Introduction to Chern-Simons theories

Abstract

Chern-Simons theories occur in Physics more often than one may think. In particular, the Hamiltonian action for any mechanical system with finite number of degrees of freedom is a CS Lagrangian. CS forms can be seen as generalizations of the coupling between the electromagnetic field and a point charge. It should come as no surprise, therefore, that the CS forms are perfectly suited to describe the consistent coupling of a (nonabelian) gauge connection to extended charged objects (branes). An application of this is provided by a spacetime region M where the electromagnetic field is described by the Lagrangian 1/4(F Λ∗ F +θF ΛF). Since the Maxwell equations are unaffected by the presence of the θ-term, this represents a slight modification of the vacuum in M (θ-vacuum). This modification is shown to correspond to the addition of a form of matter that couples through a CS form on the surface of M. The connection to superconductivity is also briefly mentioned.