×

zbMATH — the first resource for mathematics

The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory. (English) Zbl 1023.81022
Summary: We consider the prepotential of Dijkgraaf and Vafa (DV) as one more (and in fact, singular) example of the Seiberg-Witten (SW) prepotentials and discuss its properties from this perspective. Most attention is devoted to the issue of complete system of moduli, which should include not only the sizes of the cuts (in matrix model interpretation), but also their positions, i.e., the number of moduli should be almost doubled, as compared to the DV consideration. We introduce the notion of regularized DV system (not necessarily related to matrix model) and discuss the WDVV equations. These definitely hold before regularization is lifted, but an adequate limiting procedure, preserving all ingredients of the SW theory, remains to be found.

MSC:
81T10 Model quantum field theories
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
58D29 Moduli problems for topological structures
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Dijkgraaf, R.; Vafa, C.; Dijkgraaf, R.; Vafa, C.; Dijkgraaf, R.; Vafa, C., Nucl. phys. B, Nucl. phys. B, 644, 21-39, (2002)
[2] Veneziano, G.; Yankielowicz, S.; Novikov, V.; Shifman, M.; Vainshtein, A.; Zakharov, V.; Novikov, V.; Shifman, M.; Vainshtein, A.; Zakharov, V.; Konishi, K.; Seiberg, N.; Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C.; Intriligator, K.A.; Leigh, R.G.; Seiberg, N.; Intriligator, K.A.; Seiberg, N., Phys. lett. B, Nucl. phys. B, Phys. lett. B, Phys. lett. B, Phys. lett. B, Commun. math. phys., Phys. rev. D, Nucl. phys. B (proc. suppl.), 45, 1, (1996)
[3] Cachazo, F.; Intriligator, K.A.; Vafa, C.; Cachazo, F.; Vafa, C., Nucl. phys. B, 603, 3, (2001)
[4] Chekhov, L.; Mironov, A.
[5] Dorey, N.; Hollowood, T.J.; Prem Kumar, S.; Sinkovics, A.; Dorey, N.; Hollowood, T.J.; Prem Kumar, S.; Sinkovics, A.; Dorey, N.; Hollowood, T.J.; Prem Kumar, S.
[6] Dijkgraaf, R.; Gukov, S.; Kazakov, V.; Vafa, C.; Cachazo, F.; Douglas, M.R.; Seiberg, N.; Witten, E.
[7] Aganagic, M.; Vafa, C.; Ferrari, F.; Ferrari, F.; Fuji, H.; Ookouchi, Y.; Berenstein, D.; Gorsky, A.; Argurio, R.; Campos, V.L.; Ferretti, G.; Heise, R.; McGreevy, J.; Dijkgraaf, R.; Grisaru, M.T.; Lam, C.S.; Vafa, C.; Zanon, D.; Suzuki, H.; Bena, I.; Roiban, R.; Demasure, Y.; Janik, R.A.; Aganagic, M.; Klemm, A.; Marino, M.; Vafa, C.; Gopakumar, R.; Naculich, S.; Schnitzer, H.; Wyllard, N.; Dijkgraaf, R.; Neitzke, A.; Vafa, C.; Tachikawa, Y.; Feng, B.; Klemm, A.; Marino, M.; Theisen, S.
[8] Seiberg, N.; Witten, E.; Seiberg, N.; Witten, E., Nucl. phys. B, Nucl. phys. B, 430, 485, (1994), Erratum,
[9] Seiberg, N.; Witten, E.
