Spectral curves, opers and integrable systems
Publications Mathématiques de l'IHÉS, Tome 94 (2001), pp. 87-159.
@article{PMIHES_2001__94__87_0,
     author = {Ben-Zvi, David and Frenkel, Edward},
     title = {Spectral curves, opers and integrable systems},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {87--159},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {94},
     year = {2001},
     mrnumber = {1896178},
     zbl = {1113.14301},
     language = {en},
     url = {http://archive.numdam.org/item/PMIHES_2001__94__87_0/}
}
TY  - JOUR
AU  - Ben-Zvi, David
AU  - Frenkel, Edward
TI  - Spectral curves, opers and integrable systems
JO  - Publications Mathématiques de l'IHÉS
PY  - 2001
SP  - 87
EP  - 159
VL  - 94
PB  - Institut des Hautes Études Scientifiques
UR  - http://archive.numdam.org/item/PMIHES_2001__94__87_0/
LA  - en
ID  - PMIHES_2001__94__87_0
ER  - 
%0 Journal Article
%A Ben-Zvi, David
%A Frenkel, Edward
%T Spectral curves, opers and integrable systems
%J Publications Mathématiques de l'IHÉS
%D 2001
%P 87-159
%V 94
%I Institut des Hautes Études Scientifiques
%U http://archive.numdam.org/item/PMIHES_2001__94__87_0/
%G en
%F PMIHES_2001__94__87_0
Ben-Zvi, David; Frenkel, Edward. Spectral curves, opers and integrable systems. Publications Mathématiques de l'IHÉS, Tome 94 (2001), pp. 87-159. http://archive.numdam.org/item/PMIHES_2001__94__87_0/

[AB] M. Adams and M. Bergvelt, The Krichever map, vector bundles over algebraic curves, and Heisenberg algebras, Comm. Math. Phys., 154 (1993), 265-305. | MR | Zbl

[AMP] A. Álvarez, J. Muñoz and F. Plaza, The algebraic formalism of soliton equations over arbitrary base fields, Workshop on Abelian Varieties and Theta Functions (Morelia, 1996), Aportaciones Mat. Investig., 13, Soc. Mat. Mexicana (3-40), 1998 (alg-geom/9606009). | MR | Zbl

[Ba] A. BALAN, Les équations mKdV généralisées, Preprint école Polytechnique, No. 98-5, April 1998.

[BL] A. Beauville and Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys., 164 (1993), 385-419. | MR | Zbl

[BB] A. Beilinson and J. Bernstein, A Proof of Jantzen Conjectures, Advances in Soviet Mathematics, Vol. 16, Part 1 (1-50), 1993. | MR | Zbl

[BD1] A. BEILINSON and V. DRINFELD, Opers, Preprint, 1994. | MR

[BD2] A. BEILINSON and V. DRINFELD, Quantization of Hitchin's Integrable System and Hecke Eigensheaves, in preparation.

[BS] A. Blum and U. Stuhler, Drinfeld modules and elliptic sheaves, Vector bundles on curves - new directions (eds S. Kumar et al.) LNM 1649, 1997. | MR | Zbl

[Ch1] I. Cherednik, Group interpretation of Baker functions and τ-function, Uspekhi Mat. Nauk 38, No. 6 (1983), 133-134. | MR | Zbl

[Ch2] I. Cherednik, Determination of τ- functions for generalized affine Lie algebras, Funct. Anal. Appli. Prilozhen, 17, No. 3 (1983), 93-95. | MR | Zbl

[Ch3] I. Cherednik, Functional realizations of basic representations of factorizable Lie groups and Lie algebras, Funct. Anal. Appl., 19, 193-206 (1985). | MR | Zbl

[CC] C. Contou-Carrére, Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole modéré, C. R. Acad. Sci. Paris, série I 318 (1994), 743-746. | MR | Zbl

[DJKM] E. DATE, M. JIMBO, M. KASHIWARA, T. MIWA, Transformation groups for soliton equations, in: M. Jimbo, T. Miwa (eds.), Non-linear Integrable Systems - Classical Theory and Quantum Theory, pp. 39-120, Singapore: World Scientific, 1983. | MR | Zbl

