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The WDVV Equations in Pure Seiberg–Witten Theory

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Abstract

We review the relationship between pure four-dimensional Seiberg–Witten theory and the periodic Toda chain. We discuss the definition of the prepotential and give two proofs that it satisfies the generalized Witten–Dijkgraaf–Verlinde–Verlinde equations. A number of steps in the definitions and proofs that is missing in the literature is supplied.

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References

  1. Adler, M. and van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math. 38(3) (1980), 318–379.

    Google Scholar 

  2. Alishahiha, M., de Boer, J., Mosaffa, A. E. and Wijnhout, J.: N=1 G2 SYM theory and compactification to three dimensions, J. High Energy Phys. (9) (2003), 066, 24 pp. (electronic).

  3. Alvarez-Gaumé, L. and Hassan, S. F.: Introduction to S-duality in N=2 supersymmetric gauge theories (a pedagogical review of the work of Seiberg and Witten), Fortschr. Phys. 45(3–4) (1997), 159–236.

    Google Scholar 

  4. Arnold, V. I., Guse \u{\i} n-Zade, S. M. and Varchenko, A. N.: Singularities of Differentiable Maps, Vol. I, Birkhäuser, Boston, 1985. The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds.

  5. Bilal, A.: Duality in N=2 SUSY SU(2) Yang–Mills theory: A pedagogical introduction to the work of Seiberg and Witten, In: R.C.P. 25, Vol. 48, Univ. Louis Pasteur, Strasbourg, 1997, pp. 87–120.

    Google Scholar 

  6. Brieskorn, E. and Knörrer, H.: Plane Algebraic Curves, Birkhäuser, Basel, 1986. Translated from the German by John Stillwell.

    Google Scholar 

  7. Buchberger, B.: Introduction to Groebner bases, In: Logic of Computation (Marktoberdorf, 1995), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci. 157, Springer, Berlin, 1997, pp. 35–66.

    Google Scholar 

  8. De Wit, B., Lauwers, P. G. and Van Proeyen, A.: Lagrangians of N=2 supergravity – matter systems, Nuclear Phys. B 255 (1985), 569.

    Google Scholar 

  9. De Wit, B. and Marshakov, A. V.: Electric–magnetic duality and the Witten–Dijkgraaf–Verlinde–Verlinde equation, Teoret. Mat. Fiz. 129(2) (2001), 230–238.

    Google Scholar 

  10. D’Hoker, E. and Phong, D. H.: Lectures on supersymmetric Yang–Mills theory and integrable systems, In: Theoretical Physics at the End of the Twentieth Century (Banff, AB, 1999), CRM Ser. Math. Phys., Springer, New York, 2002, pp. 1–125.

    Google Scholar 

  11. Dijkgraaf, R.: Fields, strings and duality, In: Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 3–147.

    Google Scholar 

  12. Dijkgraaf, R., Verlinde, H. and Verlinde, E.: Topological strings in d<1, Nuclear Phys. B 352 (1991), 59–86.

    Google Scholar 

  13. Donagi, R.: Decomposition of spectral covers, Astérisque 218 (1993), 145–175. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992).

    Google Scholar 

  14. Dubrovin, B.: Geometry of 2D topological field theories, In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Springer, Berlin, 1996, pp. 120–348.

    Google Scholar 

  15. Eguchi, T. and Yang, S.-K.: A new description of the E(6) singularity, Phys. Lett. B 394 (1997), 315–322.

    Google Scholar 

  16. Farkas, H. M. and Kra, I.: Riemann Surfaces, 2nd edn, Springer-Verlag, New York, 1992.

    Google Scholar 

  17. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A. and Morozov, A.: Integrability and Seiberg–Witten exact solution, Phys. Lett. B 355 (1995), 466–474.

    Google Scholar 

  18. Griffiths, P. A.: On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460–495; ibid. (2) 90, 496–541.

    Google Scholar 

  19. Hoevenaars, L. K., Kersten, P. H. M. and Martini, R.: Generalized WDVV equations for F4 pure N=2 super-Yang–Mills theory, Phys. Lett. B 503(1–2) (2001), 189–196.

    Google Scholar 

  20. Hoevenaars, L. K. and Martini, R.: Generalized WDVV equations for Br and Cr pure N=2 super-Yang–Mills theory, Lett. Math. Phys. 57(2) (2001), 175–183.

    Google Scholar 

  21. Hoevenaars, L. K. and Martini, R.: On the WDVV equations in five-dimensional gauge theories, Phys. Lett. B 557(1–2) (2003), 94–104.

    Google Scholar 

  22. Hollowood, T. J.: Strong coupling N=2 gauge theory with arbitrary gauge group, Adv. Theor. Math. Phys. 2(2) (1998), 335–355.

    Google Scholar 

  23. Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, 2nd edn, Graduate Texts in Math. 9, Springer-Verlag, New York, 1978.

