on a four manifold M where Gc is a compact Lie group 1 Electromagnetism. corresponds to the simplest abelian compact Lie group U 1 It is natural to ask. whether there is a non abelian analogue of the electro magnetic duality for gauge. theories with non abelian gauge groups, The answer was proposed in the late 1970s by Montonen and Olive MO following. Goddard Nuyts and Olive GNO see also EW O A gauge theory has a coupling. constant g which plays the role of the electric charge e The conjectural non abelian. electro magnetic duality which has later become known as S duality has the form. 0 1 Gc g LGc 1 g, In other words the duality states that the gauge theory with gauge group Gc more. precisely its N 4 supersymmetric version and coupling constant g should be. equivalent to the gauge theory with the Langlands dual gauge group LGc and coupling. constant 1 g note that if Gc U 1 then LGc is also U 1 If true this duality would. have tremendous consequences for quantum gauge theory because it would relate a. theory at small values of the coupling constant weak coupling to a theory with large. values of the coupling constant strong coupling Quantum gauge theory is usually. defined as a power series expansion in g which can only converge for small values of g. It is a very hard problem to show that these series make sense beyond perturbation. theory S duality indicates that the theory does exist non perturbatively and gives us. a tool for understanding it at strong coupling That is why it has become a holy grail. of modern Quantum Field Theory, Looking at 0 1 we see that the Langlands dual group shows up again Could it be. that the Langlands duality in Mathematics is somehow related to S duality in Physics. This question has remained a mystery until about five years ago In March of 2004. DARPA sponsored a meeting of a small group of physicists and mathematicians at. the Institute for Advanced Study in Princeton which I co organized to tackle this. question At the end of this meeting Edward Witten gave a broad outline of a relation. between the two topics This was explained in more detail in his subsequent joint work. KW with Anton Kapustin This paper and the work that followed it opened new. bridges between areas of great interest for both physicists and mathematicians leading. to new ideas insights and directions of research, The goal of these notes is to describe briefly some elements of the emerging pic. ture In Sections 1 and 2 we will discuss the Langlands Program and its three flavors. putting it in the context of Andre Weil s big picture This will eventually lead us. to a formulation of the geometric Langlands correspondence as an equivalence of cer. tain categories of sheaves in Section 3 In Section 4 we will turn to the S duality in. topological twisted N 4 super Yang Mills theory Its dimensional reduction gives. rise to the Mirror Symmetry of two dimensional sigma models associated to the Hitchin. We will use the notation G for a complex Lie group and Gc for its compact form Note that physicists. usually denote by G a compact Lie group and by GC its complexification. moduli spaces of Higgs bundles In Section 5 we will describe a connection between. the geometric Langlands correspondence and this Mirror Symmetry following KW as. well as its ramified analogue GW1 In Section 6 we will discuss subsequent work and. open questions, Acknowledgments I thank Sergei Gukov Vincent Lafforgue Robert Langlands and. Edward Witten for inspiring discussions I also thank S Gukov and V Lafforgue for. their comments on a draft of this paper, I am grateful to DARPA and especially Benjamin Mann for generous support which. has been instrumental not only for my research but for the development of this whole. area I also thank Fondation Sciences Mathe matiques de Paris for its support during. my stay in Paris,1 LANGLANDS PROGRAM, In 1940 Andre Weil was put in jail for his refusal to serve in the army There he. wrote a letter to his sister Simone Weil a noted philosopher in response to her question. as to what really interested him in his work We This is a remarkable document in. which Weil tries to explain in fairly elementary terms presumably accessible even to. a philosopher the big picture of mathematics the way he saw it I think this sets a. great example to follow for all of us, Weil writes about the role of analogy in mathematics and he illustrates it by the. analogy that interested him the most between Number Theory and Geometry. On one side we look at the field Q of rational numbers and its algebraic closure Q. obtained by adjoining all roots of all polynomial equations in one variable with rational. coefficients like x2 1 0 The group of field automorphisms of Q is the Galois group. Gal Q Q We are interested in the structure of this group and its finite dimensional. representations We may also take a more general number field that is a finite. extension F of Q such as Q i and study its Galois group and its representations. On the other side we have Riemann surfaces smooth compact orientable surfaces. equipped with a complex structure and various geometric objects associated to them. vector bundles their endomorphisms connections etc. At first glance the two subjects are far apart However it turns out that there are. many analogies between them The key point is that there is another class of objects. which are in between the two A Riemann surface may be viewed as the set of points of. a projective algebraic curve over C In other words Riemann surfaces may be described. by algebraic equations such as the equation,1 1 y 2 x3 ax b. where a b C The set of complex solutions of this equation for generic a b for which. the polynomial on the right hand side has no multiple roots compactified by a point. at infinity is a Riemann surface of genus 1 However we may look at the equation 1 1. not only over C but also over other fields for instance over finite fields. Recall that there is a unique up to an isomorphism finite field Fq of q elements for. all q of the form pn where p is a prime In particular Fp Z pZ 0 1 p 1. with the usual arithmetic modulo p Let a b be elements of Fq Then the equation 1 1. defines a curve over Fq These objects are clearly analogous to algebraic curves over C. that is Riemann surfaces But there is also a deep analogy with number fields. Indeed let X be a curve over Fq such as an elliptic curve defined by 1 1 and F. the field of rational functions on X This function field is very similar to a number. field For instance if X is the projective line over Fq then F consists of all fractions. P t Q t where P and Q are two relatively prime polynomials in one variable with. coefficients in Fq The ring Fq t of polynomials in one variable over Fq is similar to the. ring of integers and so the fractions P t Q t are similar to the fractions p q where. Thus we find a bridge or a turntable