[dGHM] M. De Groot, T. Hollowood and L. Miramontes, Generalized Drinfeld-Sokolov hierarchies, Comm. Math. Phys., 145 (1992), 57-84. | MR | Zbl

[DF] F. Delduc and L. Fehér, Regular conjugacy classes in the Weyl group and integrable hierarchies, J. Phys., A28 (1995), 5843-5882. | MR | Zbl

[D] R. Donagi, Decomposition of spectral covers, Astérisque, 218 (1993), 145-175. | MR | Zbl

[DG] R. Donagi and D. Gaitsgory, The gerbe of Higgs bundles, Preprint math. AG/0005132. | MR | Zbl

[DM] R. Donagi and E. Markman, Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles, Integrable systems and quantum groups, Lecture Notes in Math., 1620, Springer 1995. | MR | Zbl

[Dr] V. Drinfeld, Commutative subrings of certain noncommutative rings, Funct. Anal Appl., 11, No. 1 (1977), 11-14. | MR | Zbl

[DSi] V. Drinfeld and C. Simpson, B-structures on G-bundles and local triviality, Math. Res. Lett., 2, No. 6 (1995), 823-829. | MR | Zbl

[DS] V. Drinfeld and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Journal of Soviet Mathematics, vol. 30 (1985), 1975-2035. | MR | Zbl

[EF1] B. Enriquez and E. Frenkel, Equivalence of two approaches to integrable equations of KdV type, Comm. Math. Phys., 185 (1997), 211-230. | MR | Zbl

[EF2] B. Enriquez and E. Frenkel, Geometric interpretation of the Poisson structure in affine Toda field theories, Duke Math. J., 92 (1998), 459-495. | MR | Zbl

[Fa] G. Faltings, Stable G-bundles and projective connections, J. Alg. Geom., 2 (1993), 507-568. | MR | Zbl

[Fe] L. FEHÉR, KdV type systems and W -algebras in the Drinfeld-Sokolov approach, Preprint hepth/9510001.

[FHM] L. Fehér, J. Harnad and I. Marshall, Generalized Drinfeld-Sokolov reductions and KdV type hierarchies, Comm. Math. Phys., 154 (1993), 181-214. | MR | Zbl

[FF1] B. Feigin, E. Frenkel, Integrals of motion and quantum groups, in Lect. Notes in Math., 1620, pp. 349-418, Springer Verlag, 1995. | MR | Zbl

[FF2] B. Feigin and E. Frenkel, Kac-Moody groups and integrability of soliton equations, Invent. Math., 120 (1995), 379-408. | EuDML | MR | Zbl

[FF3] B. Feigin and E. Frenkel, Integrable hierarchies and Wakimoto modules, in Differential Topology, Infinite- Dimensional Lie Algebras, and Applications: D. B. Fuchs' 60th Anniversary Collection, A. Astashkevich and S. Tabachnikov (eds), pp. 27-60, AMS, 1999. | Zbl

[F] E. Frenkel, Five lectures on soliton equations, in Integral Systems, C.-L. Terng and K. Uhlenbeck (eds.), Surveys in Differential Geometry, vol. 4, pp. 131-180, International Press, 1998. | MR | Zbl

[Gin] V. Ginzburg, Perverse Sheaves on a Loop Group and Langlands' Duality, Preprint alg-geom/9511007.

[Ha] G. Harder, Halbeinfache Gruppenschemata uber Dedekindringen, Invent. Math., 4 (1967), 165-191. | EuDML | MR | Zbl

[Hi] N. Hitchin, Stable bundles and integrable systems, Duke Math. J., 54 (1990), 91-114. | MR | Zbl

[Kac1] V. G. Kac, Infinite-dimensional algebras, Dedekind η-function, classical Möbius function and the very strange formula, Adv. Math., 30 (1978), 85-136. | MR | Zbl

[Kac2] V. G. Kac, Infinite-dimensional Lie algebras, third edition, Cambridge University Press, 1990. | MR

[KP] V. G. Kac and D. Peterson, 112 constructions of the basic representation of the loop group of E8, Proceedings of “Anomalies, geometry, topology” (Argonne, 1985), pp. 276-298, World Scientific, 1985. | Zbl

