This masterclass is organized by:

The Korteweg-de Vries institute.


The NWO research cluster GQT.


The local organizer: Eric Opdam.
The Korteweg-de Vries institute,
Science Park 904, 1098 XH Amsterdam,
The Netherlands.

Lecturers:


J. E. Andersen
Center for Geometry and Quantization of Moduli Spaces,
Institute for Mathematics, University of Aarhus,
email: andersen at imf.au.dk

N. Reshetikhin
Department of Mathematics, 917 Evans Hall, University of California, Berkeley,
email: reshetik at math.berkeley.edu
KdV Institute, C4.145, Science Park 904 1098 XH Amsterdam,
email: N.Y. Reshetikhin at uva.nl

Week 1. Class by N. Reshetikhin.

Geometry of gauge theories.

Tentative syllabus

The goal of this master-class is to introduce classical gauge theories and some basic necessary geometric notions. Basic examples are Yang-Mills and Chern-Simons theories The class will start with an overview of principal $G$-bundles, and connections on them. Then classical Lagrangian field theories will be introduced, the Yang-Mills and Chern-Simons theory as basic examples. After this we will have an overview of some basic notions from symplectic and will focus on the Hamiltonian formulation of classical field theory. Finally we will discuss of 2D, 3D, and 4D Yang-Mills theories, the 3D Chern-Simons theory, and their special features. If time permits the discretization of the classical Yang-Mills theory will be introduced.
The Lecture Notes will eventually be completed. Below is the tentative schedule of classes

May 31

Principal $G$-bundles and connections on them. Gauge transformations. The concept of Lagrangian field theory. Scalar field. Yang-Mills and 3D Chern-Simons field theories.

June 1

Boundary conditions. Analysis of 2D, 3D, and 4D Yang-Mills theories, instantons, topological field theories.

June 2

Hamiltonian formulations of classical field theory. Scalar field. Boundary conditions in the Hamiltonian formulation of classical field theory. Hamiltonian reduction.

June 3

Yang-Mills, reduced and non-reduced. Chern-Simons, reduced and non-reduced.

June 4

Discretization of classical field theories.

Week 2. Class by J.E. Andersen.

Geometric quantization.

Tentative syllabus

The aim of this master-class is an introduction to geometric quantization with applications to topological quantum field theory. It will start with geometric quantization on symplectic manifolds with either real polarizations or Kahler polarizations, introducing all necessary notions. Then I will get to the Hitchin connection which identify geometric quantizations for different choices of polarizations. Following this I will introduce the Toeplitz operators, will explain their role in geometric quantization, and will describe how they are related to the Hitchin connection. These techniques will be applied to moduli spaces of flat connections. In particular, I will prove the asymptotic faithfulness of the mapping group action on the geometric quantization spaces. If time permit I will introduce coherent states and will prove that these actions does not satisfy Kazhdan's property T.
The lecture notes eventually will be completed. During the master-class the entries below will be filled with an outline of the progress in the class.

June 7

The class progress.

June 8

The class progress.

June 9

The class progress.

June 10

The class progress.

June 11

The class progress.
Last Updated: