Supersymmetric Chern-Simons Theory in Presence of a Boundary in the Light-Like Direction

TL;DR

This paper studies a three-dimensional supersymmetric Chern-Simons theory with a light-like boundary, showing how the boundary preserves a subgroup of Lorentz symmetry, reduces supersymmetry, and can be made gauge invariant via boundary degrees of freedom.

Contribution

It analyzes the effects of a light-like boundary on supersymmetric Chern-Simons theory, revealing symmetry breaking and gauge invariance restoration mechanisms.

Findings

Boundary preserves $SIM(1)$ symmetry subgroup.

Supersymmetry is reduced from $ =1$ to $ =1/2$.

Gauge invariance is restored by boundary degrees of freedom.

Abstract

In this paper, we will analyze a three dimensional supersymmetric Chern-Simons theory on a manifold with a boundary. The boundary we will consider in this paper will be defined by $n\cdot x=0$, where $n$ is a light-like vector. It will be demonstrated that this boundary is preserved under the action of the $SIM(1)$ subgroup of the Lorentz group. Furthermore, the presence of this boundary will break half of the supersymmetry of the original theory. As the original Chern-Simons theory had $\mathcal{N} =1$ supersymmetry in absence of a boundary, it will only have $\mathcal{N}=1/2$ supersymmetry in presence of this boundary. We will also observe that the Chern-Simons theory can be made gauge invariant by introducing new degrees of freedom on the boundary. The gauge transformation of these new degrees of freedom will exactly cancel the boundary term obtained from the gauge transformation of the Chern-Simons theory.