819 Evans Hall
Email address: frenkel@math.berkeley.edu
Schedule of make-up classes:
Friday, Dec 10, 1-3 pm, 891 Evans
Monday, Dec 13, 2-3 pm, 961 Evans
Tuesday, Dec 14, 1-3 pm, 961 Evans
Course description:
We will discuss the geometric Langlands correspondence over the field of complex numbers.
In the traditional local Langlands correspondence one wishes to describe smooth representations of a reductive group G over a local non-archimedian field, such as the field of formal Laurent power series over a finite field, in terms of the Galois group of the local field and and the Langlands dual group of G. When we replace the finite field by the complex field, we are naturally led to loop groups and loop Lie algebras and the central extensions of the latter, called affine Kac-Moody algebras. So we wish to understand categories of smooth representations of affine Kac-Moody algebras and describe them in terms of categories of coherent sheaves on some varieties associated to the Langlands dual group.
The local correspondence amounts to a "spectral decomposition" of the categories of smooth representations of an affine Kac-Moody algebra with respect to the local systems of the Langlands dual group over the punctured disc. In order to construct it, we will introduce a certain class of representations called Wakimoto modules and use them to describe the center of the completed enveloping algebra of an affine Kac-Moody algebra at the critical level. We will show that the center is isomorphic to the classical W-algebra of the Langlands dual group.
Classical W-algebras first appeared in the theory of integrable systems, where they govern the Hamiltonian structures of the generalized KdV systems. It is quite remarkable that a W-algebra would also appear as the center of the enveloping algebra. Equivalently, the classical W-algebra may be described as the the algebra of functions on the space of opers on the disc. Opers are bundles with connection and an extra structure, and we will study them as well. One can also define W-algebras via the Drinfeld-Sokolov reduction. We remark that classical W-algebras may be quantized. The result is a vertex algebra which appears as a symmetry of a conformal field theory. The simplest one is the Virasoro algebra.
Having described the center, we will look at various categories of Harish-Chandra modules over affine Kac-Moody algebras with fixed central character. We will show that they may be described as categories of coherent sheaves on certain varieties. We will use the categories of Harish-Chandra modules to construct the global Langlands correspondence following the work of A.Beilinson and V.Drinfeld.
The global Langlands correspondence associates to local systems defined over a smooth projective curve X the so-called Hecke eigensheaves on the moduli space of G-bundles on X. We will consider explicit examples of this correspondence, such as the Gaudin integrable system, which arises in the Langlands correspondence in genus zero.
The lectures will be augmented by a seminar that will meet on Mondays 4-5:30, in which we will discuss the classical Langlands correspondence (so as to give some motivation to the topics discussed in the course) and other related material.
Our main sources will be my paper
Lectures on Wakimoto modules, opers and the center at the critical level
the notes of the graduate course that I taught in Berkeley in the Fall of 2002 (these notes were taken by A. Barnard), and the book
E. Frenkel, D. Ben-Zvi, "Vertex algebras and algebraic curves", Second Edition AMS 2004 (see the link at the bottom of that page).
1. Construction of Wakimoto modules over the affine Kac-Moody algebras.
2. Isomorphism between the center of the completed universal enveloping algebra of an affine algebra and the classical W-algebra.
3. Opers and Miura opers. Various realizations of W-algebras.
4. Description of the categories of Harish-Chandra modules over affine algebras in terms of the Langlands dual group. Local Langlands correspondence.
5. Global Langlands correspondence. Localization on the moduli spaces of bundles on curves. Spaces of conformal blocks. Application: Gaudin model and Bethe Ansatz.
6. Work of Beilinson and Drinfeld on the global geometric Langlands correspondence.