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## Lectures on the Langlands program and conformal field theory (2005)

Citations: | 71 - 10 self |

### Citations

917 | Quantum field theory and the Jones polynomial - Witten - 1989 |

785 | Étale cohomology - Milne - 1980 |

620 |
Infinite conformal symmetry in two-dimensional quantum field theory
- Belavin, Polyakov, et al.
- 1984
(Show Context)
Citation Context ...ntations will allow us to make the connection to conformal field theory more precise. To explain how this works, let us recall that chiral correlation functions in a (rational) conformal field theory =-=[6]-=- may be interpreted as sections of a holomorphic vector bundle on the moduli space of curves, equipped with a projectively flat connection [7]. The connection comes from the Ward identities expressing... |

608 | Modular elliptic curves and Fermat’s last theorem, The Annals of Mathematics 141
- Wiles
- 1995
(Show Context)
Citation Context ...s a finite group)s24 EDWARD FRENKEL for all but finitely many primes p! This is in fact the statement of the celebrated TaniyamaShimura conjecture that has recently been proved by A. Wiles and others =-=[38]-=-. It implies Fermat’s last theorem, see [35] and references therein. In fact, the modular cusp form fE(q) is what is called a newform (this means that it does not come from a modular form whose level ... |

501 | The self-duality equations on a Riemann surface
- Hitchin
- 1987
(Show Context)
Citation Context ...c varieties. Though they are different as algebraic (or complex) varieties, the underlying real manifolds are diffeomorphic to each other. This is the socalled non-abelian Hodge theory diffeomorphism =-=[80]-=-. In fact, the underlying real manifold is hyperkähler, and the above two incarnations correspond to two particular choices of the complex structure. It is natural to ask what, if anything, this hyper... |

316 |
Faisceaux pervers, Astérisque 100
- Beilinson, Bernstein, et al.
- 1982
(Show Context)
Citation Context ...e formula 29 more precisely, to do that we need to extend this function to the set of all Fq1-points of V , where q1 = q m , m > 0sLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 39 (see =-=[48, 49, 50, 51]-=-). 30 Examples are ℓ-adic local systems on a smooth variety V , placed in cohomological degree equal to − dim V . General perverse sheaves are “glued” from such local systems defined on the strata of ... |

289 |
p-adic Numbers, p-adic Analysis and Zeta-Functions, volume 58 of Graduate Texts in Mathematics
- Koblitz
- 1984
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Citation Context ...1, of elements of (Z/NZ) × such that pN,M(xN) = xM for all pairs N, M such that M divides N. This inverse limit may be described more concretely using the notion of p-adic numbers. Recall (see, e.g., =-=[30]-=-) that if p is a prime, then a p-adic number is an infinite series of the form (1.2) akp k + ak+1p k+1 + ak+2p k+2 + . . . , where each ak is an integer between 0 and p − 1, and we choose k ∈ Z in suc... |

275 |
Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B247
- Knizhnik, Zamolodchikov
- 1984
(Show Context)
Citation Context ...rojectively flat connection have been studied in the framework of Segal’s axioms of conformal field theory [8]. Likewise, if we have a rational conformal field theory with affine Lie algebra symmetry =-=[9]-=-, such as a Wess-Zumino-Witten (WZW) model [10], then conformal blocks give rise to sections of a holomorphic vector bundle with a projectively flat connection on the moduli space of G-bundles on X. T... |

270 | Nonabelian bosonization in two dimensions - Witten - 1984 |

236 |
Quantization of Hitchin’s integrable system and Hecke eigensheaves. Vérsion préliminaire
- Beilinson, Drinfeld
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Citation Context ... theories” depending on L G-opers on X, and we then take the corresponding D-modules of conformal blocks on the moduli space BunG of G-bundles on X. A marvelous result of A. Beilinson and V. Drinfeld =-=[15]-=- is that the D-module corresponding to a L G-oper E is nothing but the sought-after Hecke eigensheaf with “eigenvalue” E! Thus, “conformal field theory” of the critical level k = −h ∨ solves the probl... |

220 |
Infinite-dimensional Lie algebras. Third edition
- Kac
- 1990
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Citation Context ... κ0 denotes a non-degenerate invariant inner product on g. It is unique up to a non-zero scalar, and we normalize it in the standard way so that the square of length of the maximal root is equal to 2 =-=[100]-=-. So, for instance, if g = slN, we have κ0(A, B) =s74 EDWARD FRENKEL TrCN (AB). We will say that a representation M of �g has level k ∈ C if 1 acts on M by multiplication by k. The Hilbert space of th... |

189 |
Conformal field theory on universal family of stable curves with gauge symmetries
- Tsuchiya, Ueno, et al.
- 1989
(Show Context)
Citation Context ...etails). This is the same connection as the one given by formula (7.5). The projectively flat connection on the bundle of conformal blocks of the WZW theory has been constructed by various methods in =-=[85, 86, 87, 88, 89]-=-. For a general conformal field theory the notion of conformal blocks is spelled out in [20], Sect. 9.2. Consider the case of a rational conformal field theory. Then the chiral algebra A has finitely ... |

183 | Geometric Langlands duality and representations of algebraic groups over commutative rings
- Mirković, Vilonen
- 2007
(Show Context)
Citation Context ...sor product given by a convolution functor corresponding to the convolution product (5.1) at the level of functions. The definition of this tensor product, which is due to Beilinson and Drinfeld (see =-=[69]-=-), is reminiscent of the fusion product arising in conformal field theory. It uses a remarkable geometric object, the Beilinson-Drinfeld Grassmannian Gr (2) , which may be defined for any curve X. Thi... |

