Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

Nuclear Physics B

Volumn 530, Issue 3, 1998, Pages 537-610

  D'Hoker, Eric ^a,b   Phong, D H ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c NONE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032547607     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0550-3213(98)00568-9     Document Type: Article

References (66)

1. [1] N. Seiberg and E. Witten, Electro-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19, hep-th/9407087; Monopoles, duality, and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 494, hep-th/9410167.
2. [1] N. Seiberg and E. Witten, Electro-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19, hep-th/9407087; Monopoles, duality, and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 494, hep-th/9410167.
3. [2] W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Proceedings of the Spring School and Workshop in String Theory, ICTP, Trieste (1996), hep-th/9611190; Nucl. Phys. B (Proc. Suppl.) 55B (1997) 83, and references therein.
4. [2] W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Proceedings of the Spring School and Workshop in String Theory, ICTP, Trieste (1996), hep-th/9611190; Nucl. Phys. B (Proc. Suppl.) 55B (1997) 83, and references therein.
5. [3] A. Gorski, I.M. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466, hep-th/9505035; M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theories, Phys. Lett. B 357 (1996) 342, hep-th/9506102; T. Nakatsu and K. Takasaki, Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, Mod. Phys. Lett. A 11 (1996) 157-168, hep-th/9509162; Isomonodromic deformations and supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 5505, hep-th/9603069.
6. [3] A. Gorski, I.M. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466, hep-th/9505035; M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theories, Phys. Lett. B 357 (1996) 342, hep-th/9506102; T. Nakatsu and K. Takasaki, Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, Mod. Phys. Lett. A 11 (1996) 157-168, hep-th/9509162; Isomonodromic deformations and supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 5505, hep-th/9603069.
7. [3] A. Gorski, I.M. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466, hep-th/9505035; M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theories, Phys. Lett. B 357 (1996) 342, hep-th/9506102; T. Nakatsu and K. Takasaki, Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, Mod. Phys. Lett. A 11 (1996) 157-168, hep-th/9509162; Isomonodromic deformations and supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 5505, hep-th/9603069.
8. [3] A. Gorski, I.M. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466, hep-th/9505035; M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theories, Phys. Lett. B 357 (1996) 342, hep-th/9506102; T. Nakatsu and K. Takasaki, Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, Mod. Phys. Lett. A 11 (1996) 157-168, hep-th/9509162; Isomonodromic deformations and supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 5505, hep-th/9603069.
9. [4] R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299, hep-th/9510101.
10. [5] E. Martinec and N. Warner, Integrable systems and supersymmetric gauge theories, Nucl. Phys. B 459 (1996) 97, hep-th/9509161.
11. [6] E. Martinec, Integrable structures in supersymmetric gauge and string theory, hep-th/9510204.
12. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
13. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
14. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
15. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
16. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
17. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
18. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
19. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
20. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
21. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
22. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
23. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
24. [7] J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145, hep-th/9510129; T. Eguchi and S. K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183; H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529, hep-th/9512161; Integrability and Seiberg-Witten theory, hep-th/960116 8; Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; C. Ahn and S. Nam, Integrable structure in supersymmetric gauge theories with massive hypermultiplets, Phys. Lett. B 387 (1996) 304, hep-th/9603028; I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom. 45 (1997) 349, hep-th/9604199; G. Bonelli and M. Matone, Non-perturbative relations in N = 2 SUSY Yang-Mills WDVV equation, Phys. Rev. Lett. 77 (1996) 4712, hep-th/9605090; A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43, hep-th/9607109; A. Marshakov, Non-perturbative quantum theories and integrable equations, Int. J. Mod. Phys. A 12 (1997) 1607, hep-th/9610242; S. Nam, Integrable models, SUSY gauge theories and string theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612134; A. Marshakov, On integrable systems and supersymmetric gauge theories, Theor. Math. Phys. 112 (1997) 791, hep-th/9702083; I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III.
