model and quantized string field, Phys. Lett. B 255 202
106. J. Ambjørn and Y.M. Makeenko, 1990 Properties of loop equations for the hermitean matrix model and for two-dimensional quantum gravity, Mod. Phys. Lett. A 5 1753
107. A. Marshakov, A. Mironov and A. Morozov, 1991 Generalized matrix models as conformal field theories: discrete case, Phys. Lett. B 265 99
108. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and S. Pakuliak, 1993 Conformal matrix models as an alternative to conventional multimatrix models, Nucl. Phys. B 404 717 [hep-th/9208044]
109. A. Mironov and S. Pakulyak, 1993 On the continuum limit of the conformal matrix models, Theor. Math. Phys. 95 604 [hep-th/9209100]
110. A. Mironov and S. Pakulyak, 1993 Teor. Mat. Fiz. 95 317
111. A. Mironov and S. Pakulyak, 1993 Int. J. Mod. Phys. A 8 3107
112. A. Marshakov, A. Mironov and A. Morozov, 1992 On equivalence of topological and quantum 2D gravity, Phys. Lett. B 274 280 [hep-th/9201011]
113. A. Marshakov, A. Mironov and A. Morozov, 1992 From Virasoro constraints in Kontsevich's model to W constraints in two matrix model, Mod. Phys. Lett. A 7 1345 [hep-th/9201010]
114. A. Mironov, A. Morozov and G.W. Semenoff, 1996 Unitary matrix integrals in the framework of generalized Kontsevich model. 1. Brezin-Gross-Witten model, Int. J. Mod. Phys. A 11 5031 [hep-th/9404005]
115. A. Morozov and S. Shakirov, 2009 Generation of matrix models by -operators J. High Energy Phys. JHEP04(2009)064 [0902.2627]
116. M. Kontsevich, 1991 Intersection theory on the moduli space of curves, Funct. Anal. Appl. 25 123
117. M. Kontsevich, 1991 Funkt. Anal. Pril. 25 50
118. M. Kontsevich, 1992 Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 1
119. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, 1992 Unification of all string models with C < 1, Phys. Lett. B 275 311 [hep-th/9111037]
120. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, 1992 Towards unified theory of 2D gravity, Nucl. Phys. B 380 181 [hep-th/9201013]
121. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, 1993 Generalized Kontsevich model versus Toda hierarchy and discrete matrix models, Nucl. Phys. B 397 339 [hep-th/9203043]
122. P. Di Francesco, C. Itzykson and J.B. Zuber, 1993 Polynomial averages in the Kontsevich model, Commun. Math. Phys. 151 193 [hep-th/9206090]
123. A.S. Alexandrov, A. Mironov and A. Morozov, 2004 Partition functions of matrix models as the first special functions of string theory. I: finite size Hermitean 1-matrix model, Int. J. Mod. Phys. A 19 4127 [hep-th/0310113]
124. A.S. Alexandrov, A. Mironov and A. Morozov, 2005 Teor. Mat. Fiz. 142 419
125. A.S. Alexandrov, A. Mironov, A. Morozov and P. Putrov, 2009 Partition functions of matrix models as the first special functions of string theory. II. Kontsevich model, Int. J. Mod. Phys. A 24 4939 [0811.2825]
126. These papers collect results from a vast variety of other works, see reference lists therein - ref-separator.
127. A. Givental, Semisimple Frobenius structures at higher genus, [math.AG/0008067]
128. B. Eynard, 2004 Topological expansion for the 1-hermitian matrix model correlation functions J. High Energy Phys. JHEP11(2004)031 [hep-th/0407261]
129. B. Eynard, 2003 Large-N expansion of the 2-matrix model J. High Energy Phys. JHEP01(2003)051 [hep-th/0210047]
130. B. Eynard, 2003 Master loop equations, free energy and correlations for the chain of matrices J. High Energy Phys. JHEP11(2003)018 [hep-th/0309036]
131. B. Eynard and N. Orantin, 2005 Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula J. High Energy Phys. JHEP12(2005)034 [math-ph/0504058]
132. L. Chekhov and B. Eynard, 2006 Hermitean matrix model free energy: Feynman graph technique for all genera J. High Energy Phys. JHEP03(2006)014 [hep-th/0504116]
133. L. Chekhov and B. Eynard, 2006 Matrix eigenvalue model: Feynman graph technique for all genera J. High Energy Phys. JHEP12(2006)026 [math-ph/0604014]
134. B. Eynard, M. Marĩo and N. Orantin, 2007 Holomorphic anomaly and matrix models J. High Energy Phys. JHEP06(2007)058 [hep-th/0702110]
