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For constant sectional curvature $-1/a^2$, spin curvature is $[{\nabla }_{\mu },{\nabla }_{\nu }]=-{\gamma }_{\mu \nu }/(2a^2)$. The added Clifford term in $\mathcal{D}$ has the opposite commutator, making $\mathcal{D}$ flat. On a simply connected region, parallel transport of any initial spinor is path independent and constructs solutions of $\mathcal{D}ϵ=0$. Thus Anti-de Sitter spacetime and its universal cover admit the associated Killing spinors locally and, with appropriate global spin data, globally.

With ${\Gamma }_{\kappa }^2=1$, the two complementary linear projections select its two eigenspaces. A bosonic brane preserves a background Killing spinor precisely when its target-space supersymmetry variation can be compensated by kappa symmetry, giving ${\Gamma }_{\kappa }ϵ=ϵ$. This condition must hold at every worldvolume point; several branes require a common solution. The full superspace construction is described in Superspace Geometry for Supermembrane Backgrounds.

A Killing spinor is a nonzero supersymmetry parameter that makes the fermionic transformations vanish on a bosonic background. In undeformed minimal four-dimensional supergravity it is parallel, ${\nabla }_{\mu }ϵ=0$; in its cosmological deformation the equation uses ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+(m/2){\gamma }_{\mu }$. Its commuting Dirac current is a Killing vector, since the spinor equation makes the symmetric covariant derivative of the current vanish. General supergravity theories also impose the vanishing of matter-fermion transformations.

The transverse harmonic function $H$ solves the field equations away from membrane sources. Positive centres contribute $Q_a/|y-y_a{|}^6$ in eight dimensions. Its Killing spinors are $H^{-1/6}{ϵ}_0$ with one oriented membrane projector, giving sixteen supersymmetries. The three-form's sign and membrane orientation must be chosen together.

With the membrane Freund-Rubin compactification flux and the required radius ratio, the Killing spinor equation separates into AdS4 and sphere equations with correlated signs. The respective real solution dimensions are four and eight, giving thirty-two. The associated Killing-spinor connections have vanishing curvature for these choices, so arbitrary initial spinor data generate solutions locally; standard global spin structures give the usual maximally supersymmetric background.

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A D-brane is a localized RR-charged source whose long-distance fields solve supergravity. The extremal charged $p$-brane solutions carry the same RR charges, brane tension and preserved supersymmetry as the open-string construction. For the standard asymptotically flat cases $p<7$, the D-brane supergravity solution in the string frame isThe function $H$ is a positive harmonic function away from the source, in $9-p$ transverse dimensions. In an electric gauge, $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$, with its orientation chosen to match the source. Magnetic branes use the dual flux; for the D3-brane, the five-form includes both electric and magnetic pieces and is self-dual. Killing spinors obey one brane projector, leaving sixteen supersymmetries. Aligned multicenter harmonic functions describe several branes with no static binding force.

To check this identification directly, insert an equally oriented static probe into the solution. Put ${\mu }_p=(2\pi {)}^{-p}{\alpha }^{\prime -(p+1)/2}$, so the asymptotic tension is ${\mu }_p/g_s$. The induced-volume factor is $H^{-(p+1)/4}$; combined with $e^{-\Phi }$, its DBI action density is $-{\mu }_p{g}_s^{-1}{H}^{-1}$. Its Wess-Zumino brane coupling contributes $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$. Hence the total static Lagrangian density is the constant $-{\mu }_p/g_s$, independently of position. This gives the parallel D-brane no-force identity in the spacetime description.

The supergravity background is the back-reacted, coarse long-distance description, while the boundary-condition description exposes the light open-string degrees of freedom. Controlled use of supergravity requires curvature small in string units and suitable local coupling. Singular cores and the special codimension-two or domain-wall cases are not automatically resolved by the elementary harmonic ansatz.

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.

Use static gauge for a D3-brane in flat space and let $X(\mathbf{x})$ be one physical transverse displacement. Put $\mathbf{E}=2\pi {\alpha }^{\prime }{\mathbf{F}}_0$ and ${\mathbf{B}}_i=\pi {\alpha }^{\prime }{\epsilon }_{ijk}{F}_{jk}$, so the fields entering the DBI action are dimensionless. With no magnetic field, writing $\mathbf{v}=\nabla X$ givesIntroduce the dimensionless electric displacement in DBI theory, $\mathbf{d}=T_3^{-1}\partial \mathcal{L}/\partial \mathbf{E}$. Its Hamiltonian density obeys the DBI Hamiltonian bound for an electric spike:The electric bound is saturated when $\mathbf{d}=\pm \nabla X$. On that locus the constitutive relation also gives $\mathbf{E}=\mathbf{d}$. With the Gauss law constraint in gauge theory away from charges, the electric BPS equations for a one-scalar D3-brane are thereforeThe magnetic version is $\mathbf{B}=\pm \nabla X$, $\mathbf{E}=0$. More generally, one constant charge angle gives the aligned dyonic branch $\mathbf{E}=\cos \vartheta \nabla X$, $\mathbf{B}=\sin \vartheta \nabla X$ with $X$ harmonic. A single aligned electric, magnetic or dyonic spike preserves eight of the D3 vacuum's sixteen supersymmetry generators. Thus “one-half BPS” here means one-half of the linearly realized worldvolume supersymmetry; the string–D3 configuration preserves one-quarter of the thirty-two spacetime supersymmetries.

