Lectures on the langlands program and conformal field theory

Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization

Volumn , Issue , 2007, Pages 387-533

  Frenkel, Edward ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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EID: 84874421172     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-30308-4_11     Document Type: Chapter

References (122)

1. R.P. Langlands, Problems in the theory of automorphic forms, in Lect. Notes in Math. 170, pp. 18-61, Springer Verlag, 1970.
2. P. Goddard, J. Nuyts and D. Olive, Gauge Theories and Magnetic Change, Nuclear Phys. B125 (1977) 1-28.
3. E. Witten, Talk at the DARPA Workshop on the Langlands Program and Physics, IAS, March 2004; Gauge theory and the geometric Langlands Program, notes of a talk at the Third Simons Workshop, SUNY at Stony Brook, August 2005, available at http://insti.physics.sunysb.edu/itp/conf/si-monswork3/talks/Witten. pdf
4. A. Kapustin, Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality, Preprint hep-th/0501015.
5. E. Witten, Quantum field theory, Grassmannians, and algebraic curves, Comm. Math. Phys. 113 (1988) 529-600.
6. A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333-380.
7. D. Friedan and S. Shenker, The analytic geometry of two-dimensional conformal field theory, Nucl. Phys. B281 (1987) 509-545.
8. G. Segal, The definition of conformal field theory, in Topology, geometry and quantum field theory, pp. 421-577, London Math. Soc. Lecture Note Ser. 308, Cambridge University Press, 2004.
9. V. Knizhnik and A. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B247 (1984) 83-103.
10. E. Witten, Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984) 455-472.
11. 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.
12. E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005) 297-404 (math.QA/0210029).
13. V. Drinfeld and V. Sokolov, Lie algebras and KdV type equations, J. Sov. Math. 30 (1985) 1975-2036.
14. A. Beilinson and V. Drinfeld, Opers, Preprint math.AG/0501398.
15. A. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, available at http://www.math.uchicago. edu/∼arinkin/langlands
16. A. Polyakov and P. Wiegmann, Goldstone fields in two dimensions with mutivalued actions, Phys. Lett. 141B (1984) 223-228.
17. A. Beilinson and V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications 51, AMS, 2004.
18. A. Beilinson, Langlands parameters for Heisenberg modules, Preprint math.QA/0204020.
19. D. Gaitsgory, Notes on 2D conformal field theory and string theory, in Quantum fields and strings: a course for mathematicians, Vol. 2, pp. 1017-1089, AMS, 1999.
20. E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, Second Edition, AMS, 2004.
21. J. Arthur, Automorphic representations and number theory in Seminar on Harmonic Analysis (Montreal, 1980), pp. 3-51, CMS Conf. Proc. 1, AMS, 1981.
22. J. Arthur, The principle of functoriality, Bull. AMS 40 (2002) 39-53.
23. A. Borel, Automorphic L-functions, in Automorphic Forms, Representations and L-functions, Part 2, Proc. of Symp. in Pure Math. 33, pp. 27-61, AMS, 1979.
24. A.W. Knapp, Introduction to the Langlands program, in Representation theory and automorphic forms (Edinburgh, 1996), pp. 245-302, Proc. Symp. Pure Math. 61, AMS, 1997.
25. M.R. Murty, A motivated introduction to the Langlands program, in Advances in number theory (Kingston, ON, 1991), pp. 37-66, Oxford Univ. Press, 1993.
26. S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. 10 (1984) 177-219.
27. G. Laumon, Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langland, Séminaire Bourbaki, Exp. No. 906 (math.AG/0207078).
28. J. Bernstein and S. Gelbart, eds., An Introduction to the Langlands Program, Birkḧauser, 2004.
29. E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proceedings of the International Congress of Mathematical Physics, Paris, 1994, ed. D. Iagolnitzer, pp. 606-642, International Press, 1995 (q-alg/9506003).
30. E. Frenkel, Recent Advances in the Langlands Program, Bull. Amer. Math. Soc. 41 (2004) 151-184 (math.AG/0303074).
31. N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics 58, Springer-Verlag, 1977.
32. R. Langlands, Letter to A. Weil, January 1967, available at http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/functorialityhtml #weil1967
33. S. Kudla, From modular forms to automorphic representations, in pp. 133-152.
34. E. de Shalit, L-functions of elliptic curves and modular forms, in, pp. 89-108.
35. R. Taylor, Galois representations, available at http://abel.math. harvard.edu/~rtaylor
36. K. Ribet, Galois representations and modular forms, Bull. AMS 32 (1995) 375-402.
37. D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, 2004.
38. H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ecole Norm. Sup. (4) 19 (1986) 409-468.
39. A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995) 443-551.
40. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995) 553-572.
41. C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001) 843-939.
42. V.G. Drinfeld, Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2), Amer. J. Math. 105 (1983) 85-114.
43. V.G. Drinfeld, Langlands conjecture for GL(2) over function field, Proc. of Int. Congress of Math. (Helsinki, 1978), pp. 565-574; Moduli varieties of F-sheaves, Funct. Anal. Appl. 21 (1987) 107-122; The proof of Pe-tersson's conjecture for GL(2) over a global field of characteristic p, Funct. Anal. Appl. 22 (1988) 28-43.
44. L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1-241.
45. A.N. Parshin, Abelian coverings of arithmetic schemes, Sov. Math. Dokl. 19 (1978) 1438-1442.
46. K. Kato, A generalization of local class field theory by using K-groups, J. Fac. Sci. Univ. Tokyo, Sec. 1A 26 (1979) 303-376.
47. M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, in Functional analysis on the eve of 21st century, S. Gindikin, J. Lepowsky, R. Wilson (eds.), vol. 1, Progress in Math. 131, p. 119-151, Birkḧauser, 1995.
48. E. Frenkel and D. Gaitsgory, Local geometric Langlands correspondence and affine Kac-Moody algebras, Preprint math.RT/0508382. ́
49. J.S. Milne, Etale cohomology, Princeton University Press, 1980.
50. E. Freitag, R. Kiehl, Etale Cohomology and the Weil conjecture, Springer, 1988.
51. G. Laumon, Transformation de Fourier, constantes d'équations fonc-tionelles et conjecture de Weil, Publ. IHES 65 (1987) 131-210.
52. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982).
53. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
54. S.I. Gelfand and Yu.I. Manin, Homological Algebra, Encyclopedia of Mathematical Sciences 38, Springer, 1994.
55. J. Bernstein, Algebraic theory of D-modules, available at http://www.math.uchicago.edu/∼arinkin/langlands
56. Ch. Sorger, Lectures on moduli of principal G-bundles over algebraic curves, in School on algebraic geometry (Trieste, 1999), ICTP Lecture Notes 1, ICTP, Trieste, pp. 1-57, available at http://www.ictp.trieste.it/∼pub off/lectures
57. G. Laumon, L. Moret-Bailly, Champs algébriques, Springer-Verlag, 2000.
58. A. Borel, e.a., Algebraic D-modules, Academic Press, 1987.
59. E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, Journal of AMS 15 (2001) 367-417.
60. D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. Math. 160 (2004) 617-682.
61. G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987) 309-359.
62. n: la premìere construction de Drinfeld, Preprint alg-geom/9511004.
63. G. Laumon, Faisceaux automorphes liés aux séries d'Eisenstein, in Auto-morphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), pp. 227-281, Perspect. Math. 10, Academic Press, 1990.
64. D. Gaitsgory, Automorphic sheaves and Eisenstein series, Ph.D. thesis, 1997.
65. A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002) 287-384.
66. V. Drinfeld, Talk at the DARPA Workshop, November 2003, notes available at http://math.northwestern.edu/langlands/Meetings/03 Chgo/Drinfeld I
67. J. Arthur, Unipotent automorphic representations: conjectures, Asterisque 171-172 (1989) 13-71.
68. J.-P. Serre, Algebraic Groups and Class Fields, Springer, 1988.
69. G. Laumon, Transformation de Fourier généralisé e, Preprint alg-geom/9603004.
70. M. Rothstein, Connections on the total Picard sheaf and the KP hierarchy, Acta Applicandae Mathematicae 42 (1996) 297-308.
71. I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, IHES Publ. Math. 18 (1963) 5-69.
72. T.A. Springer, Reductive groups, in Automorphic forms, representations and L-functions, Proc. Symp. Pure Math. 33, Part 1, pp. 3-27, AMS, 1979.
73. I. Mirkovíc, K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Preprint math.RT/0401222.
74. V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, Preprint alg-geom/9511007.
75. G. Lusztig, Singularities, character formulas, and a q-analogue of weight multiplicities, Astérisque 101 (1983) 208-229.
76. D. Ben-Zvi and E. Frenkel, Geometric Realization of the Segal-Sugawara Construction, in Topology, geometry and quantum field theory, pp. 46-97, London Math. Soc. Lecture Note Ser. 308, Cambridge University Press, 2004 (math.AG/0301206).
77. A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, Advances in Soviet Mathematics 16, Part 1, pp. 1-50, AMS, 1993.
78. A. Polishchuk and M. Rothstein, Fourier transform for D-algebras, Duke Math. J. 109 (2001) 123-146.
79. T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003) 197-229.
80. A. Kapustin, Topological strings on noncommutative manifolds, Preprint hep-th/0310057.
81. N. Hitchin, The self-duality equations on a Riemann surfaces, Proc. London Math. Soc. 55 (1987) 59-126.
82. C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. of AMS 1 (1988) 867-918; Non-abelian Hodge theory, Proceedings of ICM 1990, Kyoto, pp. 198-230, Springer, 1991.
