Integrability and Seiberg-Witten theory curves and periods

Nuclear Physics B

Volumn 477, Issue 3, 1996, Pages 855-877

  Itoyama, H ^a   Morozov, A ^b  

^a OSAKA UNIVERSITY   (Japan)

^b INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

Elliptic Calogero system; Integrability; Low energy effective action; N = 2 supersymmetry; Toda system

Indexed keywords

EID: 0030596980     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/0550-3213(96)00358-6     Document Type: Article

References (59)

1. [1] See, for example, A. Morozov, String Theory, what is it, Rus. Physics Uspekhi 35 (1992) 671 (v.162, No. 8, p. 84-175 of Russian edition); A. Morozov, Integrability and Matrix Models, ibid. 164 (1994) No. 1, 3-62 (Rus. ed.), hep-th/9303139.
2. [1] See, for example, A. Morozov, String Theory, what is it, Rus. Physics Uspekhi 35 (1992) 671 (v.162, No. 8, p. 84-175 of Russian edition); A. Morozov, Integrability and Matrix Models, ibid. 164 (1994) No. 1, 3-62 (Rus. ed.), hep-th/9303139.
3. [1] See, for example, A. Morozov, String Theory, what is it, Rus. Physics Uspekhi 35 (1992) 671 (v.162, No. 8, p. 84-175 of Russian edition); A. Morozov, Integrability and Matrix Models, ibid. 164 (1994) No. 1, 3-62 (Rus. ed.), hep-th/9303139.
4. [2] N. Seiberg and E. Witten, Electric-Magnetic Duality, Monopole Condensation and Confinement in N = 2 Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426 (1994) 19-52; Err.; ibid. B 430 (1994) 485-486, hep-th/9407087.
5. [2] N. Seiberg and E. Witten, Electric-Magnetic Duality, Monopole Condensation and Confinement in N = 2 Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426 (1994) 19-52; Err.; ibid. B 430 (1994) 485-486, hep-th/9407087.
6. [3] N. Seiberg and E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD, ibid. B 431 (1994) 484-550, Nucl. Phys. B 431 (1994) 484-550, hep-th/9408099.
7. [3] N. Seiberg and E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD, ibid. B 431 (1994) 484-550, Nucl. Phys. B 431 (1994) 484-550, hep-th/9408099.
8. [4] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov et al., Integrability and Exact Seiberg-Witten Solution, Phys. Lett. B 355 (1995) 466-474, hep-th/9505035.
9. [5] E. Martinec and N. Warner, Integrable Systems and Supersymmetric Gauge Theories, Nucl. Phys. B 459 (1996) 97-112, hep-th/9509161.
10. [6] 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.
11. [7] T. Eguchi and S. K. Yang, Prepotentials of N = 2 Supersymmetric Gauge Theories and Soliton Equations, hep-th/9510183.
12. [8] R. Donagi and E. Witten, Supersymmetric Yang-Mills Theory and Integrable Systems, IASSNS-HEP-95-78, Nucl. Phys. B 460 (1996) 299-334, hep-th/9510101.
13. [9] E. Martinec, Integrable Structures in Supersymmetric Gauge and String Theory, Phys. Lett. B 367 (1996) 91-96, hep-th/9510204.
14. [10] A. Gorsky and A. Marshakov, Towards Effective Topological Gauge Theories on Spectral Curves, ITEP/TH-9/95, hep-th/9510224.
15. [11] E. Martinec and N. Warner, Integrability in N = 2 Gauge Theory: A Proof, hep-th/9511052.
16. [12] A. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, Simple Singularities and N = 2 Supersymmetric Yang-Mills Theory, Phys. Lett. B 344 (1995) 169, hep-th/9411048.
17. [13] P. Agyres and A. Faraggi, The Vacuum Structure and Spectrum of N = 2 Supersymmetric SU(N) Gauge Theory, Phys. Rev. Lett. 73 (1995) 3931, hep-th/9411057.
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19. [15] A. Brandhuber and K. Landsteiner, On the Monodromies of N = 2 Supersymmetric Yang-Mills Theory with Gauge Group SO(2n), Phys. Lett. B 358 (1995) 73-80, hep-th/9507008.
20. c) Gauge Theories, Nucl. Phys. B 452 (1995) 283-312, hep-th/9505075.
21. [17] D. Olive and E. Witten, Supersymmetry Algebra that Includes Topological Charges, Phys. Lett. B 78 (1978) 97-101.
