Extended Supersymmetric BMS$_3$ algebras and Their Free Field Realisations

TL;DR

This work classifies N=(2,4,8) supersymmetric extensions of the three-dimensional BMS algebra (BMS3) via Inönü–Wigner contractions of two copies of extended superconformal algebras. It demonstrates that consistent SBMS3 algebras require asymmetric scaling of bosonic generators while fermionic generators are scaled symmetrically, and that the BMS/GCA correspondence does not extend to supersymmetric cases. The authors provide explicit free-field realizations using β-γ and b-c systems, revealing independent central charges in the contracted algebras and detailing N=2, N=4, and N=8 SBMS3 structures, including the presence of R-symmetries and their impact on the algebra. The results highlight fundamental differences between BMS3 and superconformal contractions, and establish concrete constructions for extended SBMS3 algebras with physically meaningful subalgebras such as the 3D super-Poincaré.

Abstract

We study $N=(2,4,8)$ supersymmetric extensions of the three dimensional BMS algebra (BMS$_3$) with most generic possible central extensions. We find that $N$-extended supersymmetric BMS$_3$ algebras can be derived by a suitable contraction of two copies of the extended superconformal algebras. Extended algebras from all the consistent contractions are obtained by scaling left-moving and right-moving supersymmetry generators symmetrically, while Virasoro and R-symmetry generators are scaled asymmetrically. On the way, we find that the BMS/GCA correspondence does not in general hold for supersymmetric systems. Using the $β$-$γ$ and the ${\mathfrak b}$-${\mathfrak c}$ systems, we construct free field realisations of all the extended super-BMS$_3$ algebras.