[10] Klemm, A.; Lerche, W.; Theisen, S.; Yankielowicz, S.; Argyres, P.; Farragi, A.; Hanany, A.; Oz, Y.; Seiberg, N., Phys. lett. B, Phys. rev. lett., Nucl. phys. B, Phys. lett. B, 388, 753-760, (1996)
[11] Gorsky, A.; Krichever, I.; Marshakov, A.; Mironov, A.; Morozov, A., Phys. lett. B, 355, 466, (1995)
[12] Jimbo, M.; Miwa, T.; Sato, M.; Jimbo, M.; Miwa, T.; Sato, M.; Jimbo, M.; Miwa, T.; Sato, M.; Jimbo, M.; Miwa, T.; Sato, M.; Jimbo, M.; Miwa, T.; Sato, M.; Date, E.; Kashiwara, M.; Jimbo, M.; Miwa, T., Transformation groups for soliton equations, (), 15, 1531, (1979) · Zbl 0526.17008
[13] Dubrovin, B.; Krichever, I.; Novikov, S.; Hitchin, N.; Markman, E.; Gorsky, A.; Nekrasov, N.; Gorsky, A.; Nekrasov, N.; Levin, A.; Olshanetsky, M.; Donagi, R.; Hurtubise, J.C.; Markman, E., (), Duke math. J., Comput. math., Nucl. phys. B, Commum. math. phys., Commun. math. phys., 223, 533-552, (2001)
[14] Ruijsenaars, S.N.; Babelon, O.; Bernard, D.; Fock, V.; Fock, V.; Rosly, A.; Fock, V.; Gorsky, A.; Nekrasov, N.; Roubtsov, V.; Gorsky, A.; Rubtsov, V.; Mironov, A.; Morozov, A.; Mironov, A.; Morozov, A.; Mironov, A.; Morozov, A.; Mironov, A.; Morozov, A., (), Am. math. soc. transl., Jhep, Phys. lett. B, Phys. lett. B, Phys. lett. B, 524, 217-226, (2002)
[15] Morozov, A.; Vinet, L.; Mironov, A.; Morozov, A.; Vinet, L.; Kharchev, S.; Mironov, A.; Morozov, A.; Mironov, A., Int. J. mod. phys. A, Theor. math. phys., Theor. math. phys., 114, 127-899, (1998) · Zbl 0875.35091
[16] Morozov, A.; Morozov, A., Russ. phys. usp., 37, 1-55, (1994)
[17] Morozov, A.; Gerasimov, A.; Lebedev, D.; Morozov, A.; Mironov, A., Russ. phys. usp., Int. J. mod. phys. A, Int. J. mod. phys. A, 9, 3306-4355, (1994)
[18] Martinec, E.; Warner, N.; Martinec, E.; Warner, N.; Nakatsu, T.; Takasaki, K.; Donagi, R.; Witten, E.; Eguchi, T.; Yang, S.; Martinec, E.; Gorsky, A.; Marshakov, A.; Gomez, C.; Hernandes, R.; Lopez, E.; Gomez, C.; Hernandes, R.; Lopez, E.; Itoyama, H.; Morozov, A.; Ahn, C.; Nam, S.; Bonelli, G.; Matone, M.; D’Hoker, E.; Krichever, I.; Phong, D., Nucl. phys. B, Mod. phys. lett. A, Nucl. phys. B, Mod. phys. lett. A, Phys. lett. B, Phys. lett. B, Phys. lett. B, Nucl. phys. B, Phys. lett. B, Nucl. phys. B, 489, 179-138, (1997)
[19] Itoyama, H.; Morozov, A., Nucl. phys. B, 477, 855, (1996)
[20] Itoyama, H.; Morozov, A., Nucl. phys. B, 491, 529, (1997)
[21] Seiberg, N.; Witten, E.; Argyres, P.C.; Shapere, A.D.; Gorsky, A.; Marshakov, A.; Mironov, A.; Morozov, A.; Gorsky, A.; Marshakov, A.; Mironov, A.; Morozov, A., Nucl. phys. B, Nucl. phys. B, Phys. lett. B, 380, 75-80, (1996)
[22] Nekrasov, N.; Marshakov, A.; Mironov, A.; Braden, H.W.; Marshakov, A.; Mironov, A.; Morozov, A., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, 558, 371-390, (1999)
