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A supersymmetry transformation mixes the bosonic and odd coordinates of superspace. A bare derivative with respect to an odd coordinate therefore acquires a spacetime-derivative term under the coordinate Jacobian and is not a covariant spinor superfield. Supersymmetric covariant derivatives add terms linear in the other odd coordinate; their graded anticommutation with the supercharges makes their application to a scalar superfield transform covariantly.

Take canonical kinetic terms, gauge coupling $g>0$, and define the Fayet–Iliopoulos term by $+\xi D$. If $\varphi$ is the scalar component of the charge-$q$ chiral superfield, the relevant auxiliary field Lagrangian isThe algebraic equation eliminates $D$ and gives the D-term scalar potentialSince the superpotential vanishes, $F=0$. In a homogeneous vacuum, the supersymmetry transformation of the gaugino includesThus $D_0\ne 0$ gives an inhomogeneous shift: no nonzero constant supersymmetry parameter leaves this vacuum invariant. The gaugino $\lambda$ is the goldstino, because it is the field shifting under the broken supercharge; there is no $F$ contribution here.

For $q\ne 0$, minimizing over $x=|\varphi {|}^2\ge 0$ separates three cases. If $q\xi <0$, the allowed value $x=-\xi /(gq)$ gives $D_0=V_0=0$: supersymmetry is unbroken while the internal gauge symmetry is Higgsed. If $q\xi >0$, the unconstrained zero of $D$ would have negative $x$, so the minimum is $\varphi =0$ with $D_0=-\xi$ and $V_0={\xi }^2/2$. If $\xi =0$, the minimum $\varphi =0$ is supersymmetric. Consequently, with this explicitly stated sign convention,Reversing the sign used to define the Fayet–Iliopoulos term reverses the printed sign criterion, not the physics. If $q=0$, the scalar is decoupled and flat: any $\xi \ne 0$ still breaks supersymmetry through $D_0=-\xi$, but the charged-field criterion does not apply.

On the broken branch, expand about $\varphi =0$:Writing $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ gives both real scalar modes the squared mass $gq\xi$. There is no superpotential fermion mass and no vacuum expectation value to induce gaugino–matter-fermion mixing. The Abelian gauge symmetry is unbroken, so its vector is also massless. The complete mass spectrum of single charged-field D-term breaking isThe original chiral multiplet therefore has the squared-mass splittingThe original vector multiplet has zero mass splitting, although the nonzero $D_0$ means it is not an unbroken-supersymmetry representation. As a check, the tree-level supertrace mass sum rule gives $\mathrm{Str}{M}^2=2gq\xi$ when the two real scalar degrees of freedom are counted. On the $q=0$ broken branch all the propagating fields are massless.

There is a necessary scope qualification: one nonzero-charge Weyl fermion has an Abelian cubic gauge anomaly proportional to $q^3$ and a mixed gravitational gauge anomaly proportional to $q$. These are the classical vacuum and spectrum calculations requested by the model; a consistent quantum gauge theory needs an anomaly-cancelling completion, whose additional matter must also be included in its vacuum analysis.

The supersymmetry algebra closes on spacetime translations. Its supercharges are Weyl spinors, andThus the composition of two odd supersymmetry transformations moves the spacetime argument. The generators also change spin and connect bosons with fermions. They extend the Poincaré group to the Super-Poincaré group, rather than forming an internal symmetry that commutes with spacetime transformations.

Use ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$ and keep each constant odd supersymmetry parameter on the left. The transformation of a fermion may be written $\delta {\psi }_1^{\mu }={ϵ}_1{\partial }_{+}{X}^{\mu }$ or ${\partial }_{+}{X}^{\mu }{ϵ}_1$, since the bosonic derivative commutes with the parameter. The variation $\delta$ itself is even, so it obeys the ordinary product rule; the signs below arise when moving an odd parameter past an odd field.

