Gauge Theory and Langlands Duality

Abstract

The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.

32 pages. Text of Bourbaki Seminar given by the author on June 13, 2009

References in corpus (12)

• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• Mirror symmetry, Langlands duality, and the Hitchin system
• Gauge Theory, Ramification, And The Geometric Langlands Program
• Transformation de Fourier generalisee
• Mirror Symmetry, Hitchin's Equations, And Langlands Duality
• Rigid Surface Operators
• Le lemme fondamental pour les algebres de Lie
• A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections
• Microlocal branes are constructible sheaves
• Constructible Sheaves and the Fukaya Category
• Geometric theta-lifting for the dual pair GSp_{2n}, GSO_{2m}

Cited by in corpus (9)

• Landscaping with fluxes and the E8 Yukawa Point in F-theory
• A Prehistory of n-Categorical Physics
• On $\mathcal{N}=1$ susy backgrounds with $AdS$ factor from nonabelian T-duality
• Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure
• S-duality in Abelian gauge theory revisited
• Non-abelian vortices, Hecke modifications and singular monopoles
• The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic Quantum Field Theory
• On the Magnetic Current Density in the Maxwell Equations Based on the Noether Theorem
• AP Theory IV: Intrinsic Topological Quantum Langlands Theory