175 |
Automorphic Forms and Representations
- Bump
- 1998
(Show Context)
Citation Context ...that irreducible representations of (gl2(C), O(2)) fall into the following categories: principal series, discrete series, the limits of the discrete series and finite-dimensional representations (see =-=[36]-=-). Consider the case when π∞ is a representation of the discrete series of (gl2(C), O(2)). In this case ρ = k(k − 2)/4, where k is an integer greater than 1. Then, as an sl2-module, π∞ is the direct s... |

153 |
Conformal algebra and multipoint correlation functions in 2D statistical models
- Dotsenko
- 1984
(Show Context)
Citation Context ... the algebra Fun Proj(D). This way we obtain the sought-after isomorphism z(sl2) � Fun Proj(D). 76 The operators, R V−1/ν(z)dz and R Vν(z)dz were introduced by V. Dotsenko and V. Fateev in their work =-=[108]-=- on the free field realization of the correlation functions in the minimal models, and the terminology “screening operators” originates from that work. The parameters ν and −1/ν correspond to α+ and α... |

151 |
Champs algébriques
- Laumon, Moret-Bailly
- 2000
(Show Context)
Citation Context ...e multiplicative group. What saves the day is the fact that the group of automorphisms is the same for all line bundles. This is not true for bundles of rank higher than 1. The technique developed in =-=[53, 15]-=- allows us to define sheaves on algebraic stacks and to operate with these sheaves in ways that we are accustomed to when working with algebraic varieties. So the moduli stack Bunn will be sufficient ... |

150 |
A proof of Jantzen conjectures
- Beilinson, Bernstein
- 1993
(Show Context)
Citation Context ... will enable us to speculate about a deformation of the non-abelian FourierMukai transform which will make it look more “symmetric”. The construction of this two-parameter deformation is explained in =-=[75]-=- and is in fact applicable in a rather general situation. Here we will only consider the specific case of Bun◦ G and Bun◦L G following [74]. Recall that we have used a non-degenerate invariant inner p... |

150 |
Geometric quantization of Chern-Simons gauge theory
- Axelrod, Pietra, et al.
- 1991
(Show Context)
Citation Context ...etails). This is the same connection as the one given by formula (7.5). The projectively flat connection on the bundle of conformal blocks of the WZW theory has been constructed by various methods in =-=[85, 86, 87, 88, 89]-=-. For a general conformal field theory the notion of conformal blocks is spelled out in [20], Sect. 9.2. Consider the case of a rational conformal field theory. Then the chiral algebra A has finitely ... |

134 |
Affine Kac-Moody algebras at the critical level and Gelfand-Dikii
- Feigin, Frenkel
- 1992
(Show Context)
Citation Context ... center of the chiral algebra has ℓ = rank g generating fields, and turns out to be canonically isomorphic to a classical limit of the W-algebra asociated to the Langlands dual group L G, as shown in =-=[11, 12]-=-. This isomorphism is obtained as a limit of a certain isomorphism of W-algebras that naturally arises in the context of T-duality of free bosonic theories compactified ons6 EDWARD FRENKEL tori. I wil... |

124 | Perverse sheaves on a loop group and langlands duality
- Ginzburg
- 1995
(Show Context)
Citation Context ...result. It has been conjectured by V. Drinfeld and proved in the most general setting by I. Mirković and K. Vilonen [69] (some important results in this direction were obtained earlier by V. Ginzburg =-=[70]-=- and G. Lusztig [71]). Theorem 6. The tensor category P G(O) is equivalent to the tensor category Rep L G. Moreover, the fiber functor from P G(O) to the category of vector spaces, corresponding to th... |

117 |
Stable G-bundles and projective connections
- Faltings
- 1993
(Show Context)
Citation Context ...etails). This is the same connection as the one given by formula (7.5). The projectively flat connection on the bundle of conformal blocks of the WZW theory has been constructed by various methods in =-=[85, 86, 87, 88, 89]-=-. For a general conformal field theory the notion of conformal blocks is spelled out in [20], Sect. 9.2. Consider the case of a rational conformal field theory. Then the chiral algebra A has finitely ... |

116 |
Flat connections and geometric quantization
- Hitchin
- 1990
(Show Context)
Citation Context |

107 |
Integrable Field Theory from Conformal Field Theory, Adv
- Zamolodchikov
- 1989
(Show Context)
Citation Context ...ator (8.22). This operator is marginal (has dimension (1, 1)) with respect to the stress tensor Tν(z) given by formula (8.20) and its anti-chiral counterpart. According to the general prescription of =-=[110]-=-, the chiral algebra of the deformed theory (in the first order of perturbation theory) is the kernel of the operator � Vν(z)dz on the chiral algebra of the free theory, which for irrational ν 2 is π0... |

106 |
Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de
- Laumon
- 1987
(Show Context)
Citation Context ...s of Frx on the stalk cohomologies of K at x: fK(x) = � (−1) i Tr(Frx, H i x(K)). i The map K → fK intertwines the natural operations on complexes of sheaves with natural operations on functions (see =-=[47]-=-, Sect. 1.2). For example, pull-back of a sheaf corresponds to the pull-back of a function, and push-forward of a sheaf with compact support corresponds to the fiberwise integration of a function. 28 ... |

98 | Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality
- Kapustin
(Show Context)
Citation Context ... dimensions and the analysis of what this reduction does to “D-branes”. In particular, Witten argued that the t’Hooft operators of the four-dimensional gauge theory recently introduced by A. Kapustin =-=[4]-=- become, after the dimensional reduction, the Hecke operators that are essential ingredients of the Langlands correspondence. Thus, a t’Hooft “eigenbrane” of the gauge theory becomes after the reducti... |

