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Unitary integrals and related matrix models. (English. Russian original) Zbl 1196.81201
Theor. Math. Phys. 162, No. 1, 1-33 (2010); translation from Teor. Mat. Fiz. 161, No. 3, 3-40 (2010).
Summary: We briefly review the basic properties of unitary matrix integrals, using three matrix models to analyze their properties: the ordinary unitary, the Brezin-Gross-Witten, and the Harish-Chandra-Itzykson-Zuber models. We especially emphasize the nontrivial aspects of the theory, from the De Witt’Hooft anomaly in unitary integrals to the problem of calculating correlators with the Itzykson-Zuber measure. We emphasize the method of character expansions as a technical tool. Unitary integrals are still insufficiently investigated, and many new results should be expected as this field attracts increased attention.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:
[1] E. P. Wigner, Ann. Math., 62, 548–564 (1955); F. J. Dyson, J. Math. Phys., 3, 140, 157, 166, 1191, 1199 (1962); E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Comm. Math. Phys., 59, 35–51 (1978); D. Bessis, Comm. Math. Phys., 69, 147–163 (1979); D. Bessis, C. Itzykson, and J.-B. Zuber, Adv. Appl. Math., 1, 109–157 (1980); M. L. Mehta, Comm. Math. Phys., 79, 327–340 (1981); Random Matrices, Acad. Press, Boston, Mass. (1991); D. Bessis, C. Itzykson, and J.-B. Zuber, Adv. Appl. Math., 1, 109–157 (1980); A. A. Migdal, Phys. Rep., 102, 199–290 (1983); F. David, Nucl. Phys. B, 257, 45–58, 543–576 (1985); J. Ambjørn, B. Durhuus, and J. Fröhlich, Nucl. Phys. B, 257, 433–449 (1985); V. A. Kazakov, I. K. Kostov, and A. A. Migdal, Phys. Lett. B, 157, 295–300 (1985); D. V. Boulatov, V. A. Kazakov, I. K. Kostov, and A. A. Migdal, Phys. Lett. B, 174, 87–93 (1986); Nucl. Phys. B, 275, 641–686 (1986); V. A. Kazakov, Phys. Lett. A, 119, 140–144 (1986); Modern Phys. Lett. A, 4, 2125–2139 (1989); E. Brézin and V. Kazakov, Phys. Lett. B, 236, 144–150 (1990); D. J. Gross and A. A. Migdal, Phys. Rev. Lett., 64, 127–130 (1990); Nucl. Phys. B, 340, 333–365 (1990); M. Douglas and S. Shenker, Nucl. Phys. B, 335, 635–654 (1990); L. Alvarez-Gaumé, Helv. Phys. Acta, 64, 359–526 (1991); M. R. Douglas, Phys. Lett. B, 238, 176–180 (1990); A. Levin and A. Morozov, Phys. Lett. B, 243, 207–214 (1990); P. Ginsparg, ”Matrix models of 2d gravity,” arXiv:hep-th/9112013v1 (1991); D. J. Gross and M. J. Newmann, Phys. Lett. B, 266, 291–297 (1991); A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 265, 99–107 (1991); R. Dijkgraaf, G. Moore, and R. Plesser, Nucl. Phys. B, 394, 356–382 (1993); arXiv:hep-th/9208031v1 (1992); J.-M. Daul, V. A. Kazakov, and I. K. Kostov, Nucl. Phys. B, 409, 311–338 (1993); M. Staudacher, Phys. Lett. B, 305, 332–338 (1993); arXiv:hep-th/9301038v1 (1993); J. Ambjørn, L. Chekhov, C. F. Kristjansen, and Yu. Makeenko, Nucl. Phys. B, 404, 127–172 (1993); Erratum, 449, 681 (1995); arXiv:hep-th/9302014v1 (1993); P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, Phys. Rep., 254, 1–133 (1995); arXiv:hep-th/9306153v2 (1993); A. Mironov, Internat. J. Mod. Phys. A, 9, 4355–4405 (1994); arXiv:hep-th/9312212v1 (1993); Phys. Part. Nucl., 33, 537–582 (2002); B. Eynard, ”Large random matrices: Eigenvalue distribution,” arXiv:hep-th/9401165v1 (1994); M. Adler, A. Morozov, T. Shiota, and P. van Moerbeke, Nucl. Phys. Proc. Suppl., 49, 201–212 (1996); arXiv:hep-th/9603066v1 (1996); G. Akemann, Nucl. Phys. B, 482, 403–430 (1996); arXiv:hep-th/9606004v2 (1996); G. Akemann, P. H. Damgaard, U. Magnea, and S. Nishigaki, Nucl. Phys. B, 487, 721–738 (1997); arXiv:hep-th/9609174v2 (1996); 519, 682–714 (1998); arXiv:hep-th/9712006v2 (1997); J. Ambjørn, M. G. Harris, and M. Weis, Nucl. Phys. B, 504, 482–510 (1997); arXiv:hep-th/9702188v1 (1997); A. Marshakov, M. Martellini, and A. Morozov, Phys. Lett. B, 418, 294–302 (1998); arXiv:hep-th/9706050v1 (1997); T. Guhr, A. Mueller-Groeling, and H. A. Weidenmüller, Phys. Rep., 299, 189–425 (1998); arXiv:cond-mat/9707301v1 (1997); V. A. Kazakov, I. K. Kostov, and N. A. Nekrasov, Nucl. Phys. B, 557, 413–442 (1999); arXiv:hep-th/9810035v2 (1998); M. Adler and P. van Moerbecke, Ann. Math., 149, 921–976 (1999); B. Eynard, ”Random matrices,” Saclay lecture notes, http://ipht.cea.fr/Docspht//articles/t01/014/public/publi.pdf (2000); A. Gerasimov, A. Morozov, and K. Selivanov, Internat. J. Mod. Phys. A, 16, 1531–1558 (2001); arXiv:hep-th/00050531 (2000); A. Okounkov, Internat. Math. Res. Notices, 2002, No. 18, 933–957 (2002); arXiv:math.AG/0101201v2 (2001); P. Ormerod and C. Mounfield, Phys. A, 280, 497–504 (2000); arXiv:cond-mat/0102357v1 (2001); P. Di Francesco, Nucl. Phys. B, 648, 461–496 (2003); arXiv:cond-mat/02080371 (2002); P. J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot, J. Phys. A, 36, R1-R10 (2003); arXiv:cond-mat/0303207v1 (2003); S. Alexandrov, V. Kazakov, and D. Kutasov, JHEP, 0309, 057 (2003); arXiv:hep-th/0306177v2 (2003); N. Seiberg and D. Shih, JHEP, 0402, 021 (2004); arXiv:hep-th/0312170v2 (2003); V. Kazakov and I. Kostov, ”Instantons in non-critical strings from the two-matrix model,” in: From Fields to Strings: Circumnavigating Theoretical Physics (M. Shifman, A. Vainshtein, and J. Wheater, eds.), Vol. 3, World Scientific, Singapore (2005), pp. 1864–1894; arXiv:hep-th/ 0403152v3 (2004); G. Akemann, Y. V. Fyodorov, and G. Vernizzi, Nucl. Phys. B, 694, 59–98 (2004); arXiv:hep-th/0404063v1 (2004); P. Di Francesco, ”2D quantum gravity, matrix models, and graph combinatorics,” in: Applications of Random Matrices in Physics (NATO Sci. Ser. II Math. Phys. Chem., Vol. 221, É. Brézin, V. Kazakov, Didina Serban, P. Wiegmann, and A. Zabrodin, eds.), Springer, Dordrecht (2006), pp. 33–88; arXiv:math-ph/0406013v2 (2004); A. Morozov, ”Challenges of matrix models,” in: String Theory: From Gauge Interactions to Cosmology (NATO Sci. Ser. II Math. Phys. Chem., Vol. 208, L. Baulieu, J. de Boer, B. Pioline, and E. Rabinovici, eds.), Vol. 208, Springer, Dordrecht (2006), pp. 129–162; arXiv:hep-th/0502010v2 (2005); M. Stephanov, J. J. M. Verbaarschot, and T. Wettig, ”Random matrices,” arXiv:hep-ph/0509286v1 (2005); Y. V. Fyodorov and B. A. Khoruzhenko, Comm. Math. Phys., 273, 561–599 (2007); arXiv:math-ph/0602032v2 (2006); Y. V. Fyodorov, Internat. Math. Res. Notices, 61570 (2006); arXiv:math-ph/0602039v2 (2006); J. Harnad and A. Yu. Orlov, Phys. D, 235, 168–206 (2007); arXiv:0704.1157v1 [math-ph] (2007). · Zbl 0067.08403
[2] A. A. Migdal, Sov. Phys. JETP, 42, 413–418 (1975); C. Itzykson and J.-B. Zuber, Comm. Math. Phys., 134, 197–207 (1990); P. Di Francesco and C. Itzykson, Ann. Inst. H.Poincaré;, 59, 117–139 (1993); V. A. Kazakov, M. Staudacher, and T. Wynter, Comm. Math. Phys., 179, 235–256 (1996); I. K. Kostov, M. Staudacher, and T. Wynter, Comm. Math. Phys., 191, 283–298 (1998); arXiv:hep-th/9703189v1 (1997).
[3] V. Kazakov, ”Solvable matrix models,” in: Random Matrix Models and Their Applications (Math. Sci. Res. Inst. Publ., Vol. 40, P. Bleher and A. Its, eds.), Cambridge Univ. Press, Cambridge (2001), pp. 271–283; arXiv:hep-th/0003064v1 (2000). · Zbl 1016.82018
[4] A. Yu. Morozov, Sov. Phys. Usp., 37, 1–55 (1994); arXiv:hep-th/9303139v2 (1993).
[5] A. Morozov, ”Matrix models as integrable systems,” in: Particles and Fields (G. Semenoff and L. Vinet, eds.), Springer, New York (1999), pp. 127–210; arXiv:hep-th/9502091v1 (1995).
