Calibration (differential geometry, $d\phi =0,\Vert \phi {\Vert }_{\ast }\le 1$)

A calibration is a closed real differential form whose comass is at most one. Restricting it to any oriented unit tangent plane therefore gives at most that plane's volume. A calibrated submanifold attains equality everywhere, and calibration implies volume minimization among homologous competitors.

In a flux-free static supergravity background with a parallel unit Killing spinor, this two-form satisfies the membrane tangent-plane bound because ${\Gamma }_0\gamma (u)\gamma (v)$ is a Hermitian involution. Its eigenvalues are $\pm 1$, so $\phi (u,v)\le 1$, with equality precisely at the kappa symmetry projector condition. Parallel transport gives $\nabla \phi =0$ and hence $d\phi =0$. This constructs an ordinary calibration; flux backgrounds instead call for a generalized calibration.