The electric BIon is the single-centre harmonic solutionThe sign fixes its electric charge and spike orientation. The scalar is an actual transverse coordinate, so $X\to \infty$ as $r\to 0$ describes a long narrow spike emerging from an asymptotically planar D3-brane. It carries $|N|$ fundamental strings ending on the brane. A magnetic BIon similarly represents D1-branes, and an aligned dyonic BIon represents a (p,q) string.

The string-tension interpretation of BIon energy follows explicitly. With $T_3=[(2\pi {)}^3{g}_s{\alpha }^{\prime 2}{]}^{-1}$, the energy above the planar-brane vacuum outside a small radius $r_0$ isIts divergence is the energy of a semi-infinite string, with the correct finite tension. For a magnetic spike, $c=\pi |N|{\alpha }^{\prime }$ instead, and the same calculation gives the D1-brane tension $1/(2\pi g_s{\alpha }^{\prime })$. The solution solves the full nonlinear DBI action, not just its weak-field expansion.

The validity of the abelian BIon approximation concerns its derivative expansion, string loops and gravitational backreaction. Large field strengths are included by the DBI action, but rapidly varying field strengths and embedding curvature require corrections. Far outside the singular core these corrections can be small. The transition region has $r\sim \sqrt{c}$; an electric throat can be parametrically controlled with $|N|g_s\gg 1$ while keeping gravitational backreaction small with $g_s^2|N|\ll 1$. For a magnetic throat the corresponding conditions are $|N|\gg 1$ and $g_s|N|\ll 1$. The weak-coupling and large-charge windows allow both requirements. The ideal $r\to 0$ tip is not uniformly controlled by the single abelian effective action, so one should not infer exact microscopic core physics solely from its formal singularity.

Less supersymmetry survives when independent, compatible geometric or charge projectors must hold simultaneously. A useful purely geometric family is a D3-brane whose spatial worldvolume is a special Lagrangian submanifold of ${\mathbb{C}}^3$. In a graph $z^i=x^i+i{Y}^i(\mathbf{x})$, the conditions areThe first condition makes ${\partial }_i{Y}^j$ symmetric, so locally $Y=\nabla f$. Pulling back $\Omega$ gives $\det (I+i{D}^{2}f)d{x}^1\wedge d{x}^2\wedge d{x}^3$. Hence the phase-zero special Lagrangian graph equation iswhere ${\sigma }_2$ is the sum of the pairwise products of the three Hessian eigenvalues. The positive real branch fixes the orientation. These curved D3 geometries minimize volume by their calibration $\mathrm{Re}\Omega$ and generically preserve four spacetime supersymmetries, one-quarter of the D3 worldvolume supersymmetry. Special flat graphs can preserve more.

The generic count follows from the common tangent-plane projectors. With complex coordinate pairs $(1,4),(2,5),(3,6)$, the phase-preserving rotations require $({\Gamma }_{14}-{\Gamma }_{25})ϵ=0$ and $({\Gamma }_{14}-{\Gamma }_{36})ϵ=0$. Multiplication by ${\Gamma }_{14}$ turns these into two independent commuting involution conditions with eigenvalue $-1$. Each halves the sixteen-dimensional planar D3 eigenspace, leaving $16/2^2=4$. This is an intersection of compatible kappa symmetry projectors, rather than a count based on the rank of a single projector at one point.

Charge-carrying examples include compatible networks of differently oriented (p,q) strings ending on D3-branes. At each string junction, charge vectors add to zero and tension vectors balance; their directions and charge phases must also give common kappa symmetry projectors. They can impose further independent conditions beyond the planar D3 projector. Geometrically, several transverse scalar profiles describe the corresponding spikes and intersections. Merely replacing one electric spike by an aligned dyonic spike does not lower its preserved fraction; additional independent conditions are what reduce the common Killing spinor space.