25. [8] E. D'Hoker and D.H. Phong, Calogero-Moser systems in SU(N) Seiberg-Witten theory, Nucl. Phys. B 513 (1998) 405, hep-th/9709053.
26. [9] R. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010; D.S. Freed, Special Kähler manifolds, hep-th/9712042; R. Carroll, Prepotentials and Riemann surfaces, hep-th/9802130.
27. [9] R. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010; D.S. Freed, Special Kähler manifolds, hep-th/9712042; R. Carroll, Prepotentials and Riemann surfaces, hep-th/9802130.
28. [9] R. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010; D.S. Freed, Special Kähler manifolds, hep-th/9712042; R. Carroll, Prepotentials and Riemann surfaces, hep-th/9802130.
29. [10] M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves, Advances in Math. 38 (1980) 267; M. Adler and P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Advances in Math. 38 (1980) 318; M. Adler and P. van Moerbeke, The Toda lattice, Dynkin diagrams, singularities and Abelian varieties, Invent. Math. 103 (1991) 223.
30. [10] M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves, Advances in Math. 38 (1980) 267; M. Adler and P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Advances in Math. 38 (1980) 318; M. Adler and P. van Moerbeke, The Toda lattice, Dynkin diagrams, singularities and Abelian varieties, Invent. Math. 103 (1991) 223.
31. [10] M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves, Advances in Math. 38 (1980) 267; M. Adler and P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Advances in Math. 38 (1980) 318; M. Adler and P. van Moerbeke, The Toda lattice, Dynkin diagrams, singularities and Abelian varieties, Invent. Math. 103 (1991) 223.
32. [11] F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419; J. Moser, Integrable systems of non-linear evolution equations, in Dynamical Systems, Theory and Applications, ed. J. Moser, Lecture Notes in Physics 38 (Springer, Berlin, 1975).
33. [11] F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419; J. Moser, Integrable systems of non-linear evolution equations, in Dynamical Systems, Theory and Applications, ed. J. Moser, Lecture Notes in Physics 38 (Springer, Berlin, 1975).
34. [12] I.M. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl. 14 (1980) 282.
35. [13] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91.
36. [14] M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71C (1981) 313; A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I (Birkhäuser, Boston, 1990) and references therein; A.N. Leznov and M.V. Saveliev, Group Theoretic Methods for Integration of Non-Linear Dynamical Systems (Birkhäuser, Boston, 1992).
37. [14] M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71C (1981) 313; A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I (Birkhäuser, Boston, 1990) and references therein; A.N. Leznov and M.V. Saveliev, Group Theoretic Methods for Integration of Non-Linear Dynamical Systems (Birkhäuser, Boston, 1992).
38. [14] M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71C (1981) 313; A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I (Birkhäuser, Boston, 1990) and references therein; A.N. Leznov and M.V. Saveliev, Group Theoretic Methods for Integration of Non-Linear Dynamical Systems (Birkhäuser, Boston, 1992).
39. [15] E. D'Hoker and D.H. Phong, Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras, Nucl. Phys. B 530 (1998) 611, next article in this issue, hep-th/9804125.
40. [16] E. D'Hoker and D.H. Phong, Spectral curves for super-Yang-Mills with adjoint hypermultiplet for general Lie algebras, hep-th/9804126.
41. [17] V.I. Inozemtsev, The finite Toda lattices, Comm. Math. Phys. 121 (1989) 629.
42. [18] V.I. Inozemtsev, Lax representation with spectral parameter on a torus for integrable particle systems, Lett. Math. Phys. 17 (1989) 11.
43. [19] A. Klemm, W. Lerche and S. Theisen, Non-perturbative actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 1929, hep-th/9505150.
44. c) gauge theories, Nucl. Phys. B 489 (1997) 211, hep-th/9609145; The renormalization group equation for N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89, hep-th/9610156; E. D'Hoker and D.H. Phong, Strong coupling expansions in SU(N) Seiberg-Witten theory, hep-th/9701151, Phys. Lett. B 397 (1997) 94.
45. c) gauge theories, Nucl. Phys. B 489 (1997) 211, hep-th/9609145; The renormalization group equation for N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89, hep-th/9610156; E. D'Hoker and D.H. Phong, Strong coupling expansions in SU(N) Seiberg-Witten theory, hep-th/9701151, Phys. Lett. B 397 (1997) 94.