First consider the ${ϵ}_1$ transformation. Suppress target indices and put $\mathcal{L}=2X_{+}\cdot X_{-}+i{\psi }_1\cdot {\partial }_{-}{\psi }_1+i{\psi }_2\cdot {\partial }_{+}{\psi }_2$. Its bosonic part varies asFor the first fermion, moving ${ϵ}_1$ to the left givesThe second fermion does not vary under ${ϵ}_1$. Since ${\partial }_{-}{X}_{+}={\partial }_{+}{X}_{-}$, these terms combine to ${\delta }_1\mathcal{L}=-i{ϵ}_1{\partial }_{+}({\psi }_1\cdot X_{-})$. The other parameter gives the same cancellation with $+\leftrightarrow -$ and $1\leftrightarrow 2$. ThereforeThis proves worldsheet supersymmetry invariance up to the rigid worldsheet supersymmetry boundary term, without using field equations.

For an open string, the boundary term is part of the invariance statement. For example, a Neumann endpoint has ${\partial }_{+}X={\partial }_{-}X$ and compatible fermion gluing ${\psi }_1=\kappa {\psi }_2$, $\kappa =\pm 1$. The spatial boundary flux vanishes when ${ϵ}_2=\kappa {ϵ}_1$. Both endpoints must admit the same preserved parameter. Opposite gluing signs, giving the Neveu–Schwarz sector, do not preserve a nonzero constant parameter at both ends. Thus the local bulk identity and the existence of a global rigid charge on a particular open-string sector are distinct. The local superconformal symmetry remains the basis of the string constraints; this is consistent with the spin-structure obstruction to constant worldsheet supersymmetry.

Now write $A_{\pm }^{\mu }={\partial }_{\pm }{X}^{\mu }$ and $J_{+}^1={\psi }_1\cdot A_{+}$, $J_{-}^2={\psi }_2\cdot A_{-}$. The other two components vanish. The action implies ${\partial }_{-}{\psi }_1=0$, ${\partial }_{+}{\psi }_2=0$ and ${\partial }_{+}{\partial }_{-}X=0$, so these supercurrents are chiral and conserved locally. At equal time, the given canonical commutation relations implyWith $q_{ϵ}={\pi }^{-1}\int d\sigma ({ϵ}_1{\psi }_1\cdot A_{+}+{ϵ}_2{\psi }_2\cdot A_{-})$, contraction with this commutator givesThe parameter-weighted charge is even. Since the parameters anticommute with the fermions,Using the PDF's normalization $\{{\psi }_i^{\mu }(\sigma ),{\psi }_j^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }{\delta }_{ij}\delta (\sigma -{\sigma }^{\prime })$ yieldsThus the charge generates all the specified supersymmetry transformations. Dropping the factor $\pi$ in the fermion anticommutator, as happens in the converted TeX, would fail this check. The original PDF has the correct factor.

To calculate the supercurrent bracket, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. The equal-time bosonic relations giveBosonic and fermionic fields commute. Expanding the two current products, commuting the bosonic factors once and using the fermion anticommutator gives the noncentral partFor an ordinary smooth bilocal coefficient, the distributional derivative identity isFor the fermion bilinear, its coincident classical value, or its coincident normal-ordered value, is zero by antisymmetry against the symmetric target metric. The second term is ${\psi }_1\cdot {\partial }_{\sigma }{\psi }_1$. Since ${\partial }_{-}{\psi }_1=0$, we have ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. It follows that the noncentral current bracket isThis is the canonical supercurrent bracket of free RNS matter. The other chirality similarly gives ${\theta }_{--}=({\partial }_{-}X{)}^2+(i/2){\psi }_2\cdot {\partial }_{-}{\psi }_2$ and ${\theta }_{+-}=0$ on shell.