[6] F. Cachazo, K. Intriligator, and C. Vafa, Nucl. Phys. B, 603, 3–41 (2001); arXiv:hep-th/0103067v2 (2001); F. Cachazo and C. Vafa, ”N=1 and N=2 geometry from fluxes,” arXiv:hep-th/0206017v1 (2002); R. Dijkgraaf and C. Vafa, Nucl. Phys. B, 644, 3–20, 21–39 (2002); arXiv:hep-th/0206255v2 (2002); arXiv:hep-th/0207106v1 (2002); ”A perturbative window into non-perturbative physics,” arXiv:hep-th/0208048v1 (2002); L. Chekhov and A. Mironov, Phys. Lett. B, 552, 293–302 (2003); arXiv:hep-th/02090852 (2002); R. Dijkgraaf, S. Gukov, V. A. Kazakov, and C. Vafa, Phys. Rev. D, 68, 045007 (2003); arXiv:hep-th/0210238v1 (2002); V. A. Kazakov and A. Marshakov, J. Phys. A, 36, 3107–3136 (2003); arXiv:hep-th/0211236v4 (2002); H. Itoyama and A. Morozov, Nucl. Phys. B, 657, 53–78 (2003); arXiv:hep-th/0211245v2 (2002); Phys. Lett. B, 555, 287–295 (2003); arXiv:hep-th/0211259v1 (2002); Progr. Theoret. Phys., 109, 433–463 (2003); arXiv:hep-th/0212032v2 (2002); Internat. J. Mod. Phys. A, 18, 5889–5906 (2003); arXiv:hep-th/0301136v1 (2003); S. G. Naculich, H. Schnitzer, and N. Wyllard, JHEP, 0301, 015 (2003); arXiv:hep-th/0211254v3 (2002); B. Feng, Nucl. Phys. B, 661, 113–138 (2003); arXiv:hep-th/0212010v3 (2002); I. Bena, S. de Haro, and R. Roiban, Nucl. Phys. B, 664, 45–58 (2003); arXiv:hep-th/0212083v1 (2002); C. Ahn, Phys. Lett. B, 560, 116–127 (2003); arXiv:hep-th/0301011v2 (2003); L. Chekhov, A. Marshakov, A. Mironov, and D. Vasiliev, Phys.Lett. B, 562, 323–338 (2003); arXiv:hep-th/0301071v (2003); L. O. Chekhov, A. V. Marshakov, A. D. Mironov, and D. Vasiliev, Proc. Steklov Inst. Math., 4(251), 254–292 (2005); arXiv:hep-th/0506075v1 (2005); A. Dymarsky and V. Pestun, Phys. Rev. D, 67, 125001 (2003); arXiv:hep-th/0301135v3 (2003); Yu. Ookouchi and Yo. Watabiki, Modern Phys. Lett. A, 18, 1113–1126 (2003); arXiv:hep-th/0301226v2 (2003); H. Itoyama and H. Kanno, Phys. Lett. B, 573, 227–234 (2003); arXiv:hep-th/0304184v1 (2003); Nucl. Phys. B, 686, 155–164 (2004); arXiv:hep-th/0312306v2 (2003); M. Matone and L. Mazzucato, JHEP, 0307, 015 (2003); arXiv:hep-th/0305225v3 (2003); R. Argurio, G. Ferretti, and R. Heise, Internat. J. Mod. Phys. A, 19, 2015–2078 (2004); arXiv:hep-th/0311066v4 (2003); M. Gómez-Reino, JHEP, 0406, 051 (2004); arXiv:hep-th/0405242v2 (2004); K. Fujiwara, H. Itoyama, and M. Sakaguchi, Progr. Theoret. Phys., 113, 429–455 (2005); arXiv:hep-th/0409060v3 (2004); Nucl. Phys. B, 723, 33–52 (2005); arXiv:hep-th/0503113v2 (2005); Progr. Theoret. Phys. Suppl., 164, 125–137 (2007); arXiv:hep-th/0602267v1 (2006); Sh. Aoyama, JHEP, 0510, 032 (2005); arXiv:hep-th/0504162v2 (2005); D. Berenstein and S. Pinansky, Nucl. Phys. B, 753, 69–91 (2006); arXiv:hep-th/0602294v2 (2006). · Zbl 0983.81050
[7] E. Brezin and S. Hikami, ”An extension of the HarishChandra-Itzykson-Zuber integral,” arXiv:math-ph/0208002v1 (2002).
[8] A. Zabrodin, Modern Phys.Lett.A, 7, 441–446 (1992); Ann.Henri Poincaré;, 4, s851-s861; arXiv:cond-mat/ 0210331v1 (2002); P. Wiegmann and A. Zabrodin, ”Large N expansion for normal and complex matrix ensembles,” in: Frontiers in Number Theory, Physics, and Geometry (P. Cartier, B. Julia, P. Moussa, and P. Vanhove, eds.), Vol. 1, Springer, Berlin (2006), pp. 213–229; arXiv:hep-th/0309253v1 (2003); R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin, and P. Wiegmann, Nucl. Phys. B, 704, 407–444 (2005); arXiv:hep-th/0401165v2 (2004). · Zbl 1020.82536
[9] A. Alexandrov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 19, 4127–4163 (2004); A. S. Alexandrov, A. D. Mironov, and A. Yu. Morozov, Theor. Math. Phys., 142, 349–411 (2005); arXiv:hep-th/0310113v1 (2003). · Zbl 1087.81051
[10] T. J. Hollowood, A. Iqbal, and C. Vafa, JHEP, 0803, 069 (2008); arXiv:hep-th/0310272v4 (2003).
[11] A. Alexandrov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 21, 2481–2517 (2006); arXiv:hep-th/0412099v1 (2004); Fortschr. Phys., 53, 512–521 (2005); arXiv:hep-th/0412205v1 (2004); A. Alexandrov, A. Mironov, A. Morozov, and P. Putrov, ”Partition functions of matrix models as the first special functions of string theory: II. Kontsevich model,” arXiv:0811.2825v2 [hep-th] (2008). · Zbl 1098.81075
[12] B. Eynard, JHEP, 0411, 031 (2004); arXiv:hep-th/0407261v1 (2004); B. Eynard and N. Orantin, JHEP, 0512, 034 (2005); arXiv:math-ph/0504058v2 (2005); Commun. Number Theory Phys., 1, 347–452 (2007); arXiv:math-ph/0702045v4 (2007); L. Chekhov and B. Eynard, JHEP, 0603, 014 (2006); arXiv:hep-th/ 0504116v1 (2005); 0612, 026 (2006); arXiv:math-ph/0604014v1 (2006); B. Eynard, N. Orantin, and M. Marinño, JHEP, 0706, 058 (2007); arXiv:hep-th/0702110v3 (2007); N. Orantin, ”From matrix models’ topological expansion to topological string theories: Counting surfaces with algebraic geometry,” arXiv:0709.2992v1 [hep-th] (2007); ”Gaussian matrix model in an external field and non-intersecting Brownian motions,” arXiv:0803.0705v1 [math-ph] (2008).
