Though the langlands program has nothing to do with physics at rst sight, the dual group has been independently considered in physics as an extension of electricmagnetic duality 16, 37 and lghas been know as a gno dual group among physicists. Gauge theory, class of quantum field theory, a mathematical theory involving both quantum mechanics and einsteins special theory of relativity that is commonly used to describe subatomic particles and their associated wave fields. Analytic and geometric aspects of gauge theory fall 2022 website. In this talk, we will introduce the main object of study of the local geometric langlands theory. The rst idea is to construct a family of 4dimensional topological eld theories parametrized by 2cp1 as topological twists of n 4 supersymmetric gauge theory with gauge group g. Lectures on electricmagnetic duality and the geometric. 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. Holomorphic differentials in mathematics and physics, august 1516, 2019 website. The plateaus of the hall conductance are described by hecke. Gauge theory and langlands duality math berkeley university of. We sketch some of the central further developments. We describe the relevant surface operators in n4 super yangmills theory, and the parameters they depend on, and analyze how s. Gauge theory and langlands duality by edward frenkel introduction in the late 1960s robert langlands launched what has become known as the langlands program with the ambitious goal of relating deep questions in number theory to harmonic analysis l.
Mills theory, which is the basis of the gauge theory approach to the geometric langlands program. A survey of literature on gauge theory approach to geometric langlands and allied topics. However, it does not shed any light on the utility of twodimensional axiomatic conformal. Gauge theoretic aspects of the geometric langlands. In mathematics, the classical langlands correspondence is a collection of results and conjectures relating number theory and representation theory. We describe the relevant surface operators in n4 super yangmills theory, and the parameters they depend on, and analyze how sduality acts on these. Witten electricmagnetic duality and the geometric langlands program, commun. More on gauge theory and geometric langlands request pdf. Geometrical approaches to the quantization of gauge theories bucker, beatrice, 2004.
More on gauge theory and geometric langlands sciencedirect. Electricmagnetic duality and the geometric langlands program 5 underlying this tduality was investigated in 22 and subsequently in 23 for any semisimple lie group g. For the application to geometric langlands, the n 4 theory is topologically twisted so as to produce, formally, a topological field theory in four dimensions. The conjectural geometric langlands correspondence is meant to be an. We then compute the moduli spaces for the kapustinwitten topological twists as its further twists. In particular, langlands conjectured that galois represen. I will discuss a set of tools which can be used to translate gauge theory constructions into concrete mathematical statements. This page aims to be an entry point of sorts for the literature on the gauge theory approach to geometric langlands and some other allied topics. The proof relies on the cohomological interpretation of orbital integrals, which makes available the deep topological tools of algebraic geometry such as hodge theory and the weil conjectures. Geometric langlands, khovanov homology, string theory.
Gauge theory, geometric langlands and vertex operator. Geometric representation theory september 02, 2014 september 05, 2014 september 04, 2014 11. January 1, 2008 abstract 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. Presumably people attending wittens talks in berkeley and cambridge will get to hear about this new story for the ramified case. Sduality interchanges the theory with gauge group gand twisting parameter with the theory with. Currently, it is just a list of papers ordering within each heading is influenced but not determined by chronology.
Msri workshop schedules gauge theory and langlands. Some of the ingredients that enter in the gauge theory approach to geometric langlands are also relevant for understanding khovanov homology via gauge theory 8. The langlands program was launched in the late 60s with the goal of relating galois representations and automorphic forms. The beautiful work of 1 established a rich dictionary between sduality in fourdimensional n 4 gauge theory and the mathematical subject of geometric langlands duality. In the rest of this section well describe the idea, and some of the history, of sduality. Recent advances in the langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc. The gauge group is a compact lie group g, which we will generally assume to be simple. The fact that electromagnetic duality links a group gwith its langlands dual group lgsuggested that electromagnetic duality might be linked to. Introduction to geometric langlands, notes from lecture at the vienna workshop on. Journal of high energy physics gauging spacetime symmetries on the worldsheet and the geometric langlands program ii to cite this article. Lectures on the langlands program and conformal field theory. Lectures on electricmagnetic duality and the geometric langlands program anton kapustin california institute of technology, pasadena, ca 91125, u. Gauge theory, ramification, and the geometric langlands.
