Algebraic structure of Galilean superconformal symmetries

Abstract

The semisimple part of $d$-dimensional Galilean conformal algebra ${g}^{(d)}$ is given by ${h}^{(d)}=\phantom{\rule{0ex}{0ex}}O(2,1)\ensuremath{\bigoplus}O(d)$, which after adding via a semidirect sum the $3d$-dimensional Abelian algebra ${t}^{(d)}$ of translations, Galilean boosts, and constant accelerations completes the construction. We obtain Galilean superconformal algebra ${G}^{(d)}$ by first defining the semisimple superalgebra ${H}^{(d)}$ which supersymmetrizes ${h}^{(d)}$, and further by considering the expansion of ${H}^{(d)}$ by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of ${t}^{(d)}$. For $d=3$ the supersymmetrization of ${h}^{(3)}$ is linked with a specific model of $\mathcal{N}=4$ extended superconformal mechanics, which is described by the superalgebra $D(2,1;\ensuremath{\alpha})$ if $\ensuremath{\alpha}=1$. We shall also present the alternative derivations of extended Galilean superconformal algebras for $1\ensuremath{\le}d\ensuremath{\le}5$ by employing the In\"on\"u-Wigner contraction method.