95 | Topological Strings on Noncommutative Manifolds
- Kapustin
(Show Context)
Citation Context ...ty in four-dimensional gauge theories and the Langlands correspondence that was mentioned in the Introduction. Examples of “non-commutative” D-branes related to D-modules have also been considered in =-=[79]-=-, and in fact they are 52 These dual Hitchin fibrations (restricted to the open subsets of stable Higgs pairs in T ∗ Bun ◦ G and T ∗ Bun ◦ L G ) have been shown by T. Hausel and M. Thaddeus [77] to be... |

94 | Chiral de Rham complex
- Malikov, Schechtman, et al.
- 1999
(Show Context)
Citation Context ...he Lie algebra sl2((t)) on the loop space of CP 1 . This is discussed in detail in [20], Ch. 11-12. Its96 EDWARD FRENKEL is closely related to the sheaf of chiral differential operators introduced in =-=[105]-=- and [17], Sect. 2.9 (this is explained in [20], Sect. 18.5.7). 75 We would like to use this free field realization at the critical level k = −2 (i.e., ν = 0). Unfortunately, if we set k = −2 in the a... |

93 |
The definition of conformal field theory. In Topology, geometry and quantum field theory, volume 308
- Segal
- 2004
(Show Context)
Citation Context ...ess tensor, which generates the Virasoro algebra symmetry of the theory. These bundles with projectively flat connection have been studied in the framework of Segal’s axioms of conformal field theory =-=[8]-=-. Likewise, if we have a rational conformal field theory with affine Lie algebra symmetry [9], such as a Wess-Zumino-Witten (WZW) model [10], then conformal blocks give rise to sections of a holomorph... |

92 | The analytic geometry of two dimensional conformal field theory, Nucl.Phys. B281
- Freidan, Shenker
- 1987
(Show Context)
Citation Context ...relation functions in a (rational) conformal field theory [6] may be interpreted as sections of a holomorphic vector bundle on the moduli space of curves, equipped with a projectively flat connection =-=[7]-=-. The connection comes from the Ward identities expressing the variation of correlation functions under deformations of the complex structure on the underlying Riemann surface via the insertion in the... |

91 |
Chtoucas de Drinfeld et correspondance de Langlands
- Lafforgue
(Show Context)
Citation Context ...e easier to handle than the correspondence in the number field case. In fact, it is now a theorem! First, V. Drinfeld [39, 40] proved it in the 80’s in the case of GL2, and more recently L. Lafforgue =-=[41]-=- proved it for GLn with an arbitrary n. In this section we explain the analogy between number fields and function fields and formulate the Langlands correspondence for function fields. 2.1. Function f... |

89 |
Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects”, arXiv:hep-th/0504078, Vasily Pestun, Edward Witten: ”The Hitchin functionals and the topological B-model at one loop
- Witten
(Show Context)
Citation Context ...ν φ(z) � 1 � ibn : = T−1/ν exp ν n n<0 z−n � � 1 � ibn exp ν n n>0 z−n � acting from π0 to π −1/ν (here T −1/ν denotes the operator sending |0〉 to | − 1/ν〉 and commuting with bn, n �= 0). 75 see also =-=[106, 107]-=- for a recent discussion of the curved βγ systems from the point of view of sigma models n∈ZsLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 97 So we consider the following screening oper... |

83 | Unipotent automorphic representations: conjectures, Astérisque
- Arthur
- 1989
(Show Context)
Citation Context ...plex, indicates that something funny is going on with the trivial representation. In fact, it has to do with the so-called “Arthur’s SL2” part of the parameter of a general automorphic representation =-=[63]-=-. The precise meaning of this is beyond the scope of the present article, but the idea is as follows. Arthur has conjectured that if we want to consider unitary automorphic representations of GLn(A) t... |

76 | Problems in the theory of automorphic forms
- Langlands
- 1970
(Show Context)
Citation Context ... both the Langlands correspondence and the dualities in physics have emerged as some sort of non-abelian Fourier transforms. Moreover, the so-called Langlands dual group introduced by R. Langlands in =-=[1]-=- that is essential in the formulation of the Langlands correspondence also plays a prominent role in the study of S-dualities in physics and was in fact also introduced by the physicists P. Goddard, J... |

73 |
Coset Construction from Functional Integrals, Nucl.Phys. B320
- Gawedzki, Kupiainen
- 1989
(Show Context)
Citation Context ...1) . . . Φ(xn)〉 with varying complex structure on X and varying points is the square |φ(X, (xi))| 2 of length of φ(X, (xi)) with respect to a “natural” hermitean inner product which is constructed in =-=[82, 83]-=- (see also [84]). 54 There is a unique unitary connection compatible with the holomorphic structure on ∆g(Lλ1 , . . . , Lλn ) and this hermitean metric. This connection is projectively flat. 55 It fol... |

73 | On Holomorphic factorization of WZW and coset models - Witten - 1992 |

68 | Automorphic L–functions - Borel - 1979 |

67 |
Theory of spherical functions on reductive algebraic groups over p-adic
- Satake
- 1963
(Show Context)
Citation Context ...→ H(T (F ), T (O)) which can be shown to be injective. Unfortunately, this restriction map is not compatible with the convolution product, and hence is not an algebra homomorphism. However, I. Satake =-=[67]-=- has constructed a different map H(G(F ), G(O)) → H(T (F ), T (O)) � C[ ˇ P ] which is an algebra homomorphism. Let N be a unipotent subgroup of G. For example, if G = GLn we may take as N the group o... |

66 | Affine algebras, Langlands duality and Bethe ansatz
- Frenkel
- 1994
(Show Context)
Citation Context ...mpossible to cover all essential aspects of the Langlands Program. To get a broader picture, I recommend the interested reader to consult the informative reviews [21]–[27]. My earlier review articles =-=[28, 29]-=- contain some of the material of the present notes in a more concise form as well as additional topics not covered here.s8 EDWARD FRENKEL Acknowledgments. These notes grew out of the lectures that I g... |