On a bosonic background, a Killing spinor is a nonzero spinor parameter that makes all fermionic supersymmetry variations vanish. It identifies an unbroken supersymmetry of the background; the number of independent solutions counts preserved supercharges. In simple ungauged minimal four-dimensional supergravity, with the gravitino set to zero, its equation is ${\nabla }_{\mu }ϵ=0$ and the bosonic field equations are $R_{\mu \nu }=0$. In theories with additional fields or a cosmological deformation, the Killing-spinor equation instead uses the corresponding modified connection.

Use commuting spinors to form ordinary geometric bilinears, and orient the Dirac current to the future with $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$ in mostly-plus signature. Since the spinor and gamma matrices are parallel,Thus $K$ is actually parallel, and in particular is a Killing vector field. In a local orthonormal frame,For every spatial unit vector $u$, $u_i{\alpha }^i$ is Hermitian and squares to one. Its eigenvalues are $\pm 1$, so $|u_i{K}^i|\le K^0$. Taking $u$ along the spatial current givesA nonzero parallel spinor cannot vanish at an isolated point, so this causal Dirac current is nonzero everywhere on a connected solution. Its squared norm is constant: it cannot alternate between timelike and null regions.

If $K$ is timelike, normalize its constant norm. Its parallel dual one-form is closed, and its orthogonal distribution is integrable. Adapted local coordinates put the metric in the product formThe time independence follows because $K$ is Killing, and the unit lapse and zero cross terms follow from parallelism. The vacuum equations give $R_{ij}(h)=0$. In three dimensions the Riemann tensor is completely determined by the Ricci tensor and its scalar contraction, so $h$ is flat. ThereforeConstant spinors in its flat frame provide all local Killing spinors. Global identifications are possible only when they preserve a compatible spin structure and the required spinors.

If $K$ is null, its parallelism first gives a Brinkmann coordinate system with $K={\partial }_v$. The parallel-spinor condition further restricts the local spin holonomy to the stabilizer of that spinor. In four dimensions, for a nonzero Weyl component represented by $(1,0{)}^T$, its stabilizer in $SL(2,\mathbb{C})$ is the upper-triangular group $\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)$, $b\in \mathbb{C}$: the two real null rotations, with no transverse rotation. Consequently the transverse frame is flat and the allowed curvature is of the wave type $R_{uiuj}$. Using the coordinate freedom to remove transverse cross terms gives the local plane-fronted gravitational wave formHere $R_{uiuj}=-H_{ij}/2$ and the only potentially nonzero Ricci component is $R_{uu}=-(H_{xx}+H_{yy})/2$, so the last equation is exactly the vacuum equation. Conversely, in the coframe $e^{+}=du$, $e^{-}=dv-Hdu/2$, $e^i=d{x}^i$, the nonzero spin-connection matrices are proportional to $H_i{\gamma }^{+i}du$. Constant spinors with ${\gamma }^{+}ϵ=0$ are therefore parallel. Every harmonic transverse profile, with arbitrary $u$ dependence, gives such a supersymmetric vacuum wave; the flat profile is included. This verifies both the form and sufficiency in the parallel-spinor classification of vacuum four-geometries.

There is a reality-condition subtlety in interpreting the branches. A single commuting four-dimensional Majorana spinor has a null current of a commuting Majorana spinor: its two-component representation gives $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. A timelike current can instead come from a Dirac Killing spinor or a sum of independent real-spinor currents. The conditional timelike classification above remains valid, but does not assert a generic timelike current for a single real Majorana parameter.

For the final question use constant sectional curvature $-1/a^2$, so $R_{\mu \nu }=-3g_{\mu \nu }/a^2$. The spinor curvature identity givesDefine the Anti-de Sitter Killing-spinor connection ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+{\gamma }_{\mu }/(2a)$. Since ${\nabla }_{\mu }{\gamma }_{\nu }=0$ and the connection is torsion-free,Thus the modified connection is flat. On a simply connected region, transport any initial spinor with this connection; flatness makes the result path independent and supplies actual nonzero solutions, not merely a necessary integrability condition. Hencehas a full local space of solutions on Anti-de Sitter spacetime, and globally on its universal cover. Global quotients require compatible spinor monodromy. These are Killing spinors of the cosmological deformation, not parallel-spinor vacua of the undeformed zero-cosmological-constant theory.

A Killing spinor is a nonzero spinor parameter for which the fermionic supersymmetry variations vanish on a bosonic background. In minimal four-dimensional supergravity without a cosmological term this meansIt is an overdetermined equation and expresses an unbroken supersymmetry of the background. In the cosmological deformation the corresponding equation is ${\mathcal{D}}_{\mu }ϵ=0$; in theories with additional bosonic fields their contributions also enter the fermion variations.

