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 $\nu =\frac{1}{2^n}$ $n\leq 5$ BPS states in M theory, at threshold and non-threshold, by an analysis of the BPS bound derived from the $\mathcal{N}=1$ D=11 SuperPoincar'e 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 $GL(32,\text{bR})$ transformations, the group of automorphisms of the corresponding SuperPoincar'e algebra. We prove that all $\nu =1\mathrm{/}2$ non-threshold bound states are SO(32) related with $\nu =1\mathrm{/}2$ BPS states at threshold having the same mass. We provide further examples of this phenomena for less supersymmetric $\nu =1\mathrm{/}4,1\mathrm{/}8$ non-threshold bound states.