Calibrated geometry

Calibrated geometry constructs volume-minimizing submanifolds through closed differential forms with a tangent-plane bound. It gives an algebraic sufficient condition for solving the nonlinear minimal surface problem.

A generalized calibration bounds the full energy of a supersymmetric brane, including its potential or gauge couplings. For a membrane, compatible spinor bilinears obey $d\Omega ={\iota }_{K}F$. In a stationary gauge ${\mathcal{L}}_{K}A=0$, Cartan's magic formula proves $d(\Omega +{\iota }_{K}A)=0$. The closed charge form then supplies the energy bound. This extends ordinary calibration from volume alone to charged branes, as developed in Topological charges for branes in M-theory.

An oriented submanifold is calibrated by a calibration when its tangent planes attain the comass bound at every point. Closedness then compares its volume with homologous competitors by Stokes theorem.

If $\Sigma$ is calibrated by $\phi$ and ${\Sigma }^{\prime }$ is homologous with the same boundary, Stokes theorem gives ${\int }_{\Sigma }\phi ={\int }_{{\Sigma }^{\prime }}\phi$. The tangent-plane bound gives ${\int }_{{\Sigma }^{\prime }}\phi \le \mathrm{Vol}({\Sigma }^{\prime })$, while calibration gives ${\int }_{\Sigma }\phi =\mathrm{Vol}(\Sigma )$. Thus $\Sigma$ minimizes volume in its relative homology class and is a minimal submanifold wherever smooth.

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.