[13] B. Eynard, JHEP, 0301, 051 (2003); arXiv:hep-th/0210047v1 (2002); 0311, 018; arXiv:hep-th/0309036v2 (2003). · Zbl 1226.82029
[14] M. Bertola and B. Eynard, J. Phys. A, 36, 7733–7750 (2003); arXiv:hep-th/0303161v1 (2003); B. Eynard, ”A short note about Morozov’s formula,” arXiv:math-ph/0406063v1 (2004); B. Eynard and A. Prats Ferrer, Comm. Math. Phys., 264, 115–144 (2005); arXiv:hep-th/0502041v1 (2005); A. Prats Ferrer, B. Eynard, P. Di Francesco, and J.-B. Zuber, J. Stat. Phys., 129, 885–935 (2009); arXiv:math-ph/0610049v5 (2006); M. Bertola and A. Prats Ferrer, Internat. Math. Res. Notices, 16, Art. ID rnn062 (2008); arXiv:0801.3452v1 [math.GR] (2008); M. Bergère and B. Eynard, J. Phys. A, 42, 265201 (2009); arXiv:0805.4482v1 [math-ph] (2008). · Zbl 1057.81060
[15] N. A. Nekrasov and A. Okounkov, ”Seiberg-Witten theory and random partitions,” in: The Unity of Mathematics (Progr. Math., Vol. 244, P. Etingof, V. Retakh, and I. M. Singer, eds.), Birkhäuser, Boston, Mass. (2006), pp. 525–596; arXiv:hep-th/0306238v2 (2003); A. Marshakov and N. A. Nekrasov, JHEP, 0701, 104 (2007); arXiv:hep-th/0612019v2 (2006); B. Eynard, J. Stat. Mech., 0807, P07023 (2008); arXiv:0804.0381v2 [math-ph] (2008); A. Klemm and P. Sulkowski, Nucl. Phys. B, 819, 400–430 (2009); arXiv:0810.4944v2 [hep-th] (2008); P. Sulkowski, Phys. Rev. D, 80, 086006 (2009); arXiv:0904.3064v1 [hep-th] (2009).
[16] A. S. Alexandrov, A. D. Mironov, and A. Yu. Morozov, Theor. Math. Phys., 150, 153–164 (2007); arXiv:hep-th/0605171v1 (2006); Phys. D, 235, 126–167 (2007); arXiv:hep-th/0608228v1 (2006); N. Orantin, ”Symplectic invariants, Virasoro constraints, and Givental decomposition,” arXiv:0808.0635v2 [math-ph] (2008). · Zbl 1118.81057
[17] J. Harer and D. Zagier, Invent. Math., 85, 457–485 (1986); S. K. Lando and A. K. Zvonkine, Graphs on Surfaces and Their Applications (Encycl. Math. Sci., Vol. 141), Springer, Berlin (2004); É. T. Akhmedov and Sh. Shakirov, ”Gluing of surfaces with polygonal boundaries,” arXiv:0712.2448v4 [math.CO] (2007); A. Morozov and Sh. Shakirov, JHEP, 0912, 003 (2009); arXiv: 0906.0036v1 [hep-th] (2009); ”Harer-Zagier correlation functions in Gaussian matrix models” (to appear). · Zbl 0616.14017
[18] A. Morozov and Sh. Shakirov, ”Combinatorial solution of Hermitian model at low genera” (to appear). · Zbl 1291.81263
[19] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565–618 (1991); Yu. Makeenko, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 356, 574–628 (1991); S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, Nucl. Phys. B, 404, 717–750 (1993); arXiv:hep-th/9208044v1 (1992).
[20] M. L. Kontsevich, Funct. Anal. Appl., 25, No. 2, 123–129 (1991); M. Kontsevich, Comm. Math. Phys., 147, 1–23 (1992). · Zbl 0742.14021
[21] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin, Phys. Lett. B, 275, 311–314 (1992); arXiv:hep-th/9111037v1 (1991); Nucl. Phys. B, 380, 181–240 (1992); arXiv:hep-th/9201013v1 (1992); S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Nucl. Phys. B, 397, 339-378 (1993); arXiv:hep-th/ 9203043v1 (1992); A. Marshakov, A. Mironov, and A. Morozov, Modern Phys. Lett. A, 7, 1345–1359 (1992); arXiv:hep-th/9201010v1 (1992); Phys. Lett. B, 274, 280–288 (1992); arXiv:hep-th/9201011v1 (1992).
[22] M. Adler and P. van Moerbeke, Comm. Math. Phys., 147, 25–56 (1992); P. Di Francesco, C. Itzykson, and J.-B. Zuber, Comm. Math. Phys., 151, 193–219 (1993); arXiv:hep-th/9206090v1 (1992); S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Modern Phys. Lett. A, 8, 1047–1061 (1993); Theor. Math. Phys., 95, 571–582 (1993); arXiv:hep-th/9208046v2 (1992). · Zbl 0756.35074
[23] M. Fukuma, H. Kawai, and R. Nakayama, Internat. J. Mod. Phys. A, 6, 1385–1406 (1991); R. Dijkgraaf, E. Verlinde, and H. Verlinde, Nucl. Phys. B, 348, 435–456 (1991); A. Mironov and A. Morozov, Phys. Lett. B, 252, 47-52 (1990); F. David, Modern Phys. Lett. A, 5, 1019–1029 (1990); J. Ambjørn and Yu. Makeenko, Modern Phys. Lett. A, 5, 1753–1763 (1990); H. Itoyama and Y. Matsuo, Phys. Lett. B, 255, 202–208 (1991).
[24] L. Chekhov and Yu. Makeenko, Phys. Lett. B, 278, 271–278 (1992); arXiv:hep-th/9202006v1 (1992); Modern Phys. Lett. A, 7, 1223–1236 (1992); arXiv:hep-th/9201033v1 (1992); J. Ambjørn, C. F. Kristjansen, and Yu. M. Makeenko, Modern Phys. Lett. A, 7, 3187–3202 (1992); arXiv:hep-th/9207020v1 (1992).