65 | Gaitsgory: ”Local geometric Langlands correspondence and affine Kac-Moody algebras”, arXiv:math.RT/0508382
- Frenkel, Dennis
(Show Context)
Citation Context ...systems. In general, we expect that vector bundles on curves with connections that have singularities, regular or irregular, also play an important role in the geometric Langlands correspondence, see =-=[44]-=-; we discuss this in Sect. 9.8 below. To summarize, we believe that we have found the right substitute for the (unramified) n-dimensional Galois representations in the case of a compact complex curve ... |

61 |
Etale Cohomology and The Weil Conjecture
- Freitag, Kiehl
- 1988
(Show Context)
Citation Context ...ves F|Vi are locally constant. 26the passage to the formal discs is justified by an analogue of the “strong approximation theorem” that was mentioned in Sect. 1.6 27The precise definition (see, e.g., =-=[45, 46]-=-) is more subtle: a typical example is a compatible system of locally constant Z/ℓ n Z-sheaves for n > 0s38 EDWARD FRENKEL The important property of the notion of an ℓ-adic sheaf F on V is that for an... |

58 | Geometric Eisenstein Series
- Braverman, Gaitsgory
- 2002
(Show Context)
Citation Context ... more complicated. For example, Hecke eigensheaves corresponding to local systems that are direct sums of n rank 1 local systems – the so-called geometric Eisenstein series – have been constructed in =-=[59, 60, 61]-=-. The best case scenario is when these rank 1 local systems are pairwise non-isomorphic. The corresponding Hecke eigensheaf is a direct sum of infinitely many irreducible perverse sheaves on Bunn, lab... |

57 |
Algebraic D-modules
- Borel
- 1987
(Show Context)
Citation Context ...that we consideredsLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 43 above) and the category of perverse sheaves on V . This equivalence is called the RiemannHilbert correspondence (see =-=[49, 50, 51, 54]-=-). 34 Therefore we may replace perverse sheaves on smooth algebraic varieties (or algebraic stacks, see [15]) over C by holonomic D-modules with regular singularities. Under this equivalence of catego... |

55 | M.: Mirror symmetry, Langlands duality, and the Hitchin system
- Hausel, Thaddeus
- 2003
(Show Context)
Citation Context ...ed in [79], and in fact they are 52 These dual Hitchin fibrations (restricted to the open subsets of stable Higgs pairs in T ∗ Bun ◦ G and T ∗ Bun ◦ L G ) have been shown by T. Hausel and M. Thaddeus =-=[77]-=- to be an example of the Strominger-YauZaslow duality.s72 EDWARD FRENKEL closely related to the deformed Fourier-Mukai equivalence in the abelian case that we mentioned above. We close this section wi... |

54 |
Determinant bundles and Virasoro algebras
- Beilinson, Schechtman
- 1988
(Show Context)
Citation Context ...tly the same as the one described in Sect. 7.3 (see formula (7.9)). The idea that the sheaves of coinvariants arising in conformal field theory may be obtained via a localization functor goes back to =-=[98, 99]-=-. For integer values of k the sheaf D ′ k is the sheaf of differential operators on a line bundle over BunG that is constructed in the following way. Note that the quotient G((t))/G[[t]] appearing in ... |

52 | Lectures on curved beta-gamma systems, pure spinors, and anomalies
- Nekrasov
(Show Context)
Citation Context ...ν φ(z) � 1 � ibn : = T−1/ν exp ν n n<0 z−n � � 1 � ibn exp ν n n>0 z−n � acting from π0 to π −1/ν (here T −1/ν denotes the operator sending |0〉 to | − 1/ν〉 and commuting with bn, n �= 0). 75 see also =-=[106, 107]-=- for a recent discussion of the curved βγ systems from the point of view of sigma models n∈ZsLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 97 So we consider the following screening oper... |

50 |
Vertex Algebras and Algebraic
- Frenkel, Ben-Zvi
- 2004
(Show Context)
Citation Context ...eigenvalue” E. I will not discuss this construction in detail in this survey referring the reader instead to [17], Sect. 4.9, and [18] where the abelian case is considered and the reviews in [19] and =-=[20]-=-, Sect. 20.5. These two examples show that the methods of two-dimensional conformal field theory are powerful and flexible enough to give us important examples of the geometric Langlands correspondenc... |

49 |
Sur les représentations ℓ-adiques associées aux formes modulaires de
- Carayol
- 1986
(Show Context)
Citation Context ...er, the Galois representation σE and the automorphic representation π are unramified at exactly the same primes (namely, those which do not divide NE), and formula (1.13) holds at all of those primes =-=[37]-=-. This way one obtains a bijection between isogeny classes of elliptic curves defined over Q with conductor NE and newforms of weight 2 and level NE with integer Fourier coefficients. One obtains simi... |

48 | K.: On the geometric Langlands conjecture
- Frenkel, Gaitsgory, et al.
(Show Context)
Citation Context ...31 for n = 1, the Picard variety of X may be viewed as the moduli space of line bundles 32 actually, one can show that each cuspidal automorphic function vanishes on a subset of unstable bundles (see =-=[55]-=-, Lemma 6.11), and this opens up the possibility that somehow moduli spaces of semistable bundles would suffices40 EDWARD FRENKEL Thus, we have now identified the geometric objects which should replac... |

46 |
An elementary introduction to the Langlands program
- Gelbart
- 1984
(Show Context)
Citation Context ...sented as the sum of squares of two integers. 4 The above formula now tells us that this representation is possible if and only if p ≡ 1 mod 4, which is the statement of one of Fermat’s theorems (see =-=[25]-=- for more details). For example, 5 can be written as 1 2 + 2 2 , but 7 cannot be written as the sum of squares of two integers. A statement like this is usually referred to as a reciprocity law, as it... |