The printed current identity is the classical, or noncentral, part of the algebra. Fully quantized coincident products require normal ordering; the matter algebra then also has a central contact term proportional to ${\delta }^{\prime \prime }$ in the current anticommutator. The 10 bosons and 10 real fermions contribute matter central charge $c=10+10/2=15$. The central term cannot be discarded in a literal quantum operator identity, although it is irrelevant to identifying the local noncentral stress tensor above.

The other brackets close on these same currents. The stress-tensor commutator contains $\theta {\delta }^{\prime }$ and ${\theta }^{\prime }\delta$ and, quantum mechanically, a central ${\delta }^{\prime \prime \prime }$ term. The stress-tensor–supercurrent commutator contains $J{\delta }^{\prime }$ and $J^{\prime }\delta$, expressing the conformal weight $3/2$ of $J$. The supercurrent anticommutator contains $\theta \delta$ and its central ${\delta }^{\prime \prime }$ term. In a conventional normalization of Fourier modes these form the N=1 super-Virasoro algebra:Here $C_{mn}$ and $D_{rs}$ are central terms supported at zero total mode number; their detailed coefficients are not needed for this request. The indices $r$ are half-integers in the Neveu–Schwarz sector and integers in the Ramond sector. Opposite chiralities commute or anticommute as appropriate.

For the closed RNS string, impose a copy of the constraints in each chirality. In the Neveu–Schwarz sector, the physical conditions areIn the Ramond sector, use vacuum-normal-ordered generators with zero Ramond intercept. ThenThe Ramond zero-mode equation is essential: its square gives the mass-shell condition in this convention, $G_0^2=L_0$, and on the oscillator ground sector it is the spacetime Dirac constraint. In a convention whose unshifted plane generator has $G_0^2=L_0-c/24$, the zero-mode equation instead sets that shifted combination to zero; these are the same condition, not competing intercepts.

For either choice, the corresponding tilded constraints hold on the other side. Both chiral mass-shell conditions and closed-string level matching are required:The state sectors are NS–NS, NS–R, R–NS and R–R. Quotienting physical null states removes unphysical longitudinal polarizations; equivalently, use the appropriate BRST cohomology including the superconformal ghosts. A consistent superstring also specifies the GSO projection. It is an additional sector selection, not a replacement for these local super-Virasoro constraints. Physical states satisfy both chiral positive-mode and zero-mode constraints, level matching, and the null-state quotient; Ramond states additionally obey the fermionic zero-mode constraint.

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.

Assume global four-dimensional N=1 supersymmetry with canonical kinetic terms and no explicit gaugino mass. Write $W_i={\partial }_{i}W$ and $W_{ij}={\partial }_i{\partial }_{j}W$. To match the displayed stationarity equation, use $F^i=\overline{W_i}$; this differs by a minus sign from the eliminated auxiliary field in the component convention of Solution 2. The real D-terms are $D^a=-g^a({\bar{\varphi }}_i(T^a{)}^i{}_j{\varphi }^j+{\xi }^a)$, with Hermitian gauge generators. A constant Fayet–Iliopoulos term is permitted only for an appropriate Abelian factor. The scalar potential isDifferentiate with respect to ${\varphi }^i$, holding $\bar{\varphi }$ independent:Thus stationary point conditions gives the first requested identity. Gauge invariance of the superpotential gives $W_i(T^a{)}^i{}_j{\varphi }^j=0$. Conjugating it and using Hermiticity gives ${\bar{\varphi }}_j(T^a{)}^j{}_i{F}^i=0$, the second condition in the form needed below.

