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Strings, susy gauge theories, and integrable systems

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Abstract

We consider supersymmetric Yang-Mills theories in the general framework of string theory (M-theory) and describe particular compactifications leading to the Seiberg-Witten (SW) exact effective theories. We show strong parallels between SW and integrable system theories.

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References

  1. M. Green, J. Schwarz and E. Witten,Superstring Theory, Vol. 2, Loop Amplitudes, Anomalies, and Phenomenology, Cambridge Univ. Press, Cambridge (1988).

    Google Scholar 

  2. R. Dijkgraaf, E. Verlinde, and H. Verlinde,Commun. Math. Phys.,115, 649 (1988); P. Ginsparg,Nucl. Phys. B. 295 [FS21], 153 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Polchinski,Phys. Rev. Lett.,75, 4724 (1995); Preprint hep-th/9510017 (1995);Progr. Theor. Phys. Suppl.,123, 9 (1996); Preprint hep-th/9511157 (1995);Rev. Mod. Phys.,68, 1245 (1996); Preprint hep-th/9607050 (1996); “TASI lectures on D-branes,” Preprint hep-th/9611050 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. P. Townsend,Phys. Lett. B,350, 184 (1995); Preprint hep-th/9501068 (1995); E. Witten,Nucl. Phys. B. 443, 85 (1995); Preprint hep-th/9503124 (1995); P. Townsend, “Four lectures on M-theory,” Preprint hep-th/9612121 (1996); J. Schwarz, “Lectures on superstring and M-theory dualities,” Preprint hep-th/9607201 (1996); C. Vafa. “Lectures on strings and dualities,” Preprint hep-th/9702201 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  5. E. Witten, “Vector bundles and F-theory,” Preprint hep-th/9703166 (1997).

  6. E. Witten,Nucl. Phys. B,460, 335 (1996); Preprint hep-th/9510135 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. D.-E. Diaconescu. “D-branes, monopoles, and Nahm equations,” Preprint hep-th/9608163 (1996).

  8. A. Hanany and E. Witten,Nucl. Phys. B,492, 152 (1997); Preprint hep-th/9611230 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  9. R. Donagi and E. Witten,Nucl. Phys. B,460, 299 (1996); Preprint hep-th/9510101 (1995); E. Martinec.Phys. Lett. B,367, 91 (1996); Preprint hep-th/9510204 (1995); A. Gorsky and A. Marshakov,Phys. Lett. B. 375, 127 (1996); Preprint hep-th/9510224 (1995); H. Itoyama and A. Morozov,Nucl. Phys. B,477, 855 (1996); Preprint hep-th/9511125 (1995); E. D'Hoker and D. Phong, “Calogero-Moser systems inSU(N) Seiberg-Witten theory,” Preprint hep-th/9709053 (1997); “Spectral curves for super-Yang-Mills with adjoint hypermultiplet for general Lie algebras,” Preprint hep-th/9804126 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. N. Seiberg and E. Witten,Nucl. Phys. B,426, 19 (1994); Preprint hep-th/9407087 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. N. Seiberg and E. Witten,Nucl. Phys. B,431, 484 (1994); Preprint hep-th/9408099 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. A. Klemm, W. Lerche, S. Theisen, and S. Yankielowicz,Phys. Lett. B,344, 169 (1995); Preprint hep-th/9411048 (1994); P. Argyres and A. Faraggi,Phys. Rev. Lett.,73, 3931 (1995); Preprint hep-th/9411057 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  13. K. Wilson and J. Kogut,Phys. Rep. C,12, 75 (1974).

    Article  ADS  Google Scholar 

  14. M. Shifman and A. Vainshtein,Nucl. Phys. B,277, 456 (1986).

    Article  ADS  Google Scholar 

  15. N. Seiberg, “The power of holomorphy—exact results in 4D SUSY field theory,” Preprint hep-th/9408013 (1994).

  16. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov,Phys. Lett. B,355, 466 (1995); Preprint hep-th/9505035 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii,Theory of Solitons [in Russian], Nauka, Moscow (1980); English transl.; S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. ZakharovTheory of Solitons: The Inverse Scattering Method, Plenum, New York (1984).

    MATH  Google Scholar 

  18. B. Dubrovin,Russ. Math. Surv.,36, 11 (1981).

    Article  MathSciNet  Google Scholar 

  19. I. Krichever,Funct. Anal. Appl.,11, 12 (1977);Russ. Math. Surv.,32, 185 (1977).

    Article  MATH  Google Scholar 

  20. J. Fay,Theta-Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, New York (1973).

    MATH  Google Scholar 

  21. D. Mumford,Tata Lectures on Theta, Vols. I and II, Birkhäuser, Boston (1983, 1984).

    MATH  Google Scholar 

  22. B. Dubrovin, I. Krichever, and S. Novikov, “Integrable systems: I,” in:Contemporary Problems in Mathematics (R. V. Gamkrelidze, ed.) [in Russian], Vol. 4,Dynamical Systems—4, VINITI, Moscow (1985), p. 179.