[25] E. J. Martinec, Comm. Math. Phys., 138, 437–449 (1991); V. Periwal and D. Shevitz, Phys. Rev. Lett., 64, 1326–1329 (1990); Nucl. Phys. B, 344, 731–746 (1990); K. Demeterfi and C.-I. Tan, Modern Phys. Lett. A, 5, 1563–1574 (1990). · Zbl 0732.35100
[26] M. J. Bowick, A. Morozov, and D. Shevitz, Nucl. Phys. B, 354, 496–530 (1991).
[27] V. A. Kazakov and A. A. Migdal, Nucl. Phys. B, 397, 214–238 (1993); arXiv:hep-th/9206015v1 (1992).
[28] M. Caselle, A. D’adda, and S. Panzeri, Phys. Lett. B, 293, 161–167 (1992); arXiv:hep-th/9207086v2 (1992); 302, 80–86 (1993); arXiv:hep-th/9212074v1 (1992).
[29] I. I. Kogan, A. Morozov, G. W. Semenoff, and N. Weiss, Nucl. Phys. B, 395, 547–580 (1993); arXiv:hep-th/9208012v1 (1992); Internat. J. Mod. Phys. A, 8, 1411–1436 (1993); arXiv:hep-th/9208054v2 (1992); M. Dobroliubov, A. Morozov, G. W. Semenoff, and N. Weiss, Internat. J. Mod. Phys. A, 9, 5033–5051 (1994); arXiv:hep-th/9312145v1 (1993).
[30] A. Morozov, Modern Phys. Lett. A, 7, 3503–3507 (1992); arXiv:hep-th/9209074v2 (1992). · Zbl 1020.58502
[31] S. L. Shatashvili, Comm. Math. Phys., 154, 421–432 (1993); arXiv:hep-th/9209083v2 (1992). · Zbl 0788.22028
[32] S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2015–2051 (1995); arXiv:hep-th/9312210v1 (1993). · Zbl 0985.81639
[33] A. Mironov, A. Morozov, and G. W. Semenoff, Internat. J. Mod. Phys. A, 11, 5031–5080 (1996); arXiv:hep-th/ 9404005v2 (1994). · Zbl 1044.81723
[34] A. Hurwitz, Math. Ann., 39, 1–60 (1891); 55, 53–66 (1901). · JFM 23.0429.01
[35] R. Dijkgraaf, ”Mirror symmetry and elliptic curves,” in: The Moduli Spaces of Curves (Progr. Math., Vol. 129, R. Dijkgraaf, C. Faber, and G. van der Geer, eds.), Burkhäuser, Boston, Mass. (1995), pp. 149–163; R. Vakil, ”Enumerative geometry of curves via degeneration methods,” Doctoral dissertation, Harvard Univ., Boston, Mass. (1997); I. P. Goulden and D. M. Jackson, Proc. Amer. Math. Soc., 125, 51–60 (1997); arXiv:math/9903094v1 (1999); D. Zvonkine and S. K. Lando, Funct. Anal. Appl., 33, No. 3, 178–188 (1999); S. Lando and D. Zvonkine, Moscow Math. J., 7, 85–107 (2007); arXiv:math.AG/0303218v1 (2003); S. Natanzon and V. Turaev, Topology, 38, 889–914 (1999); I. P. Goulden, D. M. Jackson, and A. Vainshtein, Ann. Comb., 4, 27–46 (2000); A. Okounkov, Math. Rev. Lett., 7, 447–453 (2000); arXiv:ath/0004128v1 (2000); A. B. Givental, Moscow Math. J., 1, 551–568 (2001); arXiv:math/0108100v2 (2001); T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Invent. Math., 146, 297–327 (2001); S. K. Lando, Russ. Math. Surveys, 57, 463–533 (2002); A. V. Alexeevski and S. M. Natanzon, Selecta Math., 12, 307–377 (2006); arXiv:math.GT/0202164v2 (2002); ”Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups,” in: Geometry, Topology, and Mathematical Physics (Amer. Math. Soc. Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 1–25; A. V. Alekseevskii and S. M. Natanzon, Izv. Math., 72, 627–646 (2008); Russ. Math. Surveys, 61, 767–769 (2006); S. M. Natanzon, ”Disk single Hurwitz numbers,” arXiv:0804.0242v2 [math.GT] (2008); J. Zhou, ”Hodge integrals, Hurwitz numbers, and symmetric groups,” arXiv:math.AG/0308024v1 (2003); A. Okounkov and R. Pandharipande, Ann. Math., 163, 517-560 (2006); arXiv:math.AG/0204305v1 (2002); T. Graber and R. Vakil, Compos. Math., 135, 25–36 (2003); M. E. Kazarian and S. K. Lando, Izv. Math., 68, 82–113 (2004); arXiv:math.AG/0410388v1 (2004); ”An algebro-geometric proof of Witten’s conjecture,” arXiv:math/0601760v1 (2006); M. Kazarian, Adv. Math., 221, 1–21 (2009); arXiv:0809.3263v1 [math.AG] (2008); S. Lando, ”Combinatorial facets of Hurwitz numbers,” in: Applications of Group Theory to Combinatorics (J. Koolen, J. H. Kwak, and M.-Y. Xu, eds.), CRC Press, Boca Raton, Fla. (2008), pp. 109–132; V. Bouchard and M. Mariño, ”Hurwitz numbers, matrix models, and enumerative geometry,” in: From Hodge Theory to Integrability and TQFT: tt * -Geometry (Proc. Sympos. Pure Math., Vol. 78, R. Y. Donagi and K. Wendland, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 263–283; arXiv:0709.1458v2 [math.AG] (2007); A. Mironov and A. Morozov, JHEP, 0902, 024 (2009); arXiv:0807.2843v3 [hep-th] (2008).