The initial program to develop a geometric analog was due to drinfeld, drinfeldlaumonfor speci c groups a more general arbitrary g and modern program is due to. We develop techniques for describing the derived moduli spaces of solutions to the equations of motion in twists of supersymmetric gauge theories as derived algebraic stacks. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities. For informal explanations of some aspects of gauge theory and geometric langlands, see 911. Langlands program, field theory, and mirror symmetry. We describe the relevant surface operators in n4 super yangmills theory, and the parameters they depend on, and analyze how sduality acts. A highlight here is the proof by kronheimer and mrowka of property p34.
The work of kapustin and witten centres around a gaugetheoretic interpretation of the geometric langlands correspondence. Gauge theory and the geometric langlands program edward witten august 10th, 2005 talk at the third simons workshop in mathematics and physics suny at stony brook, july 25 august 26, 2005 based on notes by ram sriharsha introduction the langlands program of number theory, or what we might call langlands duality, was. Representation theoryquantum field theorygauge theory automorphic forms triumph ofexploitation of symmetry i. Gauge theory and langlands duality edward frenkel introduction. Gauging spacetime symmetries on the worldsheet and the.
Since dmodules on moduli stacks of g principal bundles play a central role in gauge quantum field theory in. A physics perspective on geometric langlands duality. The gauge theory approach to the geometric langlands program is extended to the case of wild ramification. Gauge theory, vertex algebras and the geometric langlands. Geometry and physics of alx metrics in gauge theory aug 16 20, 2021. The workshop will explore the relation between boundary conditions in fourdimensional gauge theory, the geometric langlands program and vertex operator algebras. Aswin balasubramanian geometric langlands from 4d n. Formulated by robert langlands in the late 1960s, the langlands correspondence is related to important conjectures in number theory such as the taniyamashimura conjecture, which includes fermats last theorem as a special case. Quantization of the hitchin moduli spaces, liouville theory and the geometric langlands correspondence i teschner, j. We introduce a holomorphic twist of n4 supersymmetric gauge theory and compute the derived moduli space. For a gauge theory with gauge group gand a coupling constant e, there exists.
Sponsosrship for this workshop has been provided by. The title of the forthcoming gukovwitten paper is supposedly gauge theory, ramification, and the geometric langlands program. Talk at the third simons workshop in mathematics and physics. In the gauge theory approach to the geometric langlands program, ramification can be described in terms of surface operators, which are supported on twodimensional surfaces somewhat as wilson or t hooft operators are supported on curves.
Then well go on to talk about geometric representation theory, and more speci cally the geometric langlands program, and the connection introduced by kapustin and witten between these two disparate elds. In 2006, edward witten, charles simonyi professor in the school of natural sciences, cowrote with anton kapustin a 225page paper, electricmagnetic duality and the geometric langlands program, on the relation of part of the geometric langlands program to ideas of the duality between electricity and magnetism. Yangmills theory and geometry imperial college london. Gauge theory and langlands duality introductory workshop. Gauge theory, ramification, and the geometric langlands program. They are associated with the dmodules that appear on the right hand side of the geometric langlands program.
Kapustin a note on quantum geometric langlands duality, gauge theory, and quantization of the moduli space of flat connections, preprint arxiv. By applying to physics, these novel perspectives endow with a unified account of the integer fractional quantum hall effect. I provide an introduction to the recent work on the montonenolive duality of cn4 superyangmills theory and the geometric langlands program. I should note that the broader langlands program is a vast research program at the intersection of. Gauge theory, mirror symmetry, and the geometric langlands. A survey of literature on gauge theory approach to geometric. Keywords gauge theory modulus space gauge group wilson loop vertex operator. Geometric langlands twists of n 4 gauge theory from. The remarks above are not meant to suggest that the impact of yangmills.
A survey of literature on gauge theory approach to. Other clues about the relation of the geometric langlands program to quantum. Topological quantum field theory and the geometric. More on geometric langlands a grand unified theory of. I provide an introduction to the recent work on the montonenolive duality of n 4 superyangmills theory and the geometric langlands program. In a gauge theory there is a group of transformations of the field variables gauge transformations that leaves the basic physics of the quantum field unchanged. Electricmagnetic duality and the geometric langlands. Kapustin and witten have proposed a gauge theory interpretation of the geometric langlands program.