BPS states of D=4 N=1 supersymmetry

TL;DR

This work classifies BPS states in D=4, N=1 supersymmetry by analyzing the positive semidefinite matrix $\\{Q,Q\\}$ in the space of real symmetric $4\\times4$ matrices, uncovering a quartic invariant $P(H)$ whose roots determine preserved fractions of SUSY and linking these states to a self-dual convex cone and Jordan algebra structure. It provides model-independent constraints for 1/4, 1/2, and 3/4 SUSY, proves the stability of BPS states via a reverse-triangle inequality, and examines intersecting domain walls, including a no-go result for 3/4 SUSY in the Wess-Zumino model. The geometry is framed in terms of convex cones and Jordan algebras, with a detailed extension to AdS$_4$ where an $osp(1|4;\mathbb{R})$-like structure yields parallel conclusions and new AdS examples. The results illuminate how BPS configurations organize themselves on stratified boundary faces corresponding to different SUSY fractions, and they offer a unifying language for potential applications in AdS/CFT and M-theory contexts.

Abstract

We find the combinations of momentum and domain-wall charges corresponding to BPS states preserving 1/4, 1/2 or 3/4 of D=4 N=1 supersymmetry, and we show how the supersymmetry algebra implies their stability. These states form the boundary of the convex cone associated with the Jordan algebra of $4\times 4$ real symmetric matrices, and we explore some implications of the associated geometry. For the Wess-Zumino model we derive the conditions for preservation of 1/4 supersymmetry when one of two parallel domain-walls is rotated and in addition show that this model does not admit any classical configurations with 3/4 supersymmetry. Our analysis also provides information about BPS states of N=1 D=4 anti-de Sitter supersymmetry.