46 |
Spectra of Wess-Zumino-Witten models with arbitrary simple groups
- Felder, Gawedzki, et al.
- 1988
(Show Context)
Citation Context ...(A, B) =s74 EDWARD FRENKEL TrCN (AB). We will say that a representation M of �g has level k ∈ C if 1 acts on M by multiplication by k. The Hilbert space of the WZW theory of level k is the direct sum =-=[81]-=- Hk = � Lλ ⊗ Lλ, λ∈ b P k + Here Lλ and Lλ are two copies of the irreducible integrable representation of the corresponding affine Lie algebra �g of level k and highest weight λ, and the set � P k + l... |

44 |
Reductive groups, in: “Automorphic forms, representations and Lfunctions
- Springer
- 1977
(Show Context)
Citation Context ...× → k × , which are classified by an integer (“winding number”), and 〈µ, ˇ λ〉 is equal to this number. The sets P and ˇ P contain subsets ∆ and ∆ ∨ of roots and coroots of G, respectively (see, e.g., =-=[68]-=- for more details). Let now X be a smooth projective curve over Fq and let us pick a point x ∈ X. Assume for simplicity that its residue field is Fq. To simplify notation we will omit the index x from... |

43 |
Un lemme de descente
- Beauville, Laszlo
- 1995
(Show Context)
Citation Context ...es on X is in bijection with the double quotient Gout\G((t))/G[[t]]. This is a “one-point” version of the Weil type adèlic uniformization given in Lemma 2. Furthermore, it follows from the results of =-=[96, 97]-=- that this identification is not only an isomorphism of the sets of points, but we actually have an isomorphism of algebraic stacks (7.8) BunG � Gout\G((t))/G[[t]], where Gout is the group of algebrai... |

42 |
Conformal Field Theory, in Quantum Fields and Strings: A course for Mathematicians Vol
- Gawedzki
- 1999
(Show Context)
Citation Context ...1) . . . Φ(xn)〉 with varying complex structure on X and varying points is the square |φ(X, (xi))| 2 of length of φ(X, (xi)) with respect to a “natural” hermitean inner product which is constructed in =-=[82, 83]-=- (see also [84]). 54 There is a unique unitary connection compatible with the holomorphic structure on ∆g(Lλ1 , . . . , Lλn ) and this hermitean metric. This connection is projectively flat. 55 It fol... |

42 |
A family of representations of affine Lie algebras
- Feigin, Frenkel
- 1988
(Show Context)
Citation Context ... with values in the dual space to the Cartan subalgebra h ⊂ g. For that we use the free field realization of �g, which was constructed by M. Wakimoto [103] for g = sl2 and by B. Feigin and the author =-=[104]-=- for an arbitrary simple Lie algebra g. We first recall the free field realization in the case of sl2. In his case we need a chiral bosonic βγ system generated by the fields β(z), γ(z) and a free chir... |

41 |
Correspondance de Langlands géométrique pour les corps de fonctions
- Laumon
- 1987
(Show Context)
Citation Context ...ic setting is a map in one direction: from local systems to Hecke eigensheaves. We are now naturally led to the geometric Langlands conjecture for GLn, whose formulation is due to Drinfeld and Laumon =-=[57]-=-. This statement makes sense when X is over Fq or over C, and it is now a theorem in both cases. Note that Bunn is a disjoint union of connected components Bun d n corresponding to vector bundles of d... |

41 |
On the Wess-Zumino-Witten models on Riemann surfaces, Nuclear Phys. B309
- Bernard
- 1988
(Show Context)
Citation Context ...7.3. Sheaves of conformal blocks on BunG. The construction of twisted D-modules on BunG is completely analogous to the corresponding construction on Mg,n outlined above. We now briefly recall it (see =-=[9, 91, 93, 94, 95, 83, 20]-=-). 56the spaces CA(M1, . . . , Mn) give rise to what is known as the modular functor of conformal field theory [8] 57it is a twisted D-module because the connection is not flat, but only projectively ... |

38 |
Quantum field theory, Grassmannians, and algebraic curves
- Witten
- 1988
(Show Context)
Citation Context ...rmal field theory to construct Hecke eigensheaves. Actually, the analogy between conformal field theory and the theory of automorphic representations was already observed a long time ago by E. Witten =-=[5]-=-. However, at that time the geometric Langlands correspondence had not yet been developed. As we will see, the geometric reformulation of the classical theory of automorphic representations will allow... |

33 | On a vanishing conjecture appearing in the geometric Langlands correspondence
- Gaitsgory
- 2004
(Show Context)
Citation Context ...for GL1 (we recall it in the next section) and by Drinfeld in the case of GL2 [39] (see [28], Sect. 6, for a review). These works motivated the conjecture in the case of GLn, which has been proved in =-=[55, 56]-=- (these works weresLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 47 also influenced by [57, 58]). In the case when X is over C we can replace “perverse sheaf” in the statement of Theore... |

33 |
Introduction to Algebraic Field Theory on Curves, unpublished manuscript
- Beilinson, Feigin, et al.
(Show Context)
Citation Context ...tly the same as the one described in Sect. 7.3 (see formula (7.9)). The idea that the sheaves of coinvariants arising in conformal field theory may be obtained via a localization functor goes back to =-=[98, 99]-=-. For integer values of k the sheaf D ′ k is the sheaf of differential operators on a line bundle over BunG that is constructed in the following way. Note that the quotient G((t))/G[[t]] appearing in ... |