    Google Scholar 

  24. Ito, K.: A-D-E singularity and the Seiberg–Witten theory, Prog. Theor. Phys. Suppl. 135 (1999), 94.

    Google Scholar 

  25. Ito, K. and Yang, S.-K.: Flat coordinates, topological Landau–Ginzburg models and the Seiberg–Witten period integrals, Phys. Lett. B 415 (1997), 45–53.

    Google Scholar 

  26. Ito, K. and Yang, S.-K.: The WDVV equations in N=2 supersymmetric Yang–Mills theory, Phys. Lett. B 433 (1998), 56–62.

    Google Scholar 

  27. Kac, V. G.: Infinite-Dimensional Lie Algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  28. Kirwan, F.: Complex Algebraic Curves, Cambridge University Press, Cambridge, 1992.

    Google Scholar 

  29. Krichever, I. M.: The τ-function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47(4) (1994), 437–475.

    Google Scholar 

  30. Ksir, A. E.: Dimensions of Prym varieties, Internat. J. Math. Math. Sci. 26(2) (2001), 107–116.

    Google Scholar 

  31. Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490.

    Google Scholar 

  32. Lerche, W. and Warner, N. P.: Exceptional SW geometry from ALE fibrations, Phys. Lett. B 423 (1998), 79–86.

    Google Scholar 

  33. Manin, J. I.: Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 737–756.

    Google Scholar 

  34. Marshakov, A., Mironov, A. and Morozov, A.: WDVV-like equations in N=2 SUSY Yang–Mills theory, Phys. Lett. B 389 (1996), 43–52.

    Google Scholar 

  35. Marshakov, A., Mironov, A. and Morozov, A.: WDVV equations from algebra of forms, Modern Phys. Lett. A 12 (1997), 773–788.

    Google Scholar 

  36. Marshakov, A., Mironov, A. and Morozov, A.: More evidence for the WDVV equations in N=2 SUSY Yang–Mills theories, Internat. J. Modern Phys. A 15 (2000), 1157.

    Google Scholar 

  37. Marshakov, A. V.: On associativity equations, Teoret. Mat. Fiz. 132(1) (2002), 3–49.

    Google Scholar 

  38. Martinec, E. and Warner, N.: Integrable systems and supersymmetric gauge theory, Nuclear Phys. B 459 (1996), 97–112.

    Google Scholar 

  39. Martini, R. and Gragert, P. K. H.: Solutions of WDVV equations in Seiberg–Witten theory from root systems, J. Nonlinear Math. Phys. 6(1) (1999), 1–4.

    Google Scholar 

  40. Martini, R. and Hoevenaars, L. K.: Trigonometric solutions of the WDVV equations from root systems, Lett. Math. Phys. 65(1) (2003), 15–18.

    Google Scholar 

  41. Mironov, A.: WDVV equations and Seiberg–Witten theory, In: Integrability: The Seiberg–Witten and Whitham Equations (Edinburgh, 1998), Gordon and Breach, Amsterdam, 2000, pp. 103–123.

    Google Scholar 

  42. Morozov, A.: Integrable structure behind the generalized WDVV equations, Phys. Lett. B 427(1–2) (1998), 93–96.

    Google Scholar 

  43. Novikov, S. P. (ed.): Dynamical Systems. VII, Encyclopaedia Math. Sci. 16, Springer-Verlag, Berlin, 1994. Integrable systems, nonholonomic dynamical systems, A translation of Current Problems in Mathematics. Fundamental Directions, Vol. 16 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987 [MR 88g:58004], Translation by A. G. Reyman [A. G. Re \u{\i} man] and M. A. Semenov-Tian-Shansky [M. A. Semenov-Tyan-Shanski \u{\i} ], Translation edited by V. I. Arnol’d and S. P. Novikov.

  44. Saito, K.: Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19(3) (1983), 1231–1264.

    Google Scholar 

  45. Seiberg, N. and Witten, E.: Electric–magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory, Nuclear Phys. B 426 (1994), 19–52.

    Google Scholar 

  46. Vafa, C.: Topological Landau–Ginzburg models, Modern Phys. Lett. A 6(4) (1991), 337–346.

    Google Scholar 

  47. Warner, N. P.: Deformations of the root systems and new solutions to generalised WDVV equations, Phys. Lett. A 261 (1999), 297–302.

    Google Scholar 

  48. Witten, E.: Two-dimensional gravity and intersection theory on moduli space, In: Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310.

    Google Scholar 

  49. Zuber, J. B.: On Dubrovin topological field theories, Modern Phys. Lett. A 9 (1994), 749–760.

    Google Scholar 

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  1. Faculteit Wiskunde en Informatica, Universiteit Utrecht, Budapestlaan 6, 3584 CD, Utrecht, The Netherlands

    L. K. Hoevenaars

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14H10, 14H20, 14H40, 14H70, 14D45.

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Hoevenaars, L.K. The WDVV Equations in Pure Seiberg–Witten Theory. Acta Appl Math 86, 49–102 (2005). https://doi.org/10.1007/s10440-005-0463-3

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