22. [18] The classical review paper on integrable systems of particles is: M. Olshanetsky and A. Perelomov, Classical Integrable Finite-Dimensional Systems Related to Lie Algebras, Phys. Rep. 71C (1981) 97-101.
23. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
24. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
25. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
26. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
27. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
28. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
29. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
30. [19] The relevant chapter of integrability theory is very old and the number of references is very large. We mention just a few papers, directly related to our consideration: I. Krichever, Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Sov. Math. Surveys, 32 (1977) 185-213; I. Krichever, The Integration of Non-linear Equations by the Methods of Algebraic Geometry, Funk. Anal. and Appl. 11 (1977) No. 1 15-31 (Rus. ed.); I. Krichever, Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable System of Particles, Funk. Anal. and Appl. 14 (1980) 282-290 (No. 4 15-31 of Rus. ed.); B. Dubrovin, Theta Functions and Non-linear Equations, Sov. Math. Surveys, 36 (1981) No. 2 11-80 (Rus. ed.); H. Flaschka and D. McLaughlin, Canonicaly Conjugate Variables for the KdV Equation and the Toda Lattice with Periodic Boundary Conditions, Progr. Theor. Phys. 55 (1976) 438-456; M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Adv. Math. 38 (1980) 267-317; M. Adler and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, ibid. 318-379. O. Babelon, E. Billey, I. Krichever and M. Talon, Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, hep-th/9411160.
31. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
32. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
33. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
34. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
35. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
36. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
37. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
38. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
39. [20] The modern-style language in this field uses the notion of Hitchin systems, which emphasizes interpretation of Lax equation as a flat connection condition. A few directly relevant references are: N. Hitchin, Stable Bundles and Integrable Systems, Duke Math. Journ. 54 (1987) 91-114; N. Hitchin, Flat Connections and Geometric Quantization, Comm.Math. Phys. 131 (1990) 347-380; E. Markman, Spectral Curves and Integrable Systems, Comp. Math. 93 (1994) 255-290; A. Beilinson and V. Drinfeld, Quantization of Hitchin's Fibration and Langlands Program, preprint (1994); B. Feigin and E. Frenkel, Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras, Int. J. Mod. Phys. A7, Suppl.1A (1992) 197-215; B. Enriquez and V. Roubtsov, Hitchin Systems, Higher Gaudin Operators and R-Matrices, alggeom/9503010; N. Nekrasov, Holomorphic Bundles and Many-Body Systems, hep-th/9503157; M. Olshanetsky Generalized Hitchin Systems and the Knizhnik-Zamolodchikov-Bernard Equation on Elliptic Curves, hep-th/9510143; O. Sheinman, hep-th/9510165; see also [8] and [9].
40. [21] See the following papers for general description and references: B. Dubrovin and S. Novikov, Sov. Math. Surveys 36 (1981) No. 2 12 (Rus. ed.); ref. [23] below.
41. [22] B. Dubrovin, Geometry of 2d Topological Field Theories, SISSA-89/94/FM; B. Dubrovin, Integrable Systems in Topological Field Theory, Nucl. Phys. B 379 (1992) 627-689; B. Dubrovin, Hamiltonian Formalism of Witham-type Hierarchies in Topological Landau-Ginzburg Model, Comm. Math. Phys. 145 (1992) 195.
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