30 |
Faisceaux automorphes liés aux séries d’Eisenstein.” Automorphic forms, Shimura varieties
- Laumon
- 1990
(Show Context)
Citation Context ... more complicated. For example, Hecke eigensheaves corresponding to local systems that are direct sums of n rank 1 local systems – the so-called geometric Eisenstein series – have been constructed in =-=[59, 60, 61]-=-. The best case scenario is when these rank 1 local systems are pairwise non-isomorphic. The corresponding Hecke eigensheaf is a direct sum of infinitely many irreducible perverse sheaves on Bunn, lab... |

30 |
character formulas, and a q–analogue of weight multiplicities, Astérisque 101
- Lusztig, Singularities
- 1983
(Show Context)
Citation Context ...conjectured by V. Drinfeld and proved in the most general setting by I. Mirković and K. Vilonen [69] (some important results in this direction were obtained earlier by V. Ginzburg [70] and G. Lusztig =-=[71]-=-). Theorem 6. The tensor category P G(O) is equivalent to the tensor category Rep L G. Moreover, the fiber functor from P G(O) to the category of vector spaces, corresponding to the forgetful functor ... |

28 |
Conformal and current algebras on a general Riemann surface, Nucl. Phys. B282
- Eguchi, Ooguri
- 1987
(Show Context)
Citation Context ...7.3. Sheaves of conformal blocks on BunG. The construction of twisted D-modules on BunG is completely analogous to the corresponding construction on Mg,n outlined above. We now briefly recall it (see =-=[9, 91, 93, 94, 95, 83, 20]-=-). 56the spaces CA(M1, . . . , Mn) give rise to what is known as the modular functor of conformal field theory [8] 57it is a twisted D-module because the connection is not flat, but only projectively ... |

27 |
Conformal field theories associated to regular chiral vertex operator algebras, I: Theories over the projective line
- Nagatomo, Tsuchiya
(Show Context)
Citation Context ...d above in verifying that the connection defined by formula (7.6) is well-defined. In a rational conformal field theory the spaces CA(M1, . . . , Mn) are expected to be finite-dimensional (see, e.g., =-=[90]-=-), and as we vary (X, (xi)), they glue into a vector bundle CA(M1, . . . , Mn) on the moduli space Mg,n. It is equipped with a projectively flat connection defined as above (see [20] for more details)... |

26 |
Lie algebras and KdV type equations
- Drinfeld, Sokolov
- 1985
(Show Context)
Citation Context ...nic sigma models. The classical W-algebra of L G is the algebra of functions on the space of gauge equivalence classes of connections on the circle introduced originally by V. Drinfeld and V. Sokolov =-=[13]-=- in their study of the generalized KdV hierarchies. The Drinfeld-Sokolov construction has been recast in a more geometric way by A. Beilinson and V. Drinfeld, who called these gauge equivalence classe... |

25 |
Transformation de Fourier generalisée, Preprint alg-geom/9603004
- Laumon
(Show Context)
Citation Context ...e ixy , x ∈ R, also form a good “basis” for functions. In other words, other functions can be written as Fourier integrals. An analogous thing happens in our situation. It has been shown by G. Laumon =-=[65]-=- and M. Rothstein [66] that the functors F and G give rise to mutually inverse (up to a sign and cohomological shift) equivalences of derived categories (4.6) derived category of O-modules on Loc1 ←→ ... |

22 |
Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2
- Drinfeld
- 1983
(Show Context)
Citation Context ...niform. The function field version of the Langlands correspondence turned out to be easier to handle than the correspondence in the number field case. In fact, it is now a theorem! First, V. Drinfeld =-=[39, 40]-=- proved it in the 80’s in the case of GL2, and more recently L. Lafforgue [41] proved it for GLn with an arbitrary n. In this section we explain the analogy between number fields and function fields a... |

22 | Lectures on moduli of principal G–bundles over algebraic curves - Sorger - 1999 |

20 |
Analogies between the Langlands correspondence and topological quantum field theory, in functional analysis on the eve of the 21st century, Birkhauser
- Kapranov
- 1995
(Show Context)
Citation Context ...al field theory. The hope is, of course, that there is a similar connection between a higher dimensional Langlands correspondence and quantum field theories in dimensions greater than two (see, e.g., =-=[43]-=- for a discussion of this analogy). 2.2. Galois representations. Let X be a smooth connected projective curve over k = Fq and F = k(X) the field of rational functions on X. Consider the Galois group G... |

19 | Global aspects of gauged Wess-Zumino-Witten models
- Hori
- 1996
(Show Context)
Citation Context ...7.3. Sheaves of conformal blocks on BunG. The construction of twisted D-modules on BunG is completely analogous to the corresponding construction on Mg,n outlined above. We now briefly recall it (see =-=[9, 91, 93, 94, 95, 83, 20]-=-). 56the spaces CA(M1, . . . , Mn) give rise to what is known as the modular functor of conformal field theory [8] 57it is a twisted D-module because the connection is not flat, but only projectively ... |

17 |
B-structures on G-bundles and local
- Drinfeld, Simpson
- 1995
(Show Context)
Citation Context ...es on X is in bijection with the double quotient Gout\G((t))/G[[t]]. This is a “one-point” version of the Weil type adèlic uniformization given in Lemma 2. Furthermore, it follows from the results of =-=[96, 97]-=- that this identification is not only an isomorphism of the sets of points, but we actually have an isomorphism of algebraic stacks (7.8) BunG � Gout\G((t))/G[[t]], where Gout is the group of algebrai... |

16 | Fourier transform for D-algebras
- Polishchuk, Rothstein
(Show Context)
Citation Context ...Bun ◦ LG ←→ derived category of D k,λ -modules on Bun ◦ G k = ˇ k −1 In fact, in the abelian case, where the Fourier-Mukai transform exists, such a deformation also exists and has been constructed in =-=[76]-=-. While the original Langlands correspondence (6.3) looks quite asymmetric: it relates flat L G-bundles on X and D-modules on Bun ◦ G , the Fourier-Mukai perspective allows us to think of it as a spec... |

15 | Notes on 2D conformal field theory and string theory, in Quantum fields and strings: a course for mathematicians
- Gaitsgory
- 1999
(Show Context)
Citation Context ...ith the “eigenvalue” E. I will not discuss this construction in detail in this survey referring the reader instead to [17], Sect. 4.9, and [18] where the abelian case is considered and the reviews in =-=[19]-=- and [20], Sect. 20.5. These two examples show that the methods of two-dimensional conformal field theory are powerful and flexible enough to give us important examples of the geometric Langlands corr... |

15 | Geometric Realization of the Segal-Sugawara Construction, in Topology, geometry and quantum field theory
- Ben-Zvi, Frenkel
- 2004
(Show Context)
Citation Context ...nstruction of this two-parameter deformation is explained in [75] and is in fact applicable in a rather general situation. Here we will only consider the specific case of Bun◦ G and Bun◦L G following =-=[74]-=-. Recall that we have used a non-degenerate invariant inner product ˇκ0 on Lg in order to identify Lg with Lg∗ . This inner product automatically induces a non-degenerate invariant inner product κ0 on... |

14 | Galois representations and modular forms
- Ribet
- 1995
(Show Context)
Citation Context ...nsider the two-dimensional representations of Gal(Q/Q) arising from elliptic curves defined over Q and look at what the Langlands correspondence means for such representations. We refer the reader to =-=[32, 33, 34, 35]-=- for more details on this subject. Roughly speaking, cuspidal automorphic representations of GL2(A) are those irreducible representations of this group which occur in the discrete spectrum of a certai... |

14 |
Faisceaux automorphes pour GLn: la première construction de Drinfeld, Preprint alg-geom/9511004
- Laumon
- 1995
(Show Context)
Citation Context ...eview). These works motivated the conjecture in the case of GLn, which has been proved in [55, 56] (these works weresLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 47 also influenced by =-=[57, 58]-=-). In the case when X is over C we can replace “perverse sheaf” in the statement of Theorem 3 by “D-module”. 36 The reader may be wondering what has become of the cuspidality condition, which was impo... |

14 |
Algebraic Groups and Class
- Serre
- 1988
(Show Context)
Citation Context ... if ni[xi] = (g), i where g is an arbitrary rational function on X. This identity is a non-trivial reciprocity law which has been proved in the abelian class field theory, by Lang and Rosenlicht (see =-=[64]-=-). It is instructive to contrast this to Deligne’s geometric proof reproduced above. When we replace functions by sheaves we can use additional information which is “invisible” at the level of functio... |

14 |
Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on
- Arinkin
- 2001
(Show Context)
Citation Context ...9.5). Additional evidence for the existence of the non-abelian Fourier transform comes from certain orthogonality relations between natural sheaves on both moduli spaces that have been established in =-=[72, 73]-=-. In the next section we speculate about a possible two-parameter deformation of the naive non-abelian Fourier-Mukai transform, loosely viewed as an equivalence between the derived categories of D-mod... |

13 |
Wakimoto modules, opers and the center at the critical
- Frenkel
- 2005
(Show Context)
Citation Context ...M AND CONFORMAL FIELD THEORY 95 dual Lie algebra is closely related to the T-duality in free bosonic conformal field theory compactified on a torus. 8.5. Free field realization. The idea of the proof =-=[11, 12]-=- of Theorem 9 is to realize the center z(g) inside the Poisson version of the chiral algebra of free bosonic field with values in the dual space to the Cartan subalgebra h ⊂ g. For that we use the fre... |

12 | Introduction to the Langlands program, Representation Theory and Automorphic Forms; An - Knapp - 1996 |

11 |
Langlands parameters for Heisenberg modules, Preprint math.QA/0204020
- Beilinson
(Show Context)
Citation Context ...ing sheaf of conformal blocks on BunG is a Hecke eigensheaf with the “eigenvalue” E. I will not discuss this construction in detail in this survey referring the reader instead to [17], Sect. 4.9, and =-=[18]-=- where the abelian case is considered and the reviews in [19] and [20], Sect. 20.5. These two examples show that the methods of two-dimensional conformal field theory are powerful and flexible enough ... |

10 |
Connections on the total Picard sheaf and the KP hierarchy, Acta Applicandae Mathematicae 42
- Rothstein
- 1996
(Show Context)
Citation Context ...rm a good “basis” for functions. In other words, other functions can be written as Fourier integrals. An analogous thing happens in our situation. It has been shown by G. Laumon [65] and M. Rothstein =-=[66]-=- that the functors F and G give rise to mutually inverse (up to a sign and cohomological shift) equivalences of derived categories (4.6) derived category of O-modules on Loc1 ←→ SE ←→ Aut 0 E p2 ↘ Jac... |

10 | Moduli of connections with a small parameter on a curve, Preprint math.AG/0409373
- Arinkin
(Show Context)
Citation Context ...he fibers of the Hitchin fibration. However, the singular fibers complicate matters (not to mention the “empty fibers”), and as far as we know, such an equivalence has not yet been established. 52 In =-=[78]-=- some results concerning this equivalence in the formal neighborhood of the point λ = 0 are obtained. 6.4. D-modules are D-branes? Derived categories of coherent O-modules on algebraic varieties have ... |

10 | The KZB Equations on Riemann Surfaces
- Felder
- 1998
(Show Context)
Citation Context |

10 |
Fock representations of affine Lie algebra A (1
- Wakimoto
(Show Context)
Citation Context ...ersion of the chiral algebra of free bosonic field with values in the dual space to the Cartan subalgebra h ⊂ g. For that we use the free field realization of �g, which was constructed by M. Wakimoto =-=[103]-=- for g = sl2 and by B. Feigin and the author [104] for an arbitrary simple Lie algebra g. We first recall the free field realization in the case of sl2. In his case we need a chiral bosonic βγ system ... |