[23] Klemm, A.; Krichever, I.; Phong, D.; D’Hoker, E.; Phong, D.; D’Hoker, E.; Phong, D.; Ketov, S.; Cappelli, A.; Valtancoli, P.; Vergnano, L.; Gomez, C.; Hernandes, R.; Kanno, H.; Ohta, Y.; Carrol, R.; Edelstein, J.; Marino, M.; Mas, J.; Edelstein, J.; Gomez-Reino, M.; Marino, M.; Mas, J.; Gorsky, A.; Gukov, S.; Mironov, A.; Morozov, A.; Morozov, A.; Braden, H.W.; Marshakov, A.; Mironov, A.; Morozov, A.; Braden, H.W.; Marshakov, A.; Marshakov, A.; Gorsky, A.; Mironov, A., Seiberg – witten theory and integrable systems, Nucl. phys., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, 595, 417-466, (1999), World Scientific Singapore
[24] Hirota, R.; Gerasimov, A.; Khoroshkin, S.; Lebedev, D.; Mironov, A.; Morozov, A., Bilinear forms of soliton equations, (), Int. J. mod. phys. A, 10, 2589-2614, (1995) · Zbl 0985.37510
[25] Krichever, I.; Krichever, I.; Dubrovin, B.; Dubrovin, B., Commun. pure appl. math., Commun. math. phys., Nucl. phys. B, Commun. math. phys., 145, 195, (1992)
[26] Gorsky, A.; Marshakov, A.; Mironov, A.; Morozov, A., Nucl. phys. B, 527, 690-716, (1998)
[27] Witten, E.; Witten, E.; Dijkgraaf, R.; Verlinde, E.; Verlinde, H.; Dubrovin, B.; Getzler, E., Nucl. phys. B, Surveys diff. geom., Nucl. phys. B, Lecture notes in mathematics, 1620, 59, (1996), Springer Berlin, pp. 120-348
[28] Marshakov, A.; Mironov, A.; Morozov, A., Phys. lett. B, 389, 43-52, (1996)
[29] Marshakov, A.; Mironov, A.; Morozov, A., Mod. phys. lett. A, 12, 773-788, (1997)
[30] Marshakov, A.; Mironov, A.; Morozov, A., Int. J. mod. phys. A, 15, 1157-1206, (2000)
[31] Braden, H.W.; Marshakov, A.; Mironov, A.; Morozov, A., Phys. lett. B, 448, 195-202, (1999)
[32] Getzler, E.; Getzler, E.; Morozov, A.; Bertoldi, G.; Matone, M.; Bertoldi, G.; Matone, M.; Mironov, A.; Morozov, A.; Ito, K.; Yang, S.-K.; Isidro, J.M.; Kalayci, J.; Nutku, Y.; Martini, R.; Gragert, P.K.H.; Veselov, A.; Veselov, A.; Natanzon, S.; Ohta, Y.; Hoevenaars, L.; Kersten, P.; Martini, R.; Dzhamay, A.; Hoevenaars, L.; Martini, R.; de Wit, B.; Marshakov, A.; Aratyn, H.; van de Leur, J.; Marshakov, A.; Braden, H.W.; Marshakov, A.; Hoevenaars, L., Phys. lett. B, Phys. lett. B, Phys. rev. D, Phys. lett. B, Phys. lett. B, Nucl. phys. B, J. phys. A, J. nonlinear math. phys., Phys. lett. A, J. geom. phys., J. math. phys., Phys. lett. B, Lett. math. phys., Theor. math. phys., Theor. math. phys., Phys. lett. B, 541, 1, 376-383, (2002)
[33] Mineev-Weinstein, M.; Zabrodin, A.; Mineev-Weinstein, M.; Wiegmann, P.; Zabrodin, A.; Kostov, I.K.; Krichever, I.; Mineev-Weinstein, M.; Wiegmann, P.; Zabrodin, A.; Boyarsky, A.; Marshakov, A.; Ruchayskiy, O.; Wiegmann, P.; Zabrodin, A.; Marshakov, A.; Wiegmann, P.; Zabrodin, A., J. nonlinear math. phys., Phys. rev. lett., Phys. lett. B, Commun. math. phys., 227, 131-153, (2002)
[34] Leigh, R.; Strassler, M.J., Nucl. phys. B, 496, 132-148, (1997)
[35] Witten, E.; Witten, E.; Witten, E.; Knizhnik, V.; Zamolodchikov, A.; Gerasimov, A.; Marshakov, A.; Morozov, A.; Olshanetsky, M.; Shatashvili, S., Nucl. phys. B, Nucl. phys. B, Commun. math. phys., Nucl. phys. B, Int. J. mod. phys. A, 5, 2495-2589, (1990)
