Supersymmetry (SUSY)

Supersymmetry extends spacetime symmetry by odd supercharges that interchange boson and fermion states. Their Super-Poincaré algebra pairs positive-energy states into supermultiplets and gives energy positivity in global supersymmetry. Superspace and superfields package the component fields into representations of this symmetry.

A supersymmetric action is invariant under supersymmetry up to boundary terms. For ordinary four-dimensional chiral superfields, a full superspace integration of a real Kähler potential and a chiral superspace integration of a superpotential provide the two-derivative ungauged action.

A two-derivative four-dimensional $\mathcal{N}=1$ gauge action is specified by a real gauge-invariant Kähler potential, a gauge-invariant holomorphic superpotential, and a holomorphic symmetric gauge kinetic function. A constant Fayet–Iliopoulos term is additionally allowed for an Abelian gauge factor. Renormalizability restricts the matter kinetic function to a quadratic form, the superpotential to degree at most three, and the gauge kinetic function to a constant invariant tensor.

The gauge kinetic function is a symmetric holomorphic function of chiral superfields multiplying the gauge field-strength F-term. Its real part controls gauge kinetic coefficients and its imaginary part controls topological-angle terms. Its Wilsonian perturbative correction is constrained by one-loop exactness of the holomorphic gauge kinetic function; canonical and one-particle-irreducible gauge couplings require separate normalization and infrared analysis.

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A holomorphic gauge coupling is the coefficient of the gauge kinetic F-term in a local Wilsonian effective action written in holomorphic variables. In a common normalization its real part is $1/g_h^2$. The continuous perturbative shift of its topological angle, together with holomorphy, restricts its beta function to one-loop order. Wave-function renormalization and anomalous canonical field rescalings distinguish it from the physical gauge coupling.

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A classical supersymmetric mechanical model takes its coordinates in a Grassmann algebra: the position is even and the fermionic coordinates are odd. Time differentiation is even, while variation with respect to an odd coordinate uses Grassmann derivatives. For example, moving an odd variation to the left and integrating by parts gives $\delta \int \psi \dot{\psi }dt=2\int \delta \psi \dot{\psi }dt$.

For the real variant $L={\dot{x}}^2/2+U^2/2-{\psi }_1{\dot{\psi }}_1/2+{\psi }_2{\dot{\psi }}_2/2+U^{\prime }{\psi }_1{\psi }_2$, the Euler-Lagrange equations areThe conserved even energy is $E={\dot{x}}^2/2-U^2/2-U^{\prime }{\psi }_1{\psi }_2$ and the odd supercharges are $Q_1=\dot{x}{\psi }_1-U{\psi }_2$, $Q_2=\dot{x}{\psi }_2-U{\psi }_1$. For example, ${\dot{Q}}_i=U^{\prime \prime }{\psi }_1{\psi }_2{\psi }_i=0$, since each odd element squares to zero. Also $d({\psi }_1{\psi }_2)/dt=U^{\prime }({\psi }_2^2+{\psi }_1^2)=0$, which proves $\dot{E}=0$.

In the real Grassmann-valued classical supersymmetric mechanics variant, impose $\dot{x}=U(x)$ and ${\psi }_2=-{\psi }_1$. Then ${\psi }_1{\psi }_2=0$, and differentiating the first-order equation gives $\ddot{x}=U{U}^{\prime }$. The remaining Euler-Lagrange equations reduce to ${\dot{\psi }}_1=-U^{\prime }{\psi }_1$. These solutions have $E=0$ but $Q_1=2U{\psi }_1=-Q_2$, which need not vanish: $d(U{\psi }_1)/dt=U^{\prime }U{\psi }_1-U{U}^{\prime }{\psi }_1=0$.

For $U=1+cx$, one obtains $x(t)=e^{ct}{x}_0+(e^{ct}-1)/c$ and ${\psi }_1(t)=e^{-ct}{\psi }_{10}$ when $c\ne 0$. At $c=0$, the limiting solution is $x(t)=x_0+t$, ${\psi }_1(t)={\psi }_{10}$. In either case $Q_1=2(1+c{x}_0){\psi }_{10}$.

In four spacetime dimensions, power-counting renormalizability restricts an ungauged chiral action to a canonical quadratic Kähler potential, up to transformations, and a superpotential of degree at most three. Terms of larger degree belong to an effective field theory.

The spinning particle constraints $\mathcal{Q}={\psi }^m{p}_m$ and $\mathcal{H}=p^2/2$ obey $\{\mathcal{Q},\mathcal{Q}{\}}_D=-2i\mathcal{H}$ and $\{\mathcal{Q},\mathcal{H}{\}}_D=0$. In the quantum theory ${\hat{\mathcal{Q}}}^2=\hat{\mathcal{H}}$, so the fermionic physical-state equation implies the mass-shell equation.

A symmetry-preserving deformation can leave protected observables unchanged while concentrating their integral on a smaller critical locus. Supersymmetric Ward identities justify the deformation; convergence and boundary terms must be controlled.