46. c) gauge theories, Nucl. Phys. B 489 (1997) 211, hep-th/9609145; The renormalization group equation for N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89, hep-th/9610156; E. D'Hoker and D.H. Phong, Strong coupling expansions in SU(N) Seiberg-Witten theory, hep-th/9701151, Phys. Lett. B 397 (1997) 94.
47. c) gauge theories, Nucl. Phys. B 489 (1997) 211, hep-th/9609145; The renormalization group equation for N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89, hep-th/9610156; E. D'Hoker and D.H. Phong, Strong coupling expansions in SU(N) Seiberg-Witten theory, hep-th/9701151, Phys. Lett. B 397 (1997) 94.
48. [21] V. Kac, Infinite-dimensional Lie Algebras (Birkhäuser, Boston, 1983.
49. [22] P. Goddard and D. Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. A 1 (1986) 303.
50. [23] I.M. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl. 14 (1980) 282.
51. [24] A. Erdelyi, ed., Higher Transcendental Functions, Bateman Manuscript Project, Vol. II (R.E. Krieger, Florida, 1981).
52. [25] W.G. McKay, J. Patera and D.W. Rand, Tables of Representations of Simple Lie Algebras, Vol. I: Exceptional Simple Lie algebras, Center de Mathématiques, Université de Montréal, 1990.
53. [26] S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69, hep-th/9505105; M. Bershadsky, K. Intrilligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215, hep-th/9605200; S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173, hep-th/9609239; S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solutions of 4D N = 2 gauge theories, Adv. Theor. Math. Phys. 1 (1998) 53, hep-th/9706110.
54. [26] S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69, hep-th/9505105; M. Bershadsky, K. Intrilligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215, hep-th/9605200; S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173, hep-th/9609239; S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solutions of 4D N = 2 gauge theories, Adv. Theor. Math. Phys. 1 (1998) 53, hep-th/9706110.
55. [26] S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69, hep-th/9505105; M. Bershadsky, K. Intrilligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215, hep-th/9605200; S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173, hep-th/9609239; S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solutions of 4D N = 2 gauge theories, Adv. Theor. Math. Phys. 1 (1998) 53, hep-th/9706110.
56. [26] S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69, hep-th/9505105; M. Bershadsky, K. Intrilligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215, hep-th/9605200; S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173, hep-th/9609239; S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solutions of 4D N = 2 gauge theories, Adv. Theor. Math. Phys. 1 (1998) 53, hep-th/9706110.
57. [27] E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3, hep-th/9703166.
58. [28] A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152.
59. [29] A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M-theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175, hep-th/9705232; K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199; A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123.
60. [29] A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M-theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175, hep-th/9705232; K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199; A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123.
61. [29] A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M-theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175, hep-th/9705232; K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199; A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123.
62. [29] A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M-theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175, hep-th/9705232; K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199; A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123.
63. [29] A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M-theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175, hep-th/9705232; K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197, hep-th/9705199; A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123.
64. [30] A. Gorskii, Branes and integrability in the N = 2 SUSY YM theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612238; A. Gorskii, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin, hep-th/9710239; S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three dimensions, hep-th/9711145.
65. [30] A. Gorskii, Branes and integrability in the N = 2 SUSY YM theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612238; A. Gorskii, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin, hep-th/9710239; S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three dimensions, hep-th/9711145.
66. [30] A. Gorskii, Branes and integrability in the N = 2 SUSY YM theory, Int. J. Mod. Phys. A 12 (1997) 1243, hep-th/9612238; A. Gorskii, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin, hep-th/9710239; S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three dimensions, hep-th/9711145.