Chern–Simons quantum field theories

Abstract

Quantum field theory is the most efficient tool we have to describe elementary particles. It is the backbone of the Standard Model that successfully explains electromagnetic, weak and strong interactions. For example, quantum electrodynamics, which describes the interactions between charged fermions mediated by photons within the field theory framework, has been tested experimentally and verified to a high level of accuracy. This has established quantum field theory as the definitive tool for studying high-energy physics. Apart from explaining the fundamental properties of matter, quantum field theories can also provide an effective description of condensed matter systems. In Chapter 6 we saw that the quantum field theory of Majorana fermions emerges from Kitaev's honeycomb lattice model. It is expected that the low-energy behaviour of the fractional quantum Hall effect can be described efficiently by certain quantum field theories, known as Chern–Simons theories (Froehlich et al. , 1997). The fractional quantum Hall effect emerges in interacting two-dimensional electron gases at low temperature in the presence of a strong magnetic field. Due to its complexity, this system evades exact theoretical analysis. An effective description with Chern–Simons theories has nevertheless proven very fruitful in understanding its topological properties. Chern–Simons theories are topological quantum field theories in the sense that all their observables are invariant under continuous coordinate transformations. In other words, relative distances, and subsequently local geometry, do not play a role in these theories.