    Google Scholar 

  23. N. Hitchin,Duke Math. J.,54, 91 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Gorsky and N. Nekrasov, “Elliptic Calogero-Moser system from two dimensional current algebra,” Preprint hep-th/9401021 (1994).

  25. N. Nekrasov, “Holomorphic bundles and many-body systems,” Preprint hep-th/9503157 (1995).

  26. A. Levin and M. Olshanetsky, “Painlevé-Calogero correspondence,” alg-geom/9706010; “Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations: Free fields approach,” Preprint hep-th/9709207 (1997).

  27. A. Marshakov, “Seiberg-Witten theory, integrable systems, and D-branes,” in:New Developments in Quantum Field Theory (P. H. Damgaard and J. Jurkiewicz, eds.) (NATO ASI Series B: Physics, Vol. 366), Plenum, New York (1999), p. 279; Preprint hep-th/9709001 (1997).

    Chapter  Google Scholar 

  28. A. Gurevich and L. Pitaevskiy,JETP,38, 291 (1974).

    ADS  Google Scholar 

  29. H. Flaschka and D. McLaughlin,Progr. Theor. Phys.,55, 438 (1976).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. H. Flaschka, M. Forest, and D. McLaughlin,Commun. Pure. Appl. Math.,33, 739 (1980).

    MATH  ADS  MathSciNet  Google Scholar 

  31. I. Krichever,Funct. Anal. Appl.,22, 200 (1988);Commun. Math. Phys.,143, 415 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  32. B. Dubrovin and S. Novikov,Russ. Math. Surv.,44, 35 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  33. I. Krichever,Commun. Pure Appl. Math.,47, 437 (1994); “The tau function of the universal Whitham hierarchy. matrix models, and topological field theories,” Preprint LPTENS-92-18, Ecole Normale Supériore, Paris (1992).

    MATH  MathSciNet  Google Scholar 

  34. B. Dubrovin, “Geometry of 2d topological field theory,” Preprint hep-th/9407018 (1994);Nucl. Phys. B. 379. 627 (1992).

  35. H. Itoyama and A. Morozov,Nucl. Phys. B,491, 529 (1997); Preprint hep-th/9512161 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov,Nucl. Phys. B,527, 690 (1998); Preprint hep-th/9802007 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. A. Marshakov, M. Martellini, and A. Morozov,Phys. Lett. B,418, 294 (1998); Preprint hep-th/9706050 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  38. A. Marshakov,Theor. Math. Phys.,112, 791 (1997); Preprint hep-th/9702187 (1997).

    MATH  MathSciNet  Google Scholar 

  39. V. Inozemtsev,Commun. Math. Phys.,121, 629 (1989).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. I. Krichever,Funct. Anal. Appl.,14, 282 (1981).

    Article  MATH  Google Scholar 

  41. E. Witten, “Branes and the dynamics of QCD,” Preprint hep-th/9706109 (1997).

  42. G. Korchemsky and I. Krichever, “Solitons in high-energy QCD,” Preprint hep-th/9704079 (1997).

  43. H. Braden, A. Marshakov, A. Mironov, and A. Morozov, “Seiberg-Witten theory for a nontrivial compatification from five to four dimensions,” Preprint hep-th/9812078 (1998); “The Ruijsenaars-Schneider model in the context of Seiberg-Witten theory,” Preprint hep-th/9902205 (1999); A. Marshakov, “Seiberg-Witten curves and integrable systems,” Preprint hep-th/9903252 (1999).

  44. I. Affleck, J. Harvey, and E. Witten,Nucl. Phys. B,206, 413 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  45. N. Seiberg and E. Witten, “Gauge dynamics and compactification to three dimensions,” Preprint hep-th/9607163 (1996).

  46. S. Katz and C. Vafa, “Geometric engineering ofN=1 quantum field theory,” Preprint hep-th/9611090 (1996). O. Aharony, A. Hanany, K. Intriligator, N. Seiberg, and M. Strassler, “Aspects ofN=1 supersymmetric gauge theories in three dimensions,” Preprint hep-th/9703110 (1997); C. Vafa. “OnN=1 Yang-Mills in four dimensions,” Preprint hep-th/9801139 (1998).