9 |
Abelian coverings of arithmetic schemes
- Parshin
- 1978
(Show Context)
Citation Context ... field of functions on an algebraic variety of dimension greater than one, and finding such a formulation is a very important open problem. There is an analogue of the abelian class field theory (see =-=[42]-=-), but not much is known beyond that. In Part III of this paper we will argue that the Langlands correspondence for the function fields of curves – transported to the realm of complex curves – is clos... |

8 |
Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands, Séminaire Bourbaki
- Laumon
(Show Context)
Citation Context ... is vacuous in this case. Hence throughout this section we will assume that the genus of X is positive. 4.1. Deligne’s proof. We present here Deligne’s proof of the n = 1 case of Theorem 3, following =-=[57, 59, 26]-=-; it works when X is over Fq and over C, but when X is over C there are additional simplifications which we will discuss below. For n = 1 the moduli stack Bunn is the Picard variety Pic of X classifyi... |

8 |
From modular forms to automorphic representations
- Kudla
(Show Context)
Citation Context ...nsider the two-dimensional representations of Gal(Q/Q) arising from elliptic curves defined over Q and look at what the Langlands correspondence means for such representations. We refer the reader to =-=[32, 33, 34, 35]-=- for more details on this subject. Roughly speaking, cuspidal automorphic representations of GL2(A) are those irreducible representations of this group which occur in the discrete spectrum of a certai... |

6 | Automorphic representations and number theory in Seminar on Harmonic Analysis (Montreal - Arthur - 1980 |

6 | Automorphic sheaves and Eisenstein series - Gaitsgory - 1997 |

6 |
Flat projective connection, unpublished manuscript
- Beilinson, Kazhdan
- 1991
(Show Context)
Citation Context |

5 | A motivated introduction to the Langlands program - Murty - 1991 |

4 |
conjecture for GL(2) over function field, Proc. of Int
- Drinfeld, Langlands
- 1978
(Show Context)
Citation Context ...niform. The function field version of the Langlands correspondence turned out to be easier to handle than the correspondence in the number field case. In fact, it is now a theorem! First, V. Drinfeld =-=[39, 40]-=- proved it in the 80’s in the case of GL2, and more recently L. Lafforgue [41] proved it for GLn with an arbitrary n. In this section we explain the analogy between number fields and function fields a... |

3 |
Algebraic theory of D–modules, available at http://www.math.uchicago.edu/∼arinkin/ langlands
- Bernstein
(Show Context)
Citation Context ...e formula 29 more precisely, to do that we need to extend this function to the set of all Fq1-points of V , where q1 = q m , m > 0sLECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 39 (see =-=[48, 49, 50, 51]-=-). 30 Examples are ℓ-adic local systems on a smooth variety V , placed in cohomological degree equal to − dim V . General perverse sheaves are “glued” from such local systems defined on the strata of ... |

3 |
On the tensionless limit of gauged WZW models, JHEP 0406 (2004) 049; Aspects of WZW models at critical level, Fortsch. Phys. 53
- Bakas, Sourdis
- 2005
(Show Context)
Citation Context ...rs acting on L ⊗k . Then the adjoint differential operators will be acting on the Serre dual line bundle 69 affine algebras at the critical level have also been considered recently by physicists, see =-=[101, 102]-=-s86 EDWARD FRENKEL K ⊗ L⊗(−k) , where K is the canonical line bundle. It is natural to guess that k should be such that the two line bundles are actually isomorphic to each other. But one can show tha... |

2 | Global geometrised Rankin-Selberg method for GL(n), Preprint math.AG/0108208
- Lysenko
(Show Context)
Citation Context ...9.5). Additional evidence for the existence of the non-abelian Fourier transform comes from certain orthogonality relations between natural sheaves on both moduli spaces that have been established in =-=[72, 73]-=-. In the next section we speculate about a possible two-parameter deformation of the naive non-abelian Fourier-Mukai transform, loosely viewed as an equivalence between the derived categories of D-mod... |

1 |
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
- Witten, Talk
(Show Context)
Citation Context ...le in the study of S-dualities in physics and was in fact also introduced by the physicists P. Goddard, J. Nuyts and D. Olive in the framework of four-dimensional gauge theory [2]. In recent lectures =-=[3]-=- E. Witten outlined a possible scenario of how the two dualities – the Langlands duality and the S-duality – could be related to each other. It is based on a dimensional reduction of a four-dimensiona... |

1 | Goldstone fields in two dimensions with mutivalued actions - Polyakov, Wiegmann - 1984 |

1 |
Shalit, L–functions of elliptic curves and modular forms
- de
(Show Context)
Citation Context ...nsider the two-dimensional representations of Gal(Q/Q) arising from elliptic curves defined over Q and look at what the Langlands correspondence means for such representations. We refer the reader to =-=[32, 33, 34, 35]-=- for more details on this subject. Roughly speaking, cuspidal automorphic representations of GL2(A) are those irreducible representations of this group which occur in the discrete spectrum of a certai... |

1 |
Galois representations, available at http://abel.math.harvard.edu/∼rtaylor
- Taylor
(Show Context)
Citation Context |

1 |
Talk at the DARPA Workshop at
- Drinfeld
- 2003
(Show Context)
Citation Context ... category AutE0 . Some interesting Hecke eigensheaves are unbounded complexes in this case, and a precise definition of the corresponding category that would include such complexes is an open problem =-=[62]-=-. Note that for X = CP1 the trivial local system is the only local system. The corresponding category AutE0 can probably be described rather explicitly. Some results in this direction are presented in... |