The Langlands Program is quickly emerging as a blueprint for a Grand Unified Theory of Mathematics. Conceived initially as a bridge between Number Theory and Automorphic Representations, it has now expanded into such areas as Geometry and Quantum Field Theory, weaving together seemingly unrelated disciplines into a web of tantalizing conjectures.
The Langlands Program seeks to establish a correspondence between objects of "Galois type" and objects of "automorphic type". One of the prevalent themes is the interplay between the local and global pictures. In the geometric context the objects on the Galois side of the correspondence are holomorphic vector bundles with connection on a complex Riemann surface in the global case, and on the punctured disc in the local case. The definition of the objects on the automorphic side of the geometric Langlands Correspondence is more subtle. It is relatively well understood in the special case when the connection has no singularities. However, in the more general case of connections with ramification, that is with singularities, the geometric Langlands Correspondence is much more mysterious, both in the local and in the global case. Actually, the impetus now shifts more to the local story, because the global correspondence is largely determined by what happens on the punctured discs around the ramification points, which is in the realm of the local correspondence. So the question really becomes: what is the geometric analogue of the local Langlands Correspondence?
A new approach to this problem has been recently developed by D. Gaitsgory and the author. To local Langlands parameters we associate representations of the formal loop group. However, in contrast to the classical theory, these representations are realized not on vector spaces, but in categories of representations of the corresponding Lie algebra, which is an affine Kac-Moody algebra.
Affine Kac-Moody algebras have a parameter, called the level. For a special value of this parameter, called the critical level, the completed enveloping algebra of an affine Kac-Moody algebra acquires an unusually large center. B. Feigin and the author have shown that this center is canonically isomorphic to the algebra of functions on the space of opers, which are bundles on the punctured disc with a flat connection and an additional datum. Remarkably, their structure group turns out to be the Langlands dual group, in agreement with the general Langlands philosophy. This is the central result which implies that the same salient features permeate both representation theory of p-adic groups and (categorical) representation theory of loop groups.
The goal of this book is to present a systematic and self-contained introduction to the local geometric Langlands Correspondence for loop groups and the related representation theory of affine Kac-Moody algebras. It is partially based on the graduate courses taught by the author at UC Berkeley in 2002 and 2004. In the book the entire theory is built from scratch, with all necessary concepts defined and the needed results proved along the way.
This is the first introductory account of the research done in this area in the last twenty years. It opens with a pedagogical overview of the classical Langlands correspondence and a motivated step-by-step passage to the geometric setting. This leads us to the study of affine Kac-Moody algebras and in particular the center of the completed enveloping algebra. Next, the book reviews the construction of a series of representations of affine Kac-Moody algebras, called Wakimoto modules. They were defined by B. Feigin and the author following the work of M. Wakimoto. These modules provide us with an effective tool for developing representation theory of affine algebras. In particular, they are crucial in the proof of the isomorphism between the spectrum of the center and opers.
A detailed exposition of the Wakimoto modules and the proof of this isomorphism constitute the main part of this book. These results allow us to establish a deep link between representation theory of affine Kac-Moody algebras of critical level and the geometry of opers. The closing chapter reviews the results and conjectures of D. Gaitsgory and the author describing the representation categories of loop groups associated to opers in the framework of the local Langlands Correspondence, as well as the implications of this for the global geometric Langlands Correspondence. These are only the first steps of a new theory, which will hopefully allow us to uncover the mysteries of the Langlands duality.
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