[36] A. Morozov and Sh. Shakirov, JHEP, 0904, 064 (2009); arXiv:0902.2627v3 [hep-th] (2009).
[37] G. Borot, B. Eynard, M. Mulase, and B. Safnuk, ”A matrix model for simple Hurwitz numbers and topological recursion,” arXiv:0906.1206v1 [math-ph] (2009); A. Morozov and Sh. Shakirov, ”On equivalence of two Hurwitz matrix models,” arXiv:0906.2573v2 [hep-th] (2009).
[38] A. Mironov, A. Morozov, and S. Natanzon, ”Complete set of cut-and-join operators in Hurwitz-Kontsevich theory,” arXiv:0904.4227v2 [hep-th] (2009). · Zbl 1312.81125
[39] A. Mironov, A. Morozov, and S. Natanzon, ”Integrability and N-point Hurwitz numbers” (to appear). · Zbl 1306.81291
[40] A. Alexandrov, A. Mironov, and A. Morozov, ”Cut-and-join operators, matrix models, and characters” (to appear).
[41] E. Brezin and D. J. Gross, Phys. Lett. B, 97, 120–124 (1980); D. J. Gross and E. Witten, Phys. Rev. D, 21, 446–453 (1980).
[42] A. Alexandrov, A. Mironov, and A. Morozov, ”BGWM as the second constituent of complex matrix model,” arXiv:0906.3305v1 [hep-th] (2009).
[43] Harish-Chandra, Amer. J. Math., 79, 87–120 (1957); 80, 241–310 (1958). · Zbl 0072.01901
[44] C. Itzykson and J.-B. Zuber, J. Math. Phys., 21, 411–421 (1980); P. Zinn-Justin and J.-B. Zuber, J. Phys. A, 36, 3173–3193 (2003); arXiv:math-ph/0209019v3 (2002). · Zbl 0997.81549
[45] B. De Wit and G.’t Hooft, Phys. Lett. B, 69, 61–64 (1977).
[46] A. B. Givental, Internat. Math. Res. Notices, 2001, No. 23, 1265–1286 (2001); arXiv:math.AG/0008067v4 (2000). · Zbl 1074.14532
[47] H. Leutwyler and A. Smilga, Phys. Rev. D, 46, 5607–5632 (1992); J. J. M. Verbaarschot, ”Universal behavior in Dirac spectra,” arXiv:hep-th/9710114v1 (1997); R. Brower, P. Rossi, and C.-I. Tan, Nucl. Phys. B, 190, 699–718 (1981); T. Akuzawa and M. Wadati, J. Phys. Soc. Japan, 67, 2151–2154 (1998).
[48] G. Frobenius, Berlin Akad. Wiss., 985–1021 (1896); F. G. Frobenius, Theory of Characteristics and of Representations of Groups [in Russian], KomKniga, Moscow (2005); D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Clarendon, Oxford (1950); F. Murnaghan, The Theory of Group Representations, Dover, New York (1963); M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley, Reading, Mass. (1962); I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, New York (1979); B. Simon, Representations of Finite and Compact Groups (Grad. Stud. Math., Vol. 10), Amer. Math. Soc., Providence, R. I. (1996); W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry (London Math. Soc. Stud. Texts, Vol. 35), Cambridge Univ. Press, Cambridge (1997); B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions (Grad. Texts Math., Vol. 203), Springer, New York (2001).
[49] A. B. Balantekin, J. Math. Phys., 25, 2028–2030 (1984); Phys. Rev. D, 62, 085017 (2000); arXiv:hep-th/ 0007161v3 (2000). · Zbl 0588.22013
[50] M. A. Semenov-Tjan-Šanskii, Izv. Math, 10, 535–563 (1976); J. J. Duistermaat and G. J. Heckman, Invent. Math., 69, 259–268 (1982); 72, 153–158 (1983); A. Alekseev, L. Faddeev, and S. Shatashvili, J. Geom. Phys., 5, 391–406 (1988); M. Blau, E. Keski-Vakkuri, and A. Niemi, Phys. Lett. B, 246, 92–98 (1990); E. Keski-Vakkuri, A. Niemi, G. Semenoff, and O. Tirkkonen, Phys. Rev. D, 44, 3899-3905 (1991); A. Hietamäki, A. Yu. Morozov, A. J. Niemi, and K. Palo, Phys. Lett. B, 263, 417–424 (1991); A. Yu. Morozov, A. J. Niemi, and K. Palo, Phys. Lett. B, 271, 365–371 (1991); Nucl. Phys. B, 377, 295–338 (1992); R. J. Szabo, ”Equivariant localization of path integrals,” arXiv:hep-th/9608068v1 (1996); Y. Karshon, Lecture notes on group action on manifolds (1996–1997); M. Stone, Nucl. Phys. B, 314, 557-586 (1989). · Zbl 0367.35044
[51] A. Losev, N. Nekrasov, and S. Shatashvili, Nucl. Phys. B, 534, 549–611 (1998); arXiv:hep-th/9711108v2 (1997); ”The freckled instantons,” arXiv:hep-th/9908204v2 (1999); Class. Q. Grav., 17, 1181–1187 (2000); arXiv:hep-th/9911099v1 (1999); N. A. Nekrasov, Adv. Theor. Math. Phys., 7, 831–864 (2003); arXiv:hep-th/0206161v1 (2002); R. Flume and R. Poghossian, Internat. J. Mod. Phys. A, 18, 2541–2563 (2003); arXiv:hep-th/0208176v2 (2002); E. Frenkel, A. Losev, and N. Nekrasov, Nucl. Phys. Proc. Suppl., 171, 215–230 (2007); arXiv:hep-th/ 0702137v1 (2007). · Zbl 0954.57013