135. N. Orantin, Gaussian matrix model in an external field and non-intersecting Brownian motions, [0803.0705]
136. Symplectic invariants, Virasoro constraints and Givental decomposition, [0808.0635] - ref-separator.
137. I.M. Krichever and S.P. Novikov, 1987 Algebras of Virasoro type, Riemann surfaces and the structure of soliton theory Funct. Anal. Appl. 21 126
138. I.M. Krichever and S.P. Novikov, 1987 Funkt. Anal. Pril. 21 47
139. I.M. Krichever and S.P. Novikov, 1988 Virasoro-Gelfand-Fuks type algebras, Riemann surfaces, operator's theory of closed strings, J. Geom. Phys. 5 631
140. I.M. Krichever and S.P. Novikov, 1987 Virasoro-type algebras, Riemann surfaces and strings in Minkowsky space, Funct. Anal. Appl. 21 294
141. I.M. Krichever and S.P. Novikov, 1989 Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces, Funct. Anal. Appl. 23 19
142. T. Morris, 1991 Checkered surfaces and complex matrices, Nucl. Phys. B 356 703
143. Y. Makeenko, 1990 The complex matrix model and two-dimensional quantum gravity Pisma Zh. Eksp. Teor. Fiz. 52 885
144. Y. Makeenko, 1990 JETP Lett. 52 259
145. A. Anderson, R.C. Myers and V. Periwal, 1991 Complex random surfaces, Phys. Lett. B 254 89
146. Y. Makeenko, A. Marshakov, A. Mironov and A. Morozov, 1991 Continuum versus discrete Virasoro in one matrix models, Nucl. Phys. B 356 574
147. J. Ambjørn, C.F. Kristjansen and Y.M. Makeenko, 1992 Higher genus correlators for the complex matrix model, Mod. Phys. Lett. A 7 3187 [hep-th/9207020]
148. E. Brezin and D. Gross, 1980 The external field problem in the large N limit of QCD, Phys. Lett. B 97 120
149. D. Gross and E. Witten, 1980 Possible third-order phase transition in the large-N lattice gauge theory, Phys. Rev. D 21 446
150. D. Gross and M. Newman, 1991 Unitary and hermitian matrices in an external field, Phys. Lett. B 266 291
151. K.G. Wilson, 1974 Confinement of quarks, Phys. Rev. D 10 2445
152. A. Polyakov, 1987 Gauge fields and strings
153. V.A. Kazakov and A.A. Migdal, 1993 Induced QCD at large-N, Nucl. Phys. B 397 214 [hep-th/9206015]
154. I.I. Kogan, A. Morozov, G.W. Semenoff and N. Weiss, 1993 Area law and continuum limit in 'induced QCD', Nucl. Phys. B 395 547 [hep-th/9208012]
155. I.I. Kogan, A. Morozov, G.W. Semenoff and N. Weiss, 1993 Continuum limits of 'induced QCD': lessons of the Gaussian model at D=1 and beyond, Int. J. Mod. Phys. A 8 1411 [hep-th/9208054]
156. A. Mironov, A. Morozov and T.N. Tomaras, 2005 On the need for phenomenological theory of P-vortices or does spaghetti confinement pattern admit condensed-matter analogies?, J. Exp. Theor. Phys. 101 331 [hep-th/0503212]
157. Harish-Chandra, 1957 Differential operators on a semisimple Lie algebra Am. J. Math. 79 87
158. C. Itzykson and J.B. Zuber, 1980 The planar approximation. 2, J. Math. Phys. 21 411
159. J. Duistermaat and G. Heckman, 1982 On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 259
160. A. Hietamaki, A.Y. Morozov, A.J. Niemi and K. Palo, 1991 Geometry of N=1/2 supersymmetry and the Atiyah-Singer index theorem, Phys. Lett. B 263 417
161. A.Y. Morozov, A. Niemi and K. Palo, 1992 Supersymplectic geometry of supersymmetric quantum field theories, Nucl. Phys. B 377 295
162. M.J. Bowick, A. Morozov and D. Shevitz, 1991 Reduced unitary matrix models and the hierarchy of τ functions, Nucl. Phys. B 354 496
163. A. Morozov, 1992 Pair correlator in the Itzykson-Zuber integral, Mod. Phys. Lett. A 7 3503 [hep-th/9209074]
164. S.L. Shatashvili, 1993 Correlation functions in the Itzykson-Zuber model, Commun. Math. Phys. 154 421 [hep-th/9209083]
165. B. Eynard, A short note about Morozov's formula, [math-ph/0406063]
166. A.P. Ferrer, B. Eynard, P. Di Francesco and J.B. Zuber, 2009 Correlation functions of Harish-Chandra integrals over the orthogonal and the symplectic groups, J. Stat. Phys. 129 885 [math-ph/0610049]
167. M. Bergere and B. Eynard, 2009 Some properties of angular integrals, J. Phys. A 42 265201 [0805.4482]
168. S. Kharchev and A. Mironov, 1992 Integrable structures of unitary matrix models, Int. J. Mod. Phys. A 7 4803
169. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, 1995 Generalized Kazakov-Migdal-Kontsevich model: group theory aspects, Int. J. Mod. Phys. A 10 2015 [hep-th/9312210]
170. S. Kharchev, A. Marshakov, A. Mironov, A. Orlov and A. Zabrodin, 1991 Matrix models among integrable theories: forced hierarchies and operator formalism, Nucl. Phys. B 366 569
171. G. Segal and G. Wilson, 1985 Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61 5
172. D. Friedan and S.H. Shenker, 1986 The integrable analytic geometry of quantum string, Phys. Lett. B 175 287
173. D. Friedan and S.H. Shenker, 1987 The analytic geometry of two-dimensional conformal field theory, Nucl. Phys. B 281 509
174. N. Ishibashi, Y. Matsuo and H. Ooguri, 1987 Soliton equations and free fermions on Riemann surfaces, Mod. Phys. Lett. A 2 119
175. L. Alvarez-Gaume, C. Gomez and C. Reina, 1987 Loop groups, Grassmannians and string theory, Phys. Lett. B 190 55
176. A. Morozov, 1987 String theory and the structure of universal module space JETP Lett. 45 585
177. A. Morozov, 1987 Pisma Zh. Eksp. Teor. Fiz. 45 457
178. A. Morozov, 1987 Phys. Lett. B 196 325
179. A.S. Schwarz, 1989 Fermionic string and universal moduli space, Nucl. Phys. B 317 323
180. J. Harer and D. Zagier, 1986 The Euler characteristic of the moduli space of curves I, Invent. Math. 85 457
181. S.K. Lando and A.K. Zvonkine, 2003 Graphs on surfaces and their applications, Springer, U.S.A.
182. E. Akhmedov and S. Shakirov, Gluing of surfaces with polygonal boundaries, to appear in Funkt. Anal. Pril. [0712.2448]
183. A. Morozov and S. Shakirov, Exact 2-point function in Hermitian matrix model, [0906.0036]
184. K. Ueno and K. Takasaki, 1984 Toda lattice hierarchy Adv. Stud. Pure Math. 4 1
185. M. Semenov-Tian-Shansky, 1985 Dressing transformations and Poisson group actions Publ. Res. Inst. Math. Sci. Kyoto 21 1237
186. S. Novikov, S. Manakov, L. Pitaevskii and V.E. Zakharov, 1984 Theory of solitons. The inverse scattering method, Plenum Press, New York U.S.A.
