Mass deformation of Bagger-Lambert theory in 3D with reducedN=1supersymmetry

Abstract

We introduce $SO(7{)}_{\mathrm{global}}$ symmetric mass terms into Bagger-Lambert theory in three dimensions. The scalar field $X_{a}{}^{I}$ and its fermionic partner ${\ensuremath{\psi}}_{a}$ are provided with the mass $m$, while a quartic interaction term is induced. These new terms explicitly break the original $SO(8{)}_{\mathrm{global}}$ symmetry down to $SO(7{)}_{\mathrm{global}}$ in terms of octonion structure constants. The original supersymmetry parameter in the ${\mathbf{8}}_{\mathrm{S}}$ is reduced to a singlet, implying that the original $N=8$ supersymmetry is reduced to $N=1$ supersymmetry. As illustrations, we present some nontrivial vacuum configurations with the breaking $SO(7{)}_{\mathrm{global}}\ensuremath{\rightarrow}SO(4{)}_{\mathrm{global}}$ or smaller symmetry groups. Interestingly, we also find that after a nontrivial vacuum expectation value $⟨X_{a}{}^{I}⟩$ is developed, the vector field $A_{\ensuremath{\mu}}{}^{ab}$ satisfies a ``self-duality'' condition, and starts propagating with a mass. These results are due to the special nature of the octonion structure constants.