[36] Gerasimov, A.; Marshakov, A.; Mironov, A.; Morozov, A.; Orlov, A.; Makeenko, Yu.; Marshakov, A.; Mironov, A.; Morozov, A.; Bowick, M.; Morozov, A.; Shevitz, D.; Marshakov, A.; Mironov, A.; Morozov, A.; Marshakov, A.; Mironov, A.; Morozov, A.; Marshakov, A.; Mironov, A.; Morozov, A.; Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Zabrodin, A.; Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Zabrodin, A.; Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A.; Pakuliak, S., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, Phys. lett. B, Mod. phys. lett. A, Phys. lett. B, Phys. lett. B, Nucl. phys. B, Nucl. phys. B, Int. J. mod. phys. A, Nucl. phys. B, 404, 717-240, (1993) · Zbl 1043.81692
[37] Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A., Mod. phys. lett. A, 8, 1047-1062, (1993)
[38] P. Griffits, J. Harris, Principles of Algebraic Geometry, 1978
[39] Losev, A.; Moore, G.; Nekrason, N.; Shatashvili, S.; Losev, A.; Nekrasov, N.; Shatashvili, S.; Marino, M.; Moore, G.; Losev, A.; Nekrasov, N.; Shatashvili, S.; Losev, A.; Nekrasov, N.; Shatashvili, S.; Edelstein, J.; Gomez-Reino, M.; Marino, M.; Edelstein, J.; Gomez-Reino, M.; Marino, M.; Edelstein, J.; Gomez-Reino, M.; Marino, M., Nucl. phys. (proc. suppl.), Nucl. phys. B, Commun. math. phys., Class. quantum grav., Adv. theor. math. phys., Jhep, Jhep, 0101, 1-16, (2001)
[40] Nekrasov, N.
[41] Flume, R.; Poghossian, R.
[42] Brezin, E.; Itzykson, C.; Parisi, G.; Zuber, J.B.; Ambjorn, J.; Chekhov, L.; Kristjansen, S.; Makeenko, Yu.; Ambjorn, J.; Chekhov, L.; Kristjansen, S.; Makeenko, Yu.; Akemann, G.; Bonnet, G.; David, F.; Eynard, B., Commun. math. phys., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, J. phys. A, 33, 6739-35, (2000), Erratum,
[43] Mironov, A.; Morozov, A.; Ambjorn, J.; Jurkiewicz, J.; Makeenko, Yu.; Itoyama, H.; Matsuo, Y.; David, F., Phys. lett. B, Phys. lett. B, Phys. lett. B, Mod. phys. lett. A, 5, 1019, (1990)
[44] Witten, E.; Diaconescu, D.-E.; Hanany, A.; Witten, E.; Gorsky, A.; Witten, E.; Witten, E., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, Phys. lett. B, Nucl. phys. B, J. geom. phys., 22, 103-133, (1997)
[45] Marshakov, A.; Martellini, M.; Morozov, A., Phys. lett. B, 418, 294-302, (1998)
[46] Cremmer, E.; Kounnas, C.; van Proeyen, A.; Derendinger, J.P.; Ferrara, S.; de With, B.; Girardello, L.; de Wit, B.; Lauwers, P.G.; Van Proeyen, A.; Strominger, A.; de Wit, B.; Lerche, W.; Mayr, P.; Warner, N., Nucl. phys. B, Nucl. phys. B, Commun. math. phys., 133, 163, (1990)
[47] Morozov, A.; Shatashvili, S.; Mironov, A.; Morozov, A.; Semenoff, G., Mod. phys. lett. A, Commun. math. phys., Int. J. mod. phys. A, 11, 5031-5080, (1996)
[48] Fay, J.; Knizhnik, V.; Lebedev, D.; Morozov, A., Theta functions on Riemann surfaces, Lecture notes in mathematics, Sov. phys. usp., Nucl. phys. B, 302, 163-971, (1988), Springer New York
[49] Itoyama, H.; Morozov, A.
[50] Chekhov, L.; Marshakov, A.; Mironov, A.; Vassiliev, D.
[51] Itoyama, H.; Morozov, A.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.