187. L. Chekhov and Y. Makeenko, 1992 A hint on the external field problem for matrix models, Phys. Lett. B 278 271 [hep-th/9202006]
188. L. Chekhov and Y. Makeenko, 1992 The multicritical Kontsevich-Penner model, Mod. Phys. Lett. A 7 1223 [hep-th/9201033]
189. R. Penner, 1987 Perturbative series and the moduli space of Riemann surfaces J. Diff. Geom. 27 35
190. C. Itzykson and J.B. Zuber, 1980 The planar approximation. 2, J. Math. Phys. 21 411
191. L. Chekhov, A. Marshakov, A. Mironov and D. Vasiliev, 2005 Complex geometry of matrix models Proc. Steklov Inst. Math. 251 254 [hep-th/0506075]
192. A.S. Alexandrov, A. Mironov and A. Morozov, 2006 Unified description of correlators in non-Gaussian phases of Hermitean matrix model, Int. J. Mod. Phys. A 21 2481 [hep-th/0412099]
193. A.S. Alexandrov, A. Mironov and A. Morozov, 2005 Solving Virasoro constraints in matrix models, Fortsch. Phys. 53 512 [hep-th/0412205]
194. R. Dijkgraaf and C. Vafa, 2002 Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 3 [hep-th/0206255]
195. R. Dijkgraaf and C. Vafa, 2002 On geometry and matrix models, Nucl. Phys. B 644 21 [hep-th/0207106]
196. A perturbative window into non-perturbative physics, [hep-th/0208048] - ref-separator.
197. L. Chekhov and A. Mironov, 2003 Matrix models vs. Seiberg-Witten/Whitham theories, Phys. Lett. B 552 293 [hep-th/0209085]
198. R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, 2003 Perturbative analysis of gauged matrix models, Phys. Rev. D 68 045007 [hep-th/0210238]
199. V.A. Kazakov and A. Marshakov, 2003 Complex curve of the two matrix model and its τ-function, J. Phys. A 36 3107 [hep-th/0211236]
200. H. Itoyama and A. Morozov, 2003 The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory, Nucl. Phys. B 657 53 [hep-th/0211245]
201. H. Itoyama and A. Morozov, 2003 Experiments with the WDVV equations for the gluino-condensate prepotential: the cubic (two-cut) case, Phys. Lett. B 555 287 [hep-th/0211259]
202. Calculating gluino condensate prepotential, Prog. Theor. Phys. 109 2003 433 [hep-th/0212032]
203. Gluino-condensate (CIV-DV) prepotential from its Whitham-time derivatives, Int. J. Mod. Phys. A 18 2003 5889 [hep-th/0301136]
204. S.G. Naculich, H.J. Schnitzer and N. Wyllard, 2003 Matrix model approach to the N=2 U(N) gauge theory with matter in the fundamental representation J. High Energy Phys. JHEP01(2003)015 [hep-th/0211254]
205. B. Feng, 2003 Geometric dual and matrix theory for SO/Sp gauge theories, Nucl. Phys. B 661 113 [hep-th/0212010]
206. I. Bena, S. de Haro and R. Roiban, 2003 Generalized Yukawa couplings and matrix models, Nucl. Phys. B 664 45 [hep-th/0212083]
207. C.-H. Ahn, 2003 Supersymmetric SO(N)/Sp(N) gauge theory from matrix model: exact mesonic vacua, Phys. Lett. B 560 116 [hep-th/0301011]
208. L. Chekhov, A. Marshakov, A. Mironov and D. Vasiliev, 2003 DV and WDVV, Phys. Lett. B 562 323 [hep-th/0301071]
209. A. Mironov, 2006 Matrix models vs. matrix integrals, Theor. Math. Phys. 146 63 [hep-th/0506158]
210. A. Mironov, 2006 Teor. Mat. Fiz. 146 77
211. A. Dymarsky and V. Pestun, 2003 On the property of Cachazo-Intriligator- Vafa prepotential at the extremum of the superpotential, Phys. Rev. D 67 125001 [hep-th/0301135]
212. Y. Ookouchi and Y. Watabiki, 2003 Effective superpotentials for SO/Sp with flavor from matrix models, Mod. Phys. Lett. A 18 1113 [hep-th/0301226]
213. H. Itoyama and H. Kanno, 2003 Supereigenvalue model and Dijkgraaf-Vafa proposal, Phys. Lett. B 573 227 [hep-th/0304184]
214. H. Itoyama and H. Kanno, 2004 Whitham prepotential and superpotential, Nucl. Phys. B 686 155 [hep-th/0312306]