83. G. Felder, K. Gawedzki and A. Kupiainen, Spectra of Wess-Zumino-Witten models with arbitrary simple groups, Comm. Math. Phys. 117 (1988) 127-158.
84. K. Gawedzki and A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B320 (1989) 625-668.
85. K. Gawedzki, Quadrature of conformal field theories, Nucl. Phys. B328 (1989) 733-752.
86. K. Gawedzki, Lectures on conformal field theory, in Quantum fields and strings: a course for mathematicians, Vol. 2, pp. 727-805, AMS, 1999.
87. E. Witten, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. 144 (1992) 189-212.
88. A. Tsuchiya, K. Ueno, and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in Integrable systems in quantum field theory and statistical mechanics, pp. 459-566, Adv. Stud. Pure Math. 19, Academic Press, Boston, 1989.
89. N. Hitchin, Projective connections and geometric quantizations, Comm. Math. Phys. 131 (1990) 347-380.
90. S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Diff. Geom. 33 (1991) 787-902.
91. G. Faltings, Stable G-bundles and projective connections, J. Alg. Geom. 2 (1993) 507-568.
92. A. Beilinson and D. Kazhdan, Flat projective connections, unpublished manuscript.
93. K. Nagatomo and A. Tsuchiya, Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line, Duke Math. J. 128 (2005) 393-471.
94. T. Eguchi and H. Ooguri, Conformal and current algebras on a general Riemann surface, Nucl. Phys. B282 (1987) 308-328.
95. E. Witten, Quantum field theory and the Jones polynomial, Comm.Math.Phys.121 (1989) 351-399.
96. D. Bernard, On the Wess-Zumino-Witten models on Riemann surfaces, Nuclear Phys. B309 (1988) 145-174.
97. G. Felder, The KZB equations on Riemann surfaces, in Symmétries quantiques (Les Houches, 1995), pp. 687-725, North-Holland, 1998 (hep-th/9609153).
98. K. Hori, Global aspects of gauged Wess-Zumino-Witten models, Comm. Math. Phys. 182 (1996) 1-32.
99. A. Beauville and Y. Laszlo, Un lemme de descente, C.R. Acad. Sci. Paris, Sér. I Math. 320 (1995) 335-340.
100. V. Drinfeld and C. Simpson, B-structures on G-bundles and local triviality, Math. Res. Lett. 2 (1995) 823-829.
101. A. Beilinson and V. Schechtman, Determinant bundles and Virasoro algebras, Comm. Math. Phys. 118 (1988) 651-701.
102. A. Beilinson, B. Feigin and B. Mazur, Introduction to algebraic field theory on curves, unpublished manuscript.
103. V. Kac, Infinite-dimensional Lie Algebras, Third Edition. Cambridge University Press, 1990.
104. U. Lindstr̈om and M. Zabzine, Tensionless Strings, WZW Models at Critical Level and Massless Higher Spin Fields, Phys. Lett. B584 (2004) 178-185.
105. I. Bakas and C. Sourdis, On the tensionless limit of gauged WZW models, JHEP 0406 (2004) 049; Aspects of WZW models at critical level, Fortsch. Phys. 53 (2005) 409-417.
106. M. Wakimoto, Fock representations of affine Lie algebra A, Comm. Math. Phys. 104 (1986) 605-609.
107. B. Feigin and E. Frenkel, A family of representations of affine Lie algebras, Russ. Math. Surv. 43 (1988) no. 5, 221-222; Affine Kac-Moody Algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990) 161-189.
108. A. Malikov, V. Schechtman and A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999) 439-473.
109. E. Witten, Two-Dimensional Models With (0, 2) Supersymmetry: Per-turbative Aspects, Preprint hep-th/0504078.
110. N. Nekrasov, Lectures on curved beta-gamma systems, pure spinors, and anomalies, Preprint hep-th/0511008.
111. Vl.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B240 (1984) 312-348.
112. C. Vafa and E. Zaslow, eds., Mirror symmetry, Clay Mathematics Monographs, vol. 1, AMS 2004.
113. A. Zamolodchikov, Integrable field theory from conformal field theory, in Integrable systems in quantum field theory and statistical mechanics, pp. 641-674, Adv. Stud. Pure Math. 19, Academic Press, 1989.
114. n symmetry, Int. J. Mod. Phys. A3 (1988) 507-520.
115. E. Frenkel and D. Gaitsgory, D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J. 125 (2004) 279-327.
116. N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91-114.
117. G. Laumon, Un analogue global du ĉone nilpotent, Duke Math. J. 57 (1988) 647-671.
118. E. Frenkel, Gaudin model and opers, in Infinite Dimensional Algebras and Quantum Integrable Systems, eds. P. Kulish, e.a., Progress in Math. 237, pp. 1-60, Birkḧauser, 2005 (math.QA/0407524).
119. E. Frenkel, Opers on the projective line, flag manifolds and Bethe Ansatz, Mosc. Math. J. 4 (2004) 655-705 (math.QA/0308269).
120. V.G. Drinfeld, Two-dimensional l-adic representations of the Galois group of a global field of characteristic p and automorphic forms on GL(2), J. Sov. Math. 36 (1987) 93-105.
121. J. Heinloth, Coherent sheaves with parabolic structure and construction of Hecke eigensheaves for some ramified local systems, Ann. Inst. Fourier (Grenoble) 54 (2004) 2235-2325.
122. B. Enriquez, B. Feigin and V. Rubtsov, Separation of variables for Gaudin-Calogero systems, Compositio Math. 110 (1998) 1-16.