At the vacuum, define $H_{ij}=⟨W_{ij}⟩$ and $G_{ia}=g^a⟨{\bar{\varphi }}_j⟩(T^a{)}^j{}_i$. The two conditions combine asHere the upper equation is stationarity and the lower equation is conjugated gauge invariance. The chiral-superfield fermion mass matrix contributes $-\frac{1}{2}{H}_{ij}{\psi }^i{\psi }^j$, and the gauge Yukawa interaction contributes the mixed chiral-fermion/gaugino bilinear. With canonical Weyl fermion and gaugino kinetic terms, and choosing the gaugino phase to put the mixed bilinear in real-sign convention, these terms areThe factor $\sqrt{2}$ is important: the matrix directly acting on the printed $(F,D)$ is the same mass bilinear with rescaled gauginos, rather than literally the canonical matrix. In detail, for $S=\mathrm{diag}(I,\sqrt{2}I)$,Since supersymmetry breaking makes this vector nonzero, there is at least one zero mass eigenvalue. A normalized massless fermion isWith the consistent gaugino phase, the inhomogeneous vacuum supersymmetry transformation of $(\psi ,\lambda )$ is proportional to the same vector $(F,D/\sqrt{2})$. Therefore $\eta$ shifts by a nonzero constant times the transformation parameter: it is the Goldstino, the Goldstone fermion of broken global supersymmetry. The argument guarantees at least one Goldstino zero mode, not the absence of other massless fermions.

For the second calculation, first take only canonical chiral superfields and F-term breaking. At a stationary vacuum $v$, let $h=\varphi -v$, $m_{ij}=W_{ij}(v)$ andThese follow by differentiating $V={\sum }_k|W_k{|}^2$ twice. The quadratic scalar potential isThe doubled scalar mass matrix has blocksTransforming to the $2N$ canonically normalized real components of $h$ preserves its trace, so the sum of real scalar mass squares is $\mathrm{tr}{M}_s^2=2\mathrm{tr}A$. The Weyl fermion mass squares are the eigenvalues of $m^{†}m=A$ and their sum is $\mathrm{tr}A$. Each Weyl mass eigenstate has two physical spin states. The printed supertrace has scalar sign $-1$ and fermion weight $+2$, henceThe frequently used boson-minus-fermion definition has the opposite overall sign and gives the same zero. The mixing block $B$ can split scalar masses but cannot change this trace. This proves the tree-level supertrace mass sum rule; it uses canonical kinetic terms and a renormalizable global theory, rather than a general supergravity or noncanonical effective theory.

In the MSSM, trying to make all squarks and sleptons heavy through direct canonical tree-level F-term breaking encounters this constraint. In an unbroken electric-charge and color sector, the scalar mass-squared trace is twice the trace for its ordinary fermionic partners. More explicitly, take a unit vector $u$ minimizing $u^{†}Au$. For $h=e^{i\alpha }u$, choose $\alpha$ so that $\mathrm{Re}(e^{2i\alpha }{u}^{T}Bu)\le 0$. The curvature along this real scalar direction is then at most $u^{†}Au$, so at least one scalar mass square is no larger than the smallest fermion mass square in that sector. For one isolated pair the scalar masses are $m_f^2\pm |B|$. Thus light quarks can force an unacceptably light squark, or a tachyon, under these assumptions. A hidden supersymmetry-breaking sector with mediation produces effective soft supersymmetry breaking; radiative effects, noncanonical kinetic terms, or supergravity can change the assumptions and allow a viable spectrum. The trace over an arbitrary enlarged theory alone would not locate the light state in the visible sector; the MSSM conclusion uses the conserved-charge block structure.

A supersymmetry transformation is the infinitesimal action generated by an odd supercharge, with constant anticommuting spinor parameter in a global theory. In superspace it acts as $\delta \mathcal{S}=(ϵQ+\overline{ϵ}\bar{Q})\mathcal{S}$, with possible conventional overall factors of $i$. On the fermion of a chiral superfield, the transformation includes $\delta \psi =\sqrt{2}ϵF_{\mathrm{aux}}+i\sqrt{2}{\sigma }^{\mu }\overline{ϵ}{\partial }_{\mu }\varphi$. A nonzero vacuum auxiliary field therefore gives an inhomogeneous fermion shift, identifying the Goldstino direction when global supersymmetry is spontaneously broken.