215. M. Matone and L. Mazzucato, 2003 Branched matrix models and the scales of supersymmetric gauge theories J. High Energy Phys. JHEP07(2003)015 [hep-th/0305225]
216. R. Argurio, G. Ferretti and R. Heise, 2004 An introduction to supersymmetric gauge theories and matrix models, Int. J. Mod. Phys. A 19 2015 [hep-th/0311066]
217. M. Gomez-Reino, 2004 Exact superpotentials, theories with flavor and confining vacua J. High Energy Phys. JHEP06(2004)051 [hep-th/0405242]
218. K. Fujiwara, H. Itoyama and M. Sakaguchi, 2005 Supersymmetric U(N) gauge model and partial breaking of N=2 supersymmetry, Prog. Theor. Phys. 113 429 [hep-th/0409060]
219. K. Fujiwara, H. Itoyama and M. Sakaguchi, 2005 Partial breaking of N=2 supersymmetry and of gauge symmetry in the U(N) gauge model, Nucl. Phys. B 723 33 [hep-th/0503113]
220. Supersymmetric U(N) gauge model and partial breaking of N=2 supersymmetry, Prog. Theor. Phys. Suppl. 164 2007 125 [hep-th/0602267]
221. S. Aoyama, 2005 The disc amplitude of the Dijkgraaf-Vafa theory: 1/N expansion vs. complex curve analysis J. High Energy Phys. JHEP10(2005)032 [hep-th/0504162]
222. D. Berenstein and S. Pinansky, Counting conifolds and Dijkgraaf-Vafa matrix models for three matrices, [hep-th/0602294]
223. L. Landau and E. Lifshitz, Mechanics, volume 1
224. H. Bateman and A. Erdelyi, 1953 Higher transcendental functions, volume 2, London U.K.
225. F.A. Berezin and F.I. Karpelevich, 1958 Zonal spherical functions and Laplace operators on some symmetric spaces (in Russian), Dokl. Akad. Nauk SSSR 118 9
226. A.D. Jackson, M.K. Sener and J.J.M. Verbaarschot, 1996 Finite volume partition functions and Itzykson-Zuber integrals, Phys. Lett. B 387 355 [hep-th/9605183]
227. T. Guhr and T. Wettig, 1996 An Itzykson-Zuber-like integral and diffusion for complex ordinary and supermatrices, J. Math. Phys. 37 6395 [hep-th/9605110]
228. E. Brézin and V.A. Kazakov, 1990 Exactly solvable field theories of closed strings, Phys. Lett. B 236 144
229. D.J. Gross and A.A. Migdal, 1990 Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 127
230. D.J. Gross and A.A. Migdal, 1990 A nonperturbative treatment of two-dimensional quantum gravity, Nucl. Phys. B 340 333
231. M. Douglas and S. Shenker, 1990 Strings in less than one dimension, Nucl. Phys. B 335 635
232. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, 1993 Landau-Ginzburg topological theories in the framework of GKM and equivalent hierarchies Mod. Phys. Lett. A 8 1047 [hep-th/9208046]
233. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, 1993 Theor. Math. Phys. 95 571
234. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, 1993 Teor. Mat. Fiz. 95 280
235. L.K. Hua, 1963 Harmonic analysis of functions of several complex variables in the classical domain, American Mathematical Society, Providence RI U.S.A.
236. V.A. Kazakov, 1983 U() lattice gauge theory as a free lattice string theory, Phys. Lett. B 128 316
237. L. Chekhov, Matrix models and geometry of moduli spaces, [hep-th/9509001]
238. I.K. Kostov, Conformal field theory techniques in random matrix models, [hep-th/9907060]
239. V.S. Dotsenko and V.A. Fateev, 1984 Conformal algebra and multipoint correlation functions in 2 D statistical models, Nucl. Phys. B 240 312
240. (1), Commun. Math. Phys. 104 605
241. V. Knizhnik, 1989 Multiloop amplitudes in the theory of quantum strings and complex geometry Usp. Fiz. Nauk 159 401
242. V. Knizhnik, 1989 Sov. Phys. Usp. 32 945
243. A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky and S. Shatashvili, 1990 Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 2495