Topological charges for branes in M-theory

TL;DR

The paper develops a universal framework for the $M2$ and $M5$ brane supersymmetry algebras in arbitrary supersymmetric backgrounds of $11$-dimensional supergravity by constructing closed forms from Killing spinors: a 2-form $ω+\boldsymbol{ι}_K A$ for membranes and a 5-form $Σ+\boldsymbol{ι}_K C + A\wedge(ω+\tfrac{1}{2}\boldsymbol{ι}_K A)$ (extended with a worldvolume field) for five-branes. These closed forms provide topological charges that enforce BPS bounds on brane energies and generalize calibration forms beyond special holonomy. The authors derive explicit algebraic and differential relations from Killing spinor equations, show the Kovalenko–Lichnerowicz-type structure ensuring consistency of the algebras, and illustrate the construction with a concrete M5-brane background, displaying how gauge choices affect the explicit form of the charges. This work unifies the treatment of M2/M5 brane charges in general backgrounds and offers a practical route to compute brane BPS conditions in complex flux geometries.

Abstract

We propose a simple form for the superalgebra of M2 and M5-brane probes in arbitrary supersymmetric backgrounds of 11D supergravity, extending previous results in the literature. In particular, we identify the topological charges in the algebras and find BPS bounds for the energies. The charges are given by the integral over a brane's spatial worldvolume of a certain closed form built out of the Killing spinors and background fields. The existence of such closed forms for arbitrary supersymmetric backgrounds generalises the existence of calibration forms for special holonomy manifolds.