## Brownian motion and Topological Field theories

##### Summary

This thesis consists of two parts. In Part 1 we give an introduction to some of the mathematical aspects of gauge theory. In particular we will introduce the concept of a connection, which allows one to define a notion of curvature and a notion of parallel transport in an arbitrary vector bundle. In this context we will treat Yang-Mills theory and Chern-Simons theory, where the notion of a connection is a fundamental one. We will introduce the notion of a principal fiber bundle, which is the right tool to formalize the notion of local symmetries, which is central in gauge theory in physics. After this, we will consider 4-dimensional BF-theory. First from the point of view of canonical quantization and Feynman path-integral quantization, and then from a categorical perspective. From the categorical perspective we will find a simple criterion that characterizes 4-dimensional BF-theory up to isomorphism. In Part 2 we start with a description and an explanation of the integer quantum Hall effect. The explanation we give starts from a description of a single electron in a magnetic field. If the magnetic field is extremely strong, then (in a quantum mechanical description) the coordinates of the electron cease to commute. After this we introduce the independent oscillator model for Brownian motion and quantize it. It turns out that there is an interesting way to combine this quantum mechanical version of Brownian motion, with the theory of an electron in a strong magnetic field. We compute the mean squared displacement of the particle as a function of time for this system. Consequently, we analyse the result in two different regimes for the friction constant, associated with the Brownian motion. If the friction constant is very large, then the result is very similar to classical Brownian motion. If, however, the friction constant is small, then the movement of the particle is suppressed. In particular, the mean squared displacement grows linearly with time, and the coefficient is proportional to the friction constant. We conclude with a final consideration of the integer quantum Hall effect, and in particular we point out a connection between the integer quantum Hall effect and the Chern class of a vector bundle.