For the undeformed theory the connection preserves both the spinor adjoint and gamma matrices. Consequently the Dirac currentobeys ${\nabla }_{\nu }{K}_{\mu }=0$. In particular ${\nabla }_{(\nu }{K}_{\mu )}=0$, so $K$ is a Killing vector, and in fact is parallel. For the real cosmological deformation, ${\nabla }_{\mu }ϵ=-(m/2){\gamma }_{\mu }ϵ$ and ${\nabla }_{\mu }\overline{ϵ}=(m/2)\overline{ϵ}{\gamma }_{\mu }$ instead give ${\nabla }_{\mu }{K}_{\nu }=(m/2)\overline{ϵ}[{\gamma }_{\mu },{\gamma }_{\nu }]ϵ$, whose symmetric part again vanishes.

The causal property uses an ordinary commuting spinor. Take an orthonormal frame with $({\gamma }^0{)}^{†}=-{\gamma }^0$, $({\gamma }^i{)}^{†}={\gamma }^i$, and $\overline{ϵ}={ϵ}^{†}{\gamma }^0$. Then $K^0=-{ϵ}^{†}ϵ$. Reverse its time orientation and write $V=-K$, soThe matrices ${\alpha }^i$ are Hermitian and obey $\{{\alpha }^i,{\alpha }^j\}=2{\delta }^{ij}$. For every spatial unit vector $u$, $(u_i{\alpha }^i{)}^2=1$, whence $|u_i{V}^i|\le V^0$. Choose $u$ along the spatial part of $V$ to obtainA nonzero Killing spinor cannot vanish at an isolated point: its defining connection equation transports its value along every curve. Thus on a connected background this is a nonzero, nowhere-spacelike causal Dirac spinor current. The estimate does not need the Majorana condition, but it does need commuting components; Grassmann bilinears have no such ordinary positivity order.

For the positive energy theorem, take a complete asymptotically flat spacelike hypersurface with a spin structure $\Sigma$, with suitable decay, Einstein constraint equations, and matter satisfying the dominant energy condition. Assume there are no inner boundaries, or that any inner-boundary contribution has the required nonnegative sign. The auxiliary commuting spinor approaches an arbitrary constant ${ϵ}_{\infty }$ at spatial infinity. Define the real Nester two-formAntisymmetry removes second derivatives except through the Riemann curvature tensor. Differentiation and the three-gamma spinor curvature identity used above giveThe first term comes from differentiating the two spinor factors. For the second term, replace ${\nabla }_{\nu }{\nabla }_{\rho }$ by half their commutator; its spin-curvature contraction is $G^{\mu \nu }{\gamma }_{\nu }/2$ from each conjugate term. This is the reason the Einstein tensor, and hence the matter energy density, enters the boundary-charge identity.

In an orthonormal frame adapted to the future normal of $\Sigma$, put ${\chi }_i={\nabla }_iϵ$. Since ${\gamma }^0{\gamma }^{0ij}=-{\gamma }^{ij}$,Here ${\nabla }_i$ is the spacetime spin connection pulled back to $\Sigma$, including its extrinsic curvature. Choose the auxiliary spinor to solve the elliptic Witten spinor equationThe standard asymptotic-flatness and completeness hypotheses permit this boundary-value problem; the positivity identity also excludes a decaying homogeneous zero mode. It is this hypersurface equation that removes the negative square. A spacetime Killing spinor is not assumed for a general initial-data set.

Stokes' theorem and $G_{\mu \nu }={\kappa }^2{T}_{\mu \nu }$ now giveThe dominant energy condition makes the second integrand nonnegative because $n$ and $V$ are future causal. Evaluating the two-form with the asymptotic spin connection identifies its surface integral with the ADM energy and momentum:For example its energy term uses $B^{0i}=\frac{1}{2}({\partial }_j{h}_{ij}-{\partial }_i{h}_{jj})|{ϵ}_{\infty }{|}^2$ to first order in the asymptotic spatial metric tensor perturbation. With ${\kappa }^2=8\pi G$ this is exactly the ADM normalization. Since $P_i{\alpha }^i$ has eigenvalues $\pm |\mathbf{P}|$, positivity for every boundary spinor provesKilling spinors describe saturation: their spatial derivative term vanishes, and the contracted matter term must also vanish for the corresponding charge to be zero. They characterize preserved supersymmetry rather than being prerequisites for positivity. Under the usual rigidity hypotheses, zero total four-momentum makes the nonnegative integral vanish for a full basis of asymptotic spinors; their parallel extensions force a vanishing Riemann curvature tensor and give the Minkowski vacuum. A single Killing spinor with null current alone does not justify asserting that every component of the total momentum vanishes.

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.

At a junction of (p,q) strings, their oriented charge vectors sum to zero. A static junction also balances the tension vectors. Supersymmetry further correlates spatial directions with charge phases so that the arms share kappa symmetry projectors. Additional independent conditions reduce the common Killing spinor space; an aligned dyonic string alone does not reduce its fraction relative to an aligned fundamental string.

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