as Weil calls it between Number Theory. and Geometry and that is the theory of algebraic curves over finite fields. In other words we can talk about three parallel tracks. Number Theory Curves over Fq Riemann Surfaces, Weil s idea is to exploit it in the following way take a statement in one of the three. columns and translate it into statements in the other columns We my work consists. in deciphering a trilingual text of each of the three columns I have only disparate. fragments I have some ideas about each of the three languages but I know as well. there are great differences in meaning from one column to another for which nothing. has prepared me in advance In the several years I have worked at it I have found little. pieces of the dictionary Weil went on to find one of the most spectacular applications. of this Rosetta stone what we now call the Weil conjectures describing analogues. of the Riemann Hypothesis in Number Theory in the context of algebraic curves over. finite fields, It is instructive to look at the Langlands Program through the prism of Weil s big. picture Langlands original formulation L concerned the two columns on the left Part. of the Langlands Program may be framed as the question of describing n dimensional. representations of the Galois group Gal F F where F is either a number field Q or. its finite extension or the function field of a curve over Fq 2 Langlands proposed that. such representations may be described in terms of automorphic representations of the. group GLn AF where AF is the ring of ade les of F I will not attempt to explain this. here referring the reader to the surveys Ge F1 F2, However it is important for us to emphasize how the Langlands dual group appears in. this story Let us replace GLn AF by G AF where G is a general reductive algebraic. Langlands more general functoriality principle is beyond the scope of the present article. group such as orthogonal or symplectic or E8 In the case when G GLn its auto. morphic representations are related to the n dimensional representations of Gal F F. that is homomorphisms Gal F F GLn The general Langlands conjectures predict. that automorphic representations of G AF are related in a similar way to homomor. phisms Gal F F LG where LG is the Langlands dual group to G 3. It is easiest to define LG in the case when G defined over a field k is split over k that. is contains a maximal split torus T which is the product of copies of the multiplicative. group GL1 over k We associate to T two lattices the weight lattice X T of homo. morphisms T GL1 and the coweight lattice X T of homomorphisms GL1 T. They contain the sets of roots X T and coroots X T of G respectively. The quadruple X T X T is called the root data for G over k The root. data determines the split group G up to an isomorphism. Let us now exchange the lattices of weights and coweights and the sets of simple roots. and coroots Then we obtain the root data, of another reductive algebraic group over C or Q which is denoted by LG 4 Here. are some examples,Sp2n SO2n 1,Spin2n SO2n Z2, In the function field case we expect to have a correspondence between homomor. phisms 5 Gal F F LG and automorphic representations of G AF where AF is the. ring of ade les of F, Fx Fqx tx being the completion of the field of functions at a closed point x of X. and the prime means that we take the restricted product in the sense that for all. but finitely many x the element of Fx belongs to its ring of integers Ox Fx tx. We have a natural diagonal inclusion F AF an hence G F G AF Roughly. More precisely LG should be defined over Q where is relatively prime to q and we should consider. homomorphisms Gal F F LG Q which are continuous with respect to natural topology see e g. Section 2 2 of F2, In Langlands definition L LG also includes the Galois group of a finite extension of F This is. needed for non split groups but since we focus here on the split case this is not necessary. More precisely the Galois group should be replaced by its subgroup called the Weil group. speaking an irreducible representation of G AF is called automorphic if it occurs in. the decomposition of L2 G F G AF with respect to the right action of G AF. For G GLn in the function field case the Langlands correspondence is a bijec. tion between equivalence classes of irreducible n dimensional adic representations of. Gal F F more precisely the Weil group and cuspidal automorphic representations. of GLn AF It has been proved by V Drinfeld Dr1 Dr2 for n 2 and by L Lafforgue. LafL for n 2 A lot of progress has also been made recently in proving the Langlands. correspondence for GLn in the number field case, For other groups the correspondence is expected to be much more subtle for instance. it is not one to one Homomorphisms from the Weil group of F to LG and more general. parameters introduced by J Arthur see Section 6 2 should parametrize certain collec. tions of automorphic representations called L packets This has only been proved in. a few cases so far,2 GEOMETRIC LANGLANDS CORRESPONDENCE. The above discussion corresponds to the middle column in the Weil big picture What. should be its analogue in the right column that is for complex curves. In order to explain this we need a geometric reformulation of the Langlands corre. spondence which would make sense for curves defined both over a finite field and over C. Thus we need to find geometric analogues of the notions of Galois representations and. automorphic representations, The former is fairly easy Let X be a curve over a field k and F k X the field of. rational functions on X If Y X is a covering of X then the field k Y of rational. functions on Y is an extension of the field F k X of rational functions on X and the. Galois group Gal k Y k X may be viewed as the group of deck transformations. of the cover If our cover is unramified then this group is a quotient of the arithmetic. fundamental group of X For a cover ramified at points x1 xn it is a quotient of the. arithmetic fundamental group of X x1 xn From now on with the exception of. Section 5 4 we will focus on the unramified case This means that we replace Gal F F. by its maximal unramified quotient which is nothing but the arithmetic fundamental. group of X Its geometric analogue when X is defined over C is 1 X. Thus the geometric counterpart of a unramified homomorphism Gal F F LG. is a homomorphism 1 X LG, From now on let X be a smooth projective connected algebraic curve defined over C. GAUGE THEORY AND LANGLANDS DUALITY by Edward FRENKEL INTRODUCTION In the late 1960s Robert Langlands launched what has become known as the Lang lands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis L In particular Langlands conjectured that Galois represen tations and motives can be described in terms of the more tangible data of automor

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