Aspects of Chern-Simons Theory

TL;DR

This set of Les Houches lectures provides a comprehensive account of Chern-Simons gauge theory in 2+1 dimensions, emphasizing how a first-order CS term yields topological mass, anyonic statistics, and nontrivial boundary dynamics. The work develops both canonical and functional quantization, explores abelian and nonabelian CS theories, and analyzes CS vortices across relativistic and nonrelativistic settings, including their moduli spaces and dynamics. A central theme is the radiative generation of CS terms and the subtle role of finite temperature, where large-gauge invariance and nonextensive finite-T contributions require careful nonperturbative treatment. Collectively, the material illuminates the deep connections between CS theory, topological phenomena, edge states in quantum Hall systems, and the quantum structure of planar field theories with broad physical relevance.

Abstract

Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.

Figures (9)

• Figure 1: A collection of point anyons with charge $e$, and with magnetic flux lines of strength $\frac{e}{\kappa}$ tied to the charges. The charge and flux are tied together throughout the motion of the particles as a result of the Chern-Simons equations (\ref{['cscomps']}).
• Figure 2: Aharonov-Bohm interaction between the charge $e$ of an anyon and the flux $\frac{e}{\kappa}$ of another anyon under double-interchange. Under such an adiabatic transport, the multi-anyon wavefunction acquires an Aharonov-Bohm phase (\ref{['abp']}).
• Figure 3: The energy spectrum for charged particles in a uniform magnetic field consists of equally spaced 'Landau levels', separated by $\hbar \omega_c$ where $\omega_c$ is the cyclotron frequency. Each Landau level has degeneracy given by the total magnetic flux through the sample.
• Figure 4: The torus can be parametrized as a parallelogram with sides $\tau$ and $1$. There are two cycles $\alpha$ and $\beta$ representing the two independent non-contractible loops on the surface.
• Figure 5: The self-dual quartic potential $\frac{\lambda}{4}\left(|\phi|^2-v^2\right)^2$ for the Abelian-Higgs model. The vacuum manifold is $|\phi|=v$.