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For $F={\mathrm{vol}}_{012}\wedge d{H}^{-1}$ and $F=dA$, choose $A=(1-H^{-1}){\mathrm{vol}}_{012}$. A parallel probe then has $q=-1$ in the supermembrane action, so its induced-volume term $-T_2{H}^{-1}$ and potential term $-T_2(1-H^{-1})$ add to a position-independent constant. This proves the cancellation without a weak-field approximation. An oppositely charged probe has no such cancellation.

On a bosonic configuration, the target-space supersymmetry transformation changes the fermionic coordinate by ${\delta }_{ϵ}\Theta =ϵ$, where $ϵ$ is a background Killing spinor. A kappa symmetry convention isThe configuration preserves a supersymmetry if this change can be cancelled by a local kappa symmetry transformation. Multiplying $ϵ+(1+{\Gamma }_{\kappa })\kappa =0$ by $1-{\Gamma }_{\kappa }$ proves necessity ofConversely, if this holds, choosing $\kappa =-ϵ/2$ cancels the variation. Reversing the brane orientation reverses the corresponding kappa symmetry projector.

The preserved parameters must solve this equation everywhere on the brane and satisfy the background Killing spinor equations. Thus the fraction of all thirty-two eleven-dimensional supersymmetries isOne traceless constant kappa symmetry projector in flat space preserves sixteen parameters. Further independent commuting projectors often halve that number again, but a position-dependent projector requires a common global solution; its pointwise rank alone does not determine the answer. To quote a fraction of the background supersymmetry, divide instead by the number of its Killing spinors.

For the supermembrane, let $Z^{\mathcal{M}}=(X^M,{\Theta }^{\alpha })$ be the embedding in eleven-dimensional superspace. Define the pulled-back supervielbein, the induced worldvolume metric, and induced gamma matrices byFor a chosen orientation,The Clifford algebra shows that ${\Gamma }_{\kappa }^2=1$; in an orthonormal membrane frame this reduces to $({\Gamma }_0{\Gamma }_1{\Gamma }_2{)}^2=1$. Its trace is zero, so its two eigenspaces each have real dimension sixteen.

The full supermembrane action in an on-shell eleven-dimensional supergravity superspace isHere ${\mathcal{A}}_3$ is the super-three-form and $Z^{\ast }$ denotes the pullback of a differential form. Its bosonic restriction is the ordinary supergravity potential, but its fermionic components are retained in this action. The magnitude of its Wess-Zumino brane coupling equals the membrane tension; the sign $q$ fixes the orientation and the matching sign of ${\Gamma }_{\kappa }$. In flat superspace one may take ${\Pi }^a=d{X}^a-i\overline{\Theta }{\Gamma }^{a}d\Theta$, with the super-three-form chosen to obey the standard supergravity superspace constraints. Thus this is a fermionic, kappa symmetry-invariant action, rather than merely its bosonic truncation.

To construct an ordinary calibration, first take a flux-free static background $d{s}^2=-d{t}^2+g_{mn}d{x}^{m}d{x}^n$ with a unit covariantly constant spinor $ϵ$. In a compatible orientation and mostly-plus Clifford algebra, the membrane spinor calibration form isFor an oriented orthonormal spatial pair $u,v$, $\phi (u,v)={ϵ}^{†}{\Gamma }_0\gamma (u)\gamma (v)ϵ$. The matrix on the right is the static membrane ${\Gamma }_{\kappa }$, a Hermitian involution, so this value is at most one, with equality precisely when ${\Gamma }_{\kappa }ϵ=ϵ$. Parallel transport of the spinor and gamma matrices gives $\nabla \phi =0$, hence $d\phi =0$.

A calibration is a closed differential form whose value on each oriented unit tangent plane is at most one; its comass is at most one. A calibrated submanifold $\Sigma$ saturates the bound, $\phi {|}_{\Sigma }={\mathrm{vol}}_{\Sigma }$. For any homologous competitor ${\Sigma }^{\prime }$ with the same boundary, Stokes theorem givesThis proves that calibration implies volume minimization, and in particular that the membrane's spatial surface is a minimal surface. The closure condition is indispensable. A general flux-coupled Killing spinor need not produce a closed ordinary calibration; the corresponding generalized calibration bounds the full brane energy including its potential coupling. The flux-free construction above is the ordinary volume-minimizing case.

For the flux-coupled membrane, the standard spinor bilinears give a Killing vector $K$ and a two-form $\Omega$ satisfying $d\Omega ={\iota }_{K}F$, in matching supergravity conventions. In a stationary gauge ${\mathcal{L}}_{K}A=0$, Cartan's magic formula givesThis closed charge form provides the membrane generalized calibration; its potential term is what changes the volume bound into an energy bound.

For bosonic embedding coordinates $X^M(\sigma )$, the supermembrane action truncates toThe first term is its induced worldvolume volume and the second is its Wess-Zumino brane coupling. A membrane of the opposite orientation reverses $q$.

The M2-brane supergravity solution admits the harmonic functionwith arbitrary centre positions $y_a$ in eight transverse dimensions. Away from the sources, the Laplacian of each term is zero. The unrestricted positions describe static separated membranes carrying aligned charges, so their separations have no potential. Physically, their gravitational attraction is cancelled by their four-form charge repulsion. This is the M2-brane probe no-force identity, which can also be checked directly without relying only on the existence of the multicentre solution.

Keep the field-strength ordering printed in the paper and use $F=dA$. If ${\mathrm{vol}}_{012}=d{x}^0\wedge d{x}^1\wedge d{x}^2$, a gauge vanishing at infinity isThe last equality uses the sign acquired by moving a one-form past a three-form. For this convention the parallel probe has $q=-1$. In static gauge, at a fixed transverse position, its induced worldvolume metric is $h_{ij}=H^{-2/3}{\eta }_{ij}$, hence $\sqrt{-\det h}=H^{-1}$ andUsing the opposite convention for $F$, $A$ and orientation together yields the same cancellation. Changing only the probe charge produces an anti-membrane, with nonconstant potential $T_2(2H^{-1}-1)$. Thus the orientation cannot be dropped from the argument.

The cancellation also survives an expansion for slow transverse motion. With $v^2=|\dot{y}{|}^2$,There is no static potential and no position dependence in the leading kinetic coefficient.

Near an isolated positive membrane charge, write $H\sim R^6/r^6$ with $r=|y-y_a|$ and $R^6=Q_a$. The regular membrane near-horizon limit removes the asymptotically flat constant and givesSet $z=R^3/(2r^2)$. ThenThus the Anti-de Sitter spacetime radius is half the sphere radius. In orthonormal units the four-form has magnitude $|F|=6/R$ along the AdS volume form, the Freund-Rubin compactification flux.

The full asymptotically flat solution has spinors $ϵ=H^{-1/6}{ϵ}_0$ subject to one oriented membrane projector, leaving sixteen real parameters. In the throat, the Killing spinor equation separates into the standard AdS and sphere equations with correlated signs fixed by the flux. There are four real AdS${}_4$ solutions and eight real $S^7$ solutions for the required signs, soThis is maximal supersymmetry of AdS4 times S7: the additional sixteen spinors need not extend through the asymptotically flat region. A multicentre geometry therefore remains globally sixteen-supersymmetric even though each regular local throat has this enhancement. The classical supergravity description is controlled when $R$ is large compared with the eleven-dimensional Planck length; singular harmonic multipoles not representing regular positive membrane centres do not automatically have the same throat.