LECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY EDWARD FRENKEL Contents Introduction 3 Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function fields 24 2.1. Function fields 24 2.2. Galois representations 27 2.3. Automorphic representations 28 2.4. The Langlands correspondence 31 Part II. The geometric Langlands Program 33 3. The geometric Langlands conjecture 33 3.1. Galois representations as local systems 33 3.2. Ad`elesand vector bundles 35 3.3. From functions to sheaves 37 3.4. From perverse sheaves to D-modules 40 3.5. Example: a D-module on the line 41 3.6. More on D-modules 42 3.7. Hecke correspondences 43 3.8. Hecke eigensheaves and the geometric Langlands conjecture 45 4. Geometric abelian class field theory 48 4.1. Deligne's proof 49 4.2. Functions vs. sheaves 50 Date: December 2005. Based on the lectures given by the author at the Les Houches School \Number Theory and Physics" in March of 2003 and at the DARPA Workshop \Langlands Program and Physics" at the Institute for Advanced Study in March of 2004. To appear in Proceedings of the Les Houches School. Partially supported by the DARPA grant HR0011-04-1-0031 and by the NSF grant DMS-0303529. 1 2 EDWARD FRENKEL 4.3. Another take for curves over C 51 4.4. Connection to the Fourier-Mukai transform 52 4.5. A special case of the Fourier-Mukai transform 54 5. From GLn to other reductive groups 56 5.1. The spherical Hecke algebra for an arbitrary reductive group 56 5.2. Satake isomorphism 57 5.3. The Langlands correspondence for an arbitrary reductive group 59 5.4. Categorification of the spherical Hecke algebra 60 5.5. Example: the affine Grassmannian of P GL2 62 5.6. The geometric Satake equivalence 63 6. The geometric Langlands conjecture over C 64 6.1. Hecke eigensheaves 64 6.2. Non-abelian Fourier-Mukai transform? 66 6.3. A two-parameter deformation 68 6.4. D-modules are D-branes? 71 Part III. Conformal field theory approach 73 7. Conformal field theory with Kac-Moody symmetry 73 7.1. Conformal blocks 73 7.2. Sheaves of conformal blocks as D-modules on the moduli spaces of curves 75 7.3. Sheaves of conformal blocks on BunG 77 7.4. Construction of twisted D-modules 80 7.5. Twisted D-modules on BunG 82 7.6. Example: the WZW D-module 83 8. Conformal field theory at the critical level 85 8.1. The chiral algebra 86 8.2. The center of the chiral algebra 88 8.3. Opers 90 8.4. Back to the center 93 8.5. Free field realization 95 8.6. T-duality and the appearance of the dual group 98 9. Constructing Hecke eigensheaves 100 9.1. Representations parameterized by opers 101 9.2. Twisted D-modules attached to opers 104 9.3. How do conformal blocks know about the global curve? 106 9.4. The Hecke property 108 9.5. Quantization of the Hitchin system 111 9.6. Generalization to other local systems 113 9.7. Ramification and parabolic structures 116 9.8. Hecke eigensheaves for ramified local systems 119 Index 122 References 124 LECTURES ON THE LANGLANDS PROGRAM AND CONFORMAL FIELD THEORY 3 Introduction These lecture notes give an overview of recent results in geometric Langlands correspon- dence which may yield applications to quantum field theory. It has long been suspected that the Langlands duality should somehow be related to various dualities observed in quantum field theory and string theory. Indeed, 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. 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-dimensional gauge theory to two 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 reduction a Hecke \eigensheaf", an object of interest in the geometric Langlands correspondence. The work of Kapustin and Witten shows that the Langlands duality is indeed closely related to the S-duality of quantum field theory, and this opens up exciting possibilities for both subjects. The goal of these notes is two-fold: first, it is to give a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physi- cists. I have tried to make it as self-contained as possible, requiring very little mathemat- ical background. The second goal is to describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality. The classical Langlands correspondence manifests a deep connection between number theory and representation theory. In particular, it relates subtle number theoretic data (such as the numbers of points of a mod p reduction of an elliptic curve defined by a cubic equation with integer coefficients) to more easily discernible data related to automorphic forms (such as the coefficients in the Fourier series expansion of a modular form on the upper half-plane). We will consider explicit examples of this relationship (having to do with the Taniyama-Shimura conjecture and Fermat's last theorem) in Part I of this survey. So the origin of the Langlands Program is in number theory. Establishing the Langlands correspondence in this context has proved to be extremely hard. But number fields have close relatives called function fields, the fields of functions on algebraic curves defined over a finite field. The Langlands correspondence has a counterpart for function fields, which is much better understood, and this will be the main subject of our interest in this survey. 4 EDWARD FRENKEL Function fields are defined geometrically (via algebraic curves), so one can use geometric intuition and geometric technique to elucidate the meaning of the Langlands correspon- dence. This is actually the primary reason why the correspondence is easier to understand in the function field context than in the number field context. Even more ambitiously, one can now try to switch from curves defined over finite fields to curves defined over the complex field { that is to Riemann surfaces. This requires a reformulation, called the geometric Langlands correspondence. This reformulation effectively puts the Langlands correspondence in the realm of complex algebraic geometry. Roughly speaking, the geometric Langlands correspondence predicts that to each rank n holomorphic vector bundle E with a holomorphic connection on a complex algebraic curve X one can attach an object called Hecke eigensheaf on the moduli space Bunn of rank n holomorphic vector bundles on X: holomorphic rank n bundles −! Hecke eigensheaves on Bun with connection on X n A Hecke eigensheaf is a D-module on Bunn satisfying a certain property that is de- termined by E. More generally, if G is a complex reductive Lie group, and LG is the Langlands dual group, then to a holomorphic LG-bundle with a holomorphic connection on X we should attach a Hecke eigensheaf on the moduli space BunG of holomorphic G-bundles on X: holomorphic LG-bundles −! Hecke eigensheaves on Bun with connection on X G I will give precise definitions of these objects in Part II of this survey. The main point is that we can use methods of two-dimensional conformal 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 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 the variation of correlation functions under deformations of the complex structure on the underlying Riemann surface via the insertion in the correlation function of the stress 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 holomorphic vector bundle with a projectively flat connection on the moduli space of G-bundles on X.