The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0001/CNTP-2007-0001-0001-a001.pdf

Electric-Magnetic Duality And The Geometric Langlands Program

These lecture notes are based on the master class given at the Center for the Topology and Quantization of Moduli Spaces, University of Aarhus, August 2007. I provide an introduction to the recent work on the Montonen-Olive duality of N = 4 super-Yang-Mills theory and the Geometric Langlands Program.

http://www.ctqm.au.dk/research/MCS/LectureNotesKapustin.pdf

Lectures on Electric-Magnetic Duality and the Geometric Langlands Program

Vertex Algebras and Algebraic Curves

As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.

/pdf/geometric-langlands-duality-and-representations-of-algebraic-2mtr4x0ei6.pdf

Geometric Langlands duality and representations of algebraic groups over commutative rings

We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on ℝ4 gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of ℝ4, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.

Instanton counting on blowup. I. 4-dimensional pure gauge theory

A Study in Derived Algebraic Geometry, Part 1: Volume I: Correspondences and Duality

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. 
Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. 
As an application we construct certain automorphic forms for global fields of positive characteristic, whose existence is non-obvious from the point of view of classical tools.

Geometric Eisenstein series

We define the notion of singular support of a coherent sheaf on a quasi-smooth-derived scheme or Artin stack, where “quasi-smooth” means that it is a locally complete intersection in the derived sense. This develops the idea of “cohomological” support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behavior of singular support under the direct and inverse image functors for coherent sheaves. We use the theory of singular support of coherent sheaves to formulate the categorical geometric Langlands conjecture. We verify that it passes natural consistency tests: It is compatible with the geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original “naive” form of the conjecture.

Singular support of coherent sheaves and the geometric Langlands conjecture

Let $ \mathfrak{g} $ be a simple Lie algebra over ℂ and G a connected algebraic group with Lie algebra $ \mathfrak{g} $. The affine Kac-Moody algebra $ \hat {\mathfrak{g}} $ is the universal central extension of the formal loop agebra $ \mathfrak{g} $((t)). Representations of $ \hat {\mathfrak{g}} $ have a parameter, an invariant bilinear form on $ \mathfrak{g} $, which is called the level. Representations corresponding to the bilinear form which is equal to minus one half of the Killing form are called representations of critical level. Such representations can be realized in spaces of global sections of twisted D-modules on the quotient of the loop group G((t)) by its “open compact” subgroup K, such as G[[t]] or the Iwahori subgroup I.

Local geometric Langlands correspondence and affine Kac-Moody algebras

Let $G$ be a reductive group, let $Gr=G((t))/G[[t]]$ be the corresponding affine Grassmannian and let $Fl=G((t))/I$ be the affine flag variety. We construct, following an idea of Belinson, a 1-parametric deformation of the product $Gr\times G/B$ to $Fl$. We use this construction to produce perverse sheaves on $Fl$ which are central with respect to the convolution product from spherical perverse sheaves on $Gr$.

Dennis Gaitsgory

Papers

A Study in Derived Algebraic Geometry, Part 1: Volume I: Correspondences and Duality

Geometric Eisenstein series

Singular support of coherent sheaves and the geometric Langlands conjecture

Local geometric Langlands correspondence and affine Kac-Moody algebras

Construction of central elements in the affine Hecke algebra via nearby cycles