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On Associativity Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 1, pp. 3-49 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the associativity or Witten–Dijkgraaf–Verlinde–rlinde (WDVV) equations and discuss their solution class based on the existence of the residue formulas, which is most relevant for nonperturbative physics. We demonstrate that for this case, proving the associativity equations reduces to solving a stem of linear algebraic equations. Particular examples of solutions related to Landau–Ginzburg topological theories, Seiberg–Witten theories, and the tau functions of semiclassical hierarchies are discussed in detail. We also discuss related questions including the covariance of associativity equations, their relation to dispersionless Hirota relations, and the auxiliary linear problem for the WDVV equations.
Keywords: associativity equations, Seiberg–Witten theory, integrable systems, topological theories, special Kähler geometry. 
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A. V. Marshakov. On Associativity Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 1, pp. 3-49. http://geodesic.mathdoc.fr/item/TMF_2002_132_1_a0/

[1] E. Witten, Nucl. Phys. B, 340 (1990), 281 ; R. Dijkgraaf, H. Verlinde, E. Verlinde, Nucl. Phys. B, 352 (1991), 59 | DOI | MR | DOI | MR

[2] A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B, 389 (1996), 43 ; E-print hep-th/9607109 | DOI | MR

[3] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lect. Notes in Math., 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120 ; E-print hep-th/9407018 | DOI | MR | Zbl

[4] I. Krichever, Commun. Pure Appl. Math., 47 (1992), 437 ; E-print hep-th/9205110 | DOI | MR

[5] A. Marshakov, Associativity equations in effective SUSY quantum field theory, E-print hep-th/0108023

[6] A. Marshakov, A. Mironov, A. Morozov, Int. J. Mod. Phys. A, 15 (2000), 1157 ; E-print hep-th/9701123 | DOI | MR | Zbl

[7] A. Boyarsky, A. Marshakov, O. Ruchayskiy, P. Wiegmann, A. Zabrodin, Phys. Lett. B, 515 (2001), 483 ; E-print hep-th/0105260 | DOI | MR | Zbl

[8] S. Cecotti, C. Vafa, Phys. Rev. Lett., 68 (1992), 903 ; ; Commun. Math. Phys., 157 (1993), 139 ; ; 158 ; E-print hep-th/9111016E-print hep-th/9209085E-print hep-th/9211097 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[9] K. Ito, S.-K. Yang, Phys. Lett. B, 433 (1998), 56 ; E-print hep-th/9803126 | DOI | MR

[10] N. Seiberg, E. Witten, Nucl. Phys. B, 426 (1994), 19 ; E-print hep-th/9407087 | DOI | MR | Zbl

[11] A. Klemm, W. Lerche, S. Theisen, S. Yankielowicz, Phys. Lett. B, 344 (1995), 169 ; ; On the monodromies of $N=2$ sypersymmetric Yang–Mills theory, ; P. Argyres, A. Faraggi, Phys. Rev. Lett., 74 (1995), 3931 ; E-print hep-th/9411048E-print hep-th/9412158E-print hep-th/9411057 | DOI | MR | DOI | MR

[12] B. de Vit, A. V. Marshakov, TMF, 129:2 (2001), 230 ; E-print hep-th/0105289 | DOI | MR

[13] J. Fay, Theta-functions on Riemann Surfaces, Lect. Notes in Math., 352, Springer, Berlin, 1973 | DOI | MR | Zbl

[14] A. Marshakov, A. Mironov, A. Morozov, Mod. Phys. Lett. A, 12 (1997), 773 ; E-print hep-th/9701014 | DOI | MR | Zbl

[15] L. Hoevenaars, P. Kersten, R. Martini, Phys. Lett. B, 503 (2001), 189 ; E-print hep-th/0012133 | DOI | MR | Zbl

[16] I. Krichever, Funkts. anal. i ego prilozh., 14:3 (1980), 45 | MR | Zbl

[17] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B, 355 (1995), 466 ; E-print hep-th/9505035 | DOI | MR | Zbl

[18] A. Marshakov, Seiberg-Witten Theory and Integrable Systems, World Scientific, Singapore, 1999 | MR | Zbl

[19] A. V. Marshakov, TMF, 112 (1997), 3 ; ; 121, 1999 E-print hep-th/9702083 | DOI | MR | Zbl | DOI | MR | Zbl

[20] P. Wiegmann, A. Zabrodin, Commun. Math. Phys., 213 (2000), 523 ; ; I. Kostov, I. Krichever, M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, $\tau$-function for analytic curves, ; A. Marshakov, P. Wiegmann, A. Zabrodin, Commun. Math. Phys., 227 (2002), 131 ; E-print hep-th/9909147E-print hep-th/0005259E-print hep-th/0109048 | DOI | MR | Zbl | MR | DOI | MR | Zbl

[21] K. Takasaki, T. Takebe, Rev. Math. Phys, 7 (1995), 743 | DOI | MR | Zbl

[22] J. Gibbons, Y. Kodama, “Solving dispersionless Lax equations”, Singular Limits of Dispersive Waves, Proc. of NATO Advanced Research Workshop (Lyons, France, July 8–12, 1991), NATO ASI Ser., Ser. B, Phys., 320, ed. N. M. Ercolani, Plenum, New York, 1994, 61 ; R. Carroll, Y. Kodama, J. Phys. A, 28 (1995), 6373 ; E-print hep-th/9506007 | DOI | MR | Zbl | DOI | MR | Zbl

[23] A. A. Akhmetshin, Yu. S. Volvovskii, I. M. Krichever, UMN, 54:2 (1999), 167 ; E-print hep-th/9904028 | DOI | MR | Zbl

[24] L. Hoevenaars, R. Martini, Lett. Math. Phys., 57 (2001), 175 ; E-print hep-th/0102190 | DOI | MR | Zbl

[25] L. Hoevenaars, A. Marshakov, R. Martini, “New solution to generalized WDVV equations” (to appear)