  47. M. Davies, T. Hollowood, V. Khoze, and M. Mattis, “Gluino condensate and magnetic monopoles in supersymmetric gluodynamics,” Preprint hep-th/9905015 (1999).

  48. K. Li,Nucl. Phys. B,354, 711, 725 (1991); T. Eguchi and S.-K. Yang,Mod. Phys. Lett. A,5, 1693 (1990); C. Vafa,Mod. Phys. Lett. A,6, 337 (1991); E. Witten, “Mirror manifolds and topological field theory,” Preprint hep-th/9112056 (1991).

    Article  ADS  Google Scholar 

  49. E. Witten,Surv. Diff. Geom.,1, 243 (1991); R. Dijkgraaf, E. Verlinde, and H. Verlinde,Nucl. Phys. B,352, 59 (1991).

    MathSciNet  Google Scholar 

  50. S. Cecotti and C. Vafa,Phys. Rev. Lett.,68, 903 (1992); Preprint hep-th/9111016 (1991);Commun. Math. Phys.,157, 139 (1993); Preprint hep-th/9209085 (1992);Commun. Math. Phys.,158, 569 (1993). Preprint hep-th/9211097 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  51. A. Marshakov, A. Mironov, and A. Morozov,Phys. Lett. B,389, 43 (1996); Preprint hep-th/9607109 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  52. E. Witten,Commun. Math. Phys.,118, 411 (1988).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. M. Kontsevich and Yu. Manin,Commun. Math. Phys.,164, 525 (1994); Preprint hep-th/9402147 (1994); M. Kontsevich, “Enumeration of rational curves via torus actions,” Preprint hep-th/9405035 (1994); “Homological algebra of mirror symmetry,” Preprint alg-geom/9411018 (1994); M. Kontsevich and Yu. Manin. “Quantum cohomology of a product,” with an appendix by R. Kaufmann, Preprint q-alg/9502009 (1995); M. Kontsevich and Yu. Manin, “Relations between the correlators of the topological sigma-model coupled to gravity,” Preprint alg-geom/9708024 (1997).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. A. Girental and B. Kim,Commun. Math. Phys.,168, 609 (1995); Preprint hep-th/9312096 (1993); A. Givental. “Equivariant Gromov-Witten invariants,” Preprint alg-geom/9603021 (1996); “Stationary phase integrals. quantum Toda lattices, flag manifolds, and the mirror conjecture,” Preprint alg-geom/9612001 (1996); “A mirror theorem for toric complete intersections,” Preprint alg-geom/9701016 (1997).

    Article  ADS  Google Scholar 

  55. A. Marshakov, A. Mironov, and A. Morozov, “More evidence for the WDVV equations inN=2 SUSY Yang-Mills theories,” Preprint hep-th/9701123 (1997).

  56. M. Kontsevich,Funct. Anal. Appl.,25, 123 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  57. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and A. Zabrodin,Phys. Lett. B,275, 311 (1992); Preprint hep-th/9111037 (1991);Nucl. Phys. B,380, 181 (1992); Preprint hep-th/9201013 (1992); M. Kontsevich.Commun. Math. Phys.,147, 1 (1992); C. Itzykson and J.-B. Zuber,Int. J. Mod. Phys. A,7, 5661 (1992). M. Adler and P. van Moerbeke,Commun. Math. Phys.,147, 25 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  58. E. Witten, “On the Kontsevich model and other models of two-dimensional gravity,” Preprint IASSNS-HEP-91/24, Inst. of Advanced Studies, Princeton, NJ (1991).

  59. A. Marshakov, A. Mironov, and A. Morozov,Phys. Lett. B,274, 280 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  60. S. Kharchev, A. Marshakov, A. Mironov, and A. Morozov,Mod. Phys. Lett. A,8, 1047 (1993); Preprint hep-th/9208046 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  61. S. Kharchev and A. Marshakov, “Topological versus nontopological theories andp-q-duality inc<-1 2d gravity models,” in:String Theory, Quantum Gravity, and the Unification of Fundamental Interactions (M. Bianchi et al., eds.), World Scientific, Singapore (1993), p. 331; Preprint hep-th/9210072 (1992);Int. J. Mod. Phys. A. 10, 1219 (1995); Preprint hep-th/9303100 (1993).

    Google Scholar 

  62. S. Kharchev, “Kadomtsev-Petviashvili hierarchy and generalized Kontsevich model,” Preprint ITEP/TH-78/97. ITEP, Moscow (1997); Preprint hep-th/9810091 (1998).

