# Lectures on the Langlands program and conformal field theory

@article{Frenkel2005LecturesOT, title={Lectures on the Langlands program and conformal field theory}, author={E. Frenkel}, journal={arXiv: High Energy Physics - Theory}, year={2005}, pages={387-533} }

These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I tried to make it as self-contained as possible, requiring very little mathematical background. Next, we describe the connections between the Langlands Program and two-dimensional conformal field theory that have… Expand

#### 197 Citations

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#### References

SHOWING 1-10 OF 168 REFERENCES

Quantum field theory, Grassmannians, and algebraic curves

- Mathematics
- 1988

This paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite Grassmannians, and in part to pointing out the existence of a close analogy between… Expand

Affine Algebras, Langlands Duality and Bethe Ansatz

- Mathematics, Physics
- 1995

We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands… Expand

Recent advances in the Langlands Program

- Mathematics
- 2003

These are the notes for the lecture given by the author at the “Mathematical Current Events” Special Session of the AMS meeting in Baltimore on January 17, 2003. Topics reviewed include the Langlands… Expand

On holomorphic factorization of WZW and coset models

- Mathematics
- 1992

It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional conformal field theory, which previously have… Expand

Lectures on Conformal Field Theory

- Physics
- 1988

In statistical physics, in the theory of critical phenomena at the second order phase transition points, the global scaling symmetry has been known and extensively used for many years. This led to… Expand

Introduction to the Langlands Program

- Mathematics
- 2000

This article is an introduction to automorphic forms on the adeles of a linear reductive group over a number field. The first half is a summary of aspects of local and global class field theory, with… Expand

Quantum field theory and the Jones polynomial

- Mathematics
- 1989

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones… Expand

Integrable field theory from conformal field theory

- Physics
- 1989

Abstract Equations of motion for two-dimensional quantum field theory obtained as some relevant perturbation around CFT are analyzed. It is shown that for particular degenerate fields taken as the… Expand

Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries

- Mathematics
- 1989

Publisher Summary This chapter focuses on the conformal field theory (CFT) on universal family of stable curves with gauge symmetries. CFT has not only useful application to string theory and… Expand

Global aspects of gauged Wess-Zumino-Witten models

- Physics, Mathematics
- 1996

A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a… Expand