Optical properties of a $\theta$ vacuum

Chern-Simons (CS) forms generalize the minimal coupling between gauge potentials and point charges to sources represented by charged extended objects (branes). The simplest example of such a CS-brane coupling is a domain wall coupled to the electromagnetic CS three-form. This describes a topologically charged interface where the CS form $A\wedge dA$ is supported, separating two three-dimensional spatial regions in $3+1$ spacetime. Electrodynamics at either side of the brane is described by the same Maxwell’s equations, but those two regions have different vacua characterized by a different value of the $\theta$-parameter multiplying the Pontryagin form $F\wedge F$. The $\theta$-term is the Abelian version of the concept introduced by ’t Hooft for the resolution of the $U(1)$ problem in QCD. We point out that CS-generalized classical electrodynamics show new phenomena when two neighboring regions with different $\theta$-vacua are present. These topological effects result from surface effects induced by the boundary, and we explore the consequences of such boundary effects for the propagation of the electromagnetic field in Maxwell theory. Several features including optical and electrostatic/magnetostatic responses, which may be observable in condensed matter systems like topological insulators, are discussed.