OSp(N|4) group and their contractions to P(3,1)xGauge

Abstract

Starting from SO(n,m) groups, we are in search of groups that: (1.) in a simple way, include N supersymmetric generators(see \cite{Nahm}) (2.) contain as subgroup: the de Sitter group SO(4,1) or the Anti-de Sitter group SO(3,2) (3.) permit nontrivial gauge symmetry groups. The smallest groups satisfyng above conditions are the OSp(N|4) groups, which contain Sp(4)xSO(N) ($Sp(4)\sim SO(3,2)$) or OSp(1|4)xSO(N-1). Because of this, it is possible to generate P(3,1)xG using groups contraction mechanism, which may be: $SO(3,2)\to P(3,1)$ o $OSp(N|4)\to ^SP(3,1|N)$ where P(3,1) is the Poincar\'e group and G is a gauge group, say SO(N) or SO(N-1). This group contraction mechanism and its consequences upon different groups representations including SO(3,2) or SO(4,1), is clarified and extended to OSp(N|4) representations (see \cite{Nicolai}), contracted to its N-extensi\'on SuperPoincar\'e group $\ ^SP(3,1|N)$.