[KSU] T. Katsura, Y. Shimizu and K. Ueno, Formal groups and conformal field theory over Z, in integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., 19, Boston (1989), Academic Press, 347-366. | MR | Zbl

[KL] D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. of Math., 62, No. 1 (1988), 129-168. | MR | Zbl

[Kos] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81 (1959), 973-1032. | MR | Zbl

[Kr] I. KRICHEVER, Algebro-geometric construction of the Zakharov-Shabat equations and their periodic solutions, Dokl. Akad. Nauk SSSR, 227, No. 2 (1976), 291-294. | MR | Zbl

[LS] Y. Laszlo and C. Sorger, The line bundles on the stack of parabolic G-torsors over curves and their sections, Ann. Sci. Ec. Norm. Sup., 30, No. 4 (1997), 499-525. | EuDML | Numdam | MR | Zbl

[Lau1] G. LAUMON, Transformation de Fourier généralisée, Preprint alg-geom/9603004.

[Lau2] G. LAUMON, Sur les modules de Krichever, Preprint, Université de Paris-Sud, 86 T 20.

[LMB] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge 39. Springer-Verlag, Berlin, 2000. | MR | Zbl

[LM] Y. Li and M. Mulase, Prym varieties and integrable systems, Comm. Anal. Geom., 5, No. 2 (1997), 279-332. | MR | Zbl

[McK] H. Mckean, Is there an infinite-dimensional algebraic geometry? Hints from KdV, Proc. Symp. Pure Math., 49, vol. I, pp. 27-37, AMS, 1989. | MR | Zbl

[MV] I. Mirkovic and K. Vilonen, Perverse sheaves on loop Grassmannians and Langlands duality, Math. Res. Lett., 7 (2000), 13-24. | MR | Zbl

[M1] M. Mulase, Cohomological structure in soliton equations and Jacobian varieties, J. Diff. Geom., 19 (1984), 403-430. | MR | Zbl

[M2] M. Mulase, Category of vector bundles on algebraic curves and infinite-dimensional Grassmannians, Int. J. Math., 1 (1990), 293-342. | MR | Zbl

[M3] M. Mulase, Algebraic theory of the KP equations. Perspectives in mathematical physics, Conf. Proc. Lect. Notes Math. Phys. III., Cambridge MA (1994), Internat. Pres, 151-217. | MR | Zbl

[Mum] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations, Int. Symposium on Algebraic Geometry, Kyoto, pp. 115-153, Kinokuniya Book Store, 1977. | MR | Zbl

[N1] A. Nakayashiki, Structure of Baker-Akhiezer modules of pricipally polarized abelian varieties, commuting partial differential operators and associated integrable systems, Duke Math J., 62 (1991), 315-358. | MR | Zbl

[N2] A. Nakayashiki, Commuting partial differential operators and vector bundles over abelian varieties, Amer. J. Math., 116 (1994), 65-100. | MR | Zbl

[PS] A. Pressley and G. Segal, Loop Groups, Oxford University Press, 1986. | MR | Zbl

[Ro1] M. Rothstein, Connections on the total Picard sheaf and the KP hierarchy, Acta Applicandae Mathematicae, 42 (1996), 297-308. | MR | Zbl

[Ro2] M. Rothstein, Sheaves with connection on abelian varieties, Duke Math J., 84, No. 3 (1996), 565-598. | MR | Zbl

[Sor] C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves. School on Algebraic Geometry (Trieste, 1999) ICTP, Lect. Notes 1, Abdus Salam ICTP, (Trieste, 2000), 1-57. | MR | Zbl

[Tel] C. Teleman, Borel-Weil-Bott theory on the moduli stack of G-torsors over a curve, Invent. Math., 134 (1998), 1-57. | MR | Zbl

[SW] G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. IHES, 61 (1985), 5-65. | EuDML | Numdam | MR | Zbl

[W] G. Wilson, Habillage et fonctions , C. R. Acad. Sci. Paris, 299, Sér. I, No. 13 (1984), 587-590. | MR | Zbl