    Google Scholar 

  63. J. Ambjørn, B. Durhuus, and J. Fröhlich,Nucl. Phys. B,257 [FS14], 433 (1985); F. David,Nucl. Phys. B,257 [FS14], 45 (1985); V. Kazakov,Phys. Lett. B,150, 282 (1985); V. Kazakov, I. Kostov, and A. Migdal,Phys. Lett. B,157, 295 (1985).

    Article  ADS  Google Scholar 

  64. V. Kazakov,Mod. Phys. Lett. A,4, 2125 (1989); E. Brézin and V. Kazakov,Phys. Lett. B,236, 144 (1990); M. Douglas and S. Shenker,Nucl. Phys. B,335, 635 (1990); D. Gross and A. Migdal,Phys. Rev. Lett.,64, 127 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  65. M. Fukuma, H. Kawai, and R. Nakayama,Int. J. Mod. Phys. A,6, 1385 (1991); R. Dijkgraaf, E. Verlinde, and H. Verlinde,Nucl. Phys. B,348, 435 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  66. T. Eguchi, K. Hori, and C. Xiong,Int. J. Mod. Phys. A,12, 1743 (1997); Preprint hep-th/9605225 (1996);Phys. Lett. B,402, 71 (1997); Preprint hep-th/9703086 (1997); T. Eguchi, M. Jinzenji, and C. Xiong,Nucl. Phys. B 510, 608 (1998); Preprint hep-th/9709152 (1997); T. Eguchi and C. Xiong, “Quantum cohomology at higher genus: topological recursion relations and Virasoro conditions,” Preprint hep-th/9801010 (1998); E. Getzler, “Intersection theory onM1, 4 and elliptic Gromov-Witten invariants,” Preprint alg-geom/9612004 (1996); “Topological recursion relations in genus 2,” Preprint mach/9801003 (1998); E. Getzler and R. Pandharipande. “Frobenius manifolds and Virasoro constraints,” Preprint math/9805114 (1998); B. Dubrovin and Y. Zhang. “Virasoro constraints and the Chern classes of the Hodge bundle,” Preprint math/9808048 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  67. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orloy,Nucl. Phys. B,357, 565 (1991).

    Article  ADS  Google Scholar 

  68. S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, and A. Zabrodin,Nucl. Phys. B,366, 569 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  69. M. Douglas,Phys. Lett. B,238, 176 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  70. I. Krichever, “On Heisenberg relations for the ordinary linear differential operators,” Preprint, ETH. Zürich (1990); S. P. Novikov,Funct. Anal. Appl.,24, 296 (1990); G. Moore,Commun. Math. Phys.,133, 261 (1990); “Matrix models of 2D gravity and isomonodromic deformations,” Preprint, Yale Univ., New Haven, Conn. (1990); R. Schimmrigk,Phys. Rev. Lett.,65, 2483 (1990); P. Grinevich and S. Novikov, “String equation—2: Physical solution,” Preprint soly-int/9501002 (1995).

  71. A. Marshakov and A. Mironov, “Seiberg-Witten systems and Whitham hierarchies: a short riview,” Preprint FIAN/TD-04/98, FIAN, Moscow (1998); ITEP/TH-51/98, ITEP, Moscow; Preprint hep-th/9809196 (1998).

    Google Scholar 

  72. M. Matone,Phys. Lett. B,357, 342 (1995); Preprint hep-th/9506102 (1995);Phys. Rev. D,53, 7354 (1996): Preprint hep-th/9506181 (1995); J. Sonnenschein, S. Theisen, and S. Yankielowicz, “On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories,” Preprint hep-th/9510129 (1995): T. Eguchi and S. K. Yang,Mod. Phys. Lett. A,11, 131 (1996); Preprint hep-th/9510183 (1995); P. Howe and P. West,Nucl. Phys. B,486, 425 (1997); Preprint hep-th/9607239 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  73. A. Losev, N. Nekrasov, and S. Shatashvili, “Issues in topological gauge theories,” Preprint hep-th/971108 (1997).

  74. G. Moore and E. Witten, “Integration over the u-plane in Donaldson theory,” Preprint hep-th/9709193 (1997). G. Moore and M. Mariño, “Integration over the Coulomb branch inN=2 gauge theory,” Preprint hep-th/9712062 (1997); “The Donaldson-Witten function for gauge groups of rank larger then one,” Preprint hep-th/9802185 (1998).

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  1. Tamm Department of Theoretical Physics, Lebedev Physical Institute, RAS, Moscow, Russia

    A. V. Marshakov

  2. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Marshakov

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This paper was written at the request of the Editorial Board.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 179–243. November, 1999.

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Marshakov, A.V. Strings, susy gauge theories, and integrable systems. Theor Math Phys 121, 1409–1461 (1999). https://doi.org/10.1007/BF02557215

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