BPS states and automorphisms

The purpose of the present paper is twofold. In the first part, we provide an algebraic characterization of several families of $\nu {=1/2\mathrm{}}^n$ $n<~5$ Bogomol’nyi-Prasad-Sommerfield (BPS) states in M theory, at threshold and non-threshold, by an analysis of the BPS bound derived from the $\mathcal{N}=1\mathrm{}$ $D=11\mathrm{}$ super Poincaré algebra. We determine their BPS masses and their supersymmetry projection conditions, explicitly. In the second part, we develop an algebraic formulation to study the way BPS states transform under $GL(32,R)$ transformations, the group of automorphisms of the corresponding super Poincaré algebra. We prove that all $\nu =\frac{1}{2}$ non-threshold bound states are $\mathrm{SO}\left(32\right)\mathrm{}$ related with $\nu =\frac{1}{2}$ BPS states at threshold having the same mass. We provide further examples of this phenomena for less supersymmetric $\nu =\frac{1}{4},\frac{1}{8}$ non-threshold bound states.