[52] N. Seiberg and E. Witten, Nucl. Phys. B, 426, 19–52 (1994); Erratum, 430, 485–486 (1994); arXiv:hep-th/ 9407087v1 (1994); A. Gorsky, I. M. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 355, 466–474 (1995); arXiv:hep-th/9505035v2 (1995); R. Donagi and E. Witten, Nucl. Phys. B, 460, 299–334 (1996); arXiv:hep-th/9510101v2 (1995); E. J. Martinec, Phys. Lett. B, 367, 91–96 (1996); arXiv:hep-th/9510204v2 (1995); A. Gorsky and A. Marshakov, Phys. Lett. B, 375, 127–134 (1996); arXiv:hep-th/9510224v3 (1995); H. Itoyama and A. Morozov, Nucl. Phys. B, 477, 855–877 (1996); arXiv:hep-th/9601168v1 (1996); 491, 529–573 (1997); arXiv:hep-th/9512161v1 (1995); N. Nekrasov, Nucl. Phys. B, 531, 323–344 (1998); arXiv:hep-th/9609219v3 (1996); A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 389, 43–52 (1996); arXiv:hep-th/9607109v1 (1996); Modern Phys. Lett. A, 12, 773–787 (1997); arXiv:hep-th/9701014v1 (1997); Internat. J. Mod. Phys. A, 15, 1157–1206 (2000); arXiv:hep-th/9701123v1 (1997); H. W. Braden, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 448, 195–202 (1999); arXiv:hep-th/9812078v1 (1998); Nucl. Phys. B, 558, 371–390 (1999); arXiv:hep-th/9902205v1 (1999); Acta Appl. Math., 99, 223–244 (2007); arXiv:hep-th/0606035v3 (2006); A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 380, 75–80 (1996); ”A note on spectral curve for the periodic homogeneous XYZ-spin chain,” arXiv:hep-th/9604078v1 (1996); Nucl. Phys. B, 527, 690–716 (1998); arXiv:hep-th/9802007v2 (1998). · Zbl 0996.81510
[53] F. Calogero, J. Math. Phys., 10, 2191–2196 (1969); 12, 419–436 (1971); C. F. Dunkl, Trans. Amer. Math. Soc. 311, 167–183 (1989); Canad. J. Math., 43, 1213–1227 (1991); N. Yu. Reshetikhin, L. A. Takhtadzhyan and L. Faddeev, Leningrad Math. J., 1, 193–225 (1990); A. Alekseev and S. Shatashvili, Nucl. Phys. B, 323, 719–733 (1989); A. Gorsky and N. Nekrasov, Nucl. Phys. B, 414, 213–238 (1994); arXiv:hep-th/9304047v1 (1993); 436, 582–608 (1995); arXiv:hep-th/9401017v4 (1994); P. Etingof, I. B. Frenkel, and A. A. Kirillov Jr., Duke Math. J., 80, 59–90 (1995); arXiv:hep-th/9407047v4 (1994); A. Gerasimov, S. Kharchev, A. Morozov, M. Olshanetsky, A. Marshakov, and A. Mironov, Internat. J. Mod. Phys. A, 12, 2523–2583 (1997); arXiv:hep-th/9601161v1 (1996); P. Etingof, ”Lectures on Calogero-Moser systems,” arXiv:math/0606233v3 (2006).
[54] V. V. Dolotin, ”On discriminants of polylinear forms,” arXiv:alg-geom/9511010v2 (1995); A. Morozov and Sh. Shakirov, JHEP, 0912, 002 (2009); arXiv:0903.2595v1 [math-ph] (2009); K. Fujii, ”Beyond Gaussian: A comment,” arXiv:0905.1363v3 [math-ph] (2009).
[55] I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, Mass. (1994); V. Dolotin and A. Morozov, Introduction to Non-Linear Algebra, World Scientific, Hackensack, N. J. (2007); arXiv:hep-th/0609022v4 (2006); The Universal Mandelbrot Set: Beginning of the Story, World Scientific, Hackensack, N. J. (2006); ”Algebraic geometry of discrete dynamics: The case of one variable,” arXiv:hep-th/0501235v2 (2005); Internat. J. Mod. Phys. A, 23, 3613–3684 (2008); arXiv:hep-th/0701234v1 (2007); E. T. Akhmedov, V. Dolotin, and A. Morozov, JETP Lett., 81, 639–643 (2005); arXiv:hep-th/0504160v1 (2005); A. Yu. Morozov and M. N. Serbyn, Theor. Math. Phys., 154, 270’293 (2008); V. Dolotin, A. Morozov, and Sh. Shakirov, ”A-infinity-structure on simplicial complexes,” arXiv:0704.2609v5 [math.GT] (2007); Phys. Lett. B, 651, 71–73 (2007); arXiv:0704.2884v3 [hep-th] (2007); A. Morozov, JETP Lett., 86, 745-748 (2007); A. Morozov and Sh. Shakirov, ”Analogue of the identity Log Det=Trace Log for resultants,” arXiv:0804.4632v3 [math-ph] (2008); ”Resultants and contour integrals,” arXiv:0807.4539v1 [math.AG] (2008); A. S. Anokhina, A. Yu. Morozov, and Sh. R. Shakirov, Theor. Math. Phys., 160, 1203–1228 (2009); arXiv:0812.5013v3 [math-ph] (2008).
