Galilean Superconformal Symmetries

TL;DR

The paper tackles constructing a non-relativistic, $N$-extended (with $N=2k$) Galilean superconformal symmetry in $D=4$ by performing a contraction $c \to \infty$ of the relativistic $N$-extended superconformal algebra $su(2,2|N)$. It develops a systematic contraction that preserves a rich internal structure, identifying the non-relativistic internal algebra as $\mathrm{usp}(2k)$ with the coset $u(2k)/\mathrm{usp}(2k)$ giving central charges, and introducing projected supercharges $Q^a_{\pm\alpha}$ and $S^a_{\pm\alpha}$ to realize the fermionic sector in the Galilean limit. The resulting algebra combines the $D=4$ Galilean conformal spacetime sector with an internal non-semisimple symmetry, yielding explicit fermionic, mixed, and bosonic commutators; the internal sector becomes Abelian in the contracted part, while the full structure remains non-semisimple. This framework enables potential non-relativistic superconformal mechanics and quantum realizations, while clarifying constraints on $N$ and connecting to related work by Sakaguchi.

Abstract

We consider the non-relativistic c -> \infty contraction limit of the (N=2k)- extended D=4 superconformal algebra su(2,2;N), introducing in this way the non-relativistic (N=2k)-extended Galilean superconformal algebra. Such a Galilean superconformal algebra has the same number of generators as su(2,2|2k). The usp(2k) algebra describes the non-relativistic internal symmetries, and the generators from the coset u(2k)/usp(2k) become central charges after contraction.