[56] Y. Nambu, Phys. Rev. D, 7, 2405–2412 (1973); V. T. Filippov, Siberian Math. J., 26, 879–891 (1985); L. Takhtajan, Comm. Math. Phys., 160, 295–315 (1994); arXiv:hep-th/9301111v1 (1993); J. Hoppe, Helv. Phys. Acta, 70, 302–317 (1997); arXiv:hep-th/9602020v1 (1996); H. Awata, M. Li, D. Minic, and T. Yoneya, JHEP, 0102, 013 (2001); arXiv:hep-th/9906248v1 (1999); J. Schwarz, JHEP, 0411, 078 (2004); arXiv:hep-th/0411077v2 (2004); A. Basu and J. A. Harvey, Nucl. Phys. B, 713, 136–150 (2005); arXiv:hep-th/0412310v3 (2004); J. Bagger and N. Lambert, Phys. Rev. D, 75, 045020 (2007); arXiv:hep-th/0611108v3 (2006); 77, 065008 (2008); arXiv:0711.0955v2 [hep-th] (2007); 79, 025002 (2009); arXiv:0807.0163v4 [hep-th] (2008); JHEP, 0802, 105 (2008); arXiv:0712.3738v2 [hep-th] (2007); A. Gustavsson, Nucl. Phys. B, 811, 66–76 (2009); arXiv:0709.1260v5 [hep-th] (2007); JHEP, 0804, 083 (2008); arXiv:0802.3456v3 [hep-th] (2008); S. Mukhi and C. Papageorgakis, JHEP, 0805, 085 (2008); arXiv:0803.3218v2 [hep-th] (2008); M. A. Bandres, A. E. Lipstein, and J. H. Schwarz, JHEP, 0805, 025 (2008); arXiv:0803.3242v4 [hep-th] (2008); A. Morozov, JHEP, 0805, 076 (2008); arXiv:0804.0913v1 [hep-th] (2008); JETP Lett., 87, 659–662 (2008); arXiv:0805.1703v1 [hep-th] (2008); J. Gomis, G. Milanesi, and J. G. Russo, JHEP, 0806, 075 (2008); arXiv:0805.1012v3 [hep-th] (2008); S. Benvenuti, D. Rodríguez-Gómez, E. Tonni, and H. Verlinde, JHEP, 0901, 078 (2009); arXiv:0805.1087v1 [hep-th] (2008); P.-M. Ho, Y. Imamura, and Y. Matsuo, JHEP, 0807, 003 (2008); arXiv:0805.1202v2 [hep-th] (2008); O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, JHEP, 0810, 091 (2008); arXiv:0806.1218v4 [hep-th] (2008); I. A. Bandos and P. K. Townsend, Class. Q. Grav., 25, 245003 (2008); arXiv:0806.4777v2 [hep-th] (2008); JHEP, 0902, 013 (2009); arXiv:0808.1583v3 [hep-th] (2008); J. Gomis, D. Rodíguez-Gómez, M. Van Raamsdonk, and H. Verlinde, JHEP, 0809, 113 (2008); arXiv:0807.1074v2 [hep-th] (2008); J. A. Minahan, W. Schulgin, and K. Zarembo, JHEP, 0903, 057 (2009); arXiv:0901.1142v2 [hep-th] (2009); J. Figueroa-O’Farrill, ”Deformations of 3-algebras,” arXiv:0903.4871v2 [hep-th] (2009); D. Kamani, ”Evidence for the P+1-algebra for super-P-brane,” arXiv:0904.2721v3 [hep-th] (2009). · Zbl 1027.70503
[57] B. Schlittgen and T. Wettig, J. Phys. A, 36, 3195–3201 (2003); arXiv:math-ph/0209030v2 (2002). · Zbl 1032.81025
[58] C. Lehner, T. Wettig, T. Guhr, and Y. Wei, J. Math. Phys., 49, 063510 (2008); arXiv:0801.1226v3 [hep-th] (2008). · Zbl 1152.81526
[59] F. A. Berezin and F. I. Karpelevič, Dokl. Akad. Nauk SSSR, 118, 9–12 (1958); T. Guhr and T. Wettig, J. Math. Phys., 37, 6395–6413 (1996); A. D. Jackson, M. K. Sener, and J. J. M. Verbaarschot, Phys. Lett. B, 387, 355–360 (1996).
[60] R. Dijkgraaf and E. Witten, Comm. Math. Phys., 129, 393–429 (1990). · Zbl 0703.58011
[61] A. Yu. Morozov, Sov. Phys. Usp., 35, 671–714 (1992).
[62] M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ., 19, 943–1001 (1983); K. Ueno and K. Takasaki, ”Toda lattice hierarchy,” in: Group Representations and Systems of Differential Equations (Adv. Stud. Pure Math., Vol. 4, K. Okamoto, ed.), North-Holland, Amsterdam, pp. 1–95; A. Morozov and L. Vinet, Internat. J. Mod. Phys. A, 13, 1651–1707 (1998); arXiv:hep-th/9409093v1 (1994); A. Mironov, A. Morozov, and L. Vinet, Theor. Math. Phys., 100, 890–899 (1994); arXiv:hep-th/9312213v2 (1993); A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, Internat. J. Mod. Phys. A, 10, 2589–2614 (1995); arXiv:hep-th/9405011v1 (1994); S. Kharchev, A. Mironov, and A. Morozov, Theor. Math. Phys., 104, 866–878 (1995); arXiv:q-alg/9501013v1 (1995); A. Mironov, ”Quantum deformations of {\(\tau\)}-functions, bilinear identities, and representation theory,” in: Symmetries and Integrability of Difference Equations (CRM Proc. Lecture Notes, Vol. 9, D. Levi, L. Vinet, and P. Winternitz, eds.), Amer. Math. Soc., Providence, R. I. (1996), pp. 219–237; arXiv:hep-th/9409190v2 (1994); A. D. Mironov, Theor. Math. Phys., 114, 127–183 (1998); arXiv:q-alg/9711006v2 (1997). · Zbl 0557.35091
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