With $P=W^{\prime }$ a nonconstant polynomial, positive area measure and Berezin integration oriented positively, a holomorphic polynomial insertion in $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$ equals $\pi {\sum }_a{m}_{a}f(z_a)$, where $P(z_a)=0$ and $m_a$ is the multiplicity of a root. Rescaling $P$ by $t>0$ leaves the insertion fixed and localizes it as $t\to \infty$.

If an odd graded derivation $Q$ preserves the action and integration measure, then $⟨QX⟩=0$ whenever boundary contributions and anomalies vanish. This applies to finite-dimensional zero-dimensional supersymmetric field theories as well as suitably regulated field integrals.

A zero-dimensional model is a finite-dimensional integral with bosonic and Grassmann variables, rather than a field depending on spacetime. Odd graded derivations preserve its action and integration measure. Such models expose supersymmetric Ward identities and supersymmetric localization without spacetime dynamics.

For $P=W^{\prime }$ and $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$, four odd graded derivations preserve $S$: ${\theta }_1{\partial }_z-\bar{P}{\partial }_{{\theta }_2}$, ${\theta }_2{\partial }_z+\bar{P}{\partial }_{{\theta }_1}$ and their barred counterparts. Left Grassmann derivatives fix the signs.

In four-dimensional Minkowski spacetime, extended supersymmetry has more than one independent Weyl spinor supercharge, together with their adjoints. The index $A=1,\dots ,\mathcal{N}$ distinguishes these supercharges.

In a four-dimensional theory with unbroken $\mathcal{N}\ge 2$ supersymmetry, a full charged hypermultiplet contains $\mathcal{N}=1$ chiral superfields in $R$ and $\overline{R}$. Their net complex chiral gauge spectrum vanishes. The vector multiplet is in the real adjoint gauge group representation. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex gauge chirality. Thus the chiral matter of the Standard Model can have at most unbroken $\mathcal{N}=1$ supersymmetry in four dimensions.

A full four-dimensional $\mathcal{N}=2$ hypermultiplet is a matter supermultiplet equivalent to two $\mathcal{N}=1$ chiral superfields in conjugate gauge group representations. Pseudoreal representations can admit half-hypermultiplets; these do not provide genuinely complex chiral gauge spectra.

Two-dimensional $\mathcal{N}=(2,2)$ supersymmetry has independent left- and right-moving complex supercharges. Its superspace uses ${\theta }^{\pm },{\bar{\theta }}^{\pm }$ and supports both chiral and twisted chiral superfields.

A twisted chiral superfield obeys ${\bar{D}}_{+}\Sigma =D_{-}\Sigma =0$. In an Abelian gauge theory the field-strength multiplet $\Sigma ={\bar{D}}_{+}{D}_{-}V$ is gauge invariant and twisted chiral.

A twisted superpotential is a holomorphic function of twisted chiral superfields integrated over $d{\theta }^{+}d{\bar{\theta }}^{-}$ superspace. Its highest component varies by a spacetime total derivative under two-dimensional N=(2,2) supersymmetry.

Two-dimensional $\mathcal{N}=(0,2)$ supersymmetry retains one complex right-moving supercharge. Its basic matter multiplets are chiral superfields containing a scalar and right-moving fermion, and Fermi superfields containing a left-moving fermion and an auxiliary scalar.

A Fermi superfield ${\Lambda }_{-}^a$ obeys ${\bar{D}}_{+}{\Lambda }_{-}^a=f_a(\Phi )$. A supersymmetric $J$-type coupling $\int d{\theta }^{+}{\Lambda }_{-}^a{J}^a(\Phi )$ requires ${\sum }_a{f}_a{J}^a=0$.

Four-dimensional $\mathcal{N}=1$ supersymmetry has one Weyl-spinor supercharge and four real supercharges.

The Super-Poincaré group combines Lorentz transformations and supertranslations. Its even part acts by spin-group transformations and ordinary spacetime translations; odd parameters generate the supercharges. Their graded commutator closes on translations through the Super-Poincaré algebra. A superspace tensor built from invariant coframes and Lorentz-invariant contractions is invariant under the full group.

The supercurrent is the spinor-valued Noether current of rigid supersymmetry. Its spatial integral is a supercharge. Localizing the supersymmetry parameter produces ${\partial }_{\mu }\overline{ϵ}{S}^{\mu }$ in the variation of the action; a gravitino coupling cancels this term and starts the Noether gauging procedure.

The super-Poincaré algebra extends the Poincare algebra by odd spinor generators. In four-dimensional $\mathcal{N}=1$ supersymmetry, $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ and the equal-chirality anticommutators vanish.

A supersymmetric domain wall contributes a boundary or tensorial charge to the extended anticommutator of supercharges. It is not the Lorentz-scalar central charge of an extended particle algebra. The wall's energy per unit area can cancel this charge in half the real charge combinations, giving the BPS tension bound and half-supersymmetry preservation. This does not contradict the rank classification for the ordinary algebra without wall charges.

A supertranslation is generated by translations and supercharges, excluding the Lorentz transformations. For flat Majorana superspace with ${\Pi }^m=d{X}^m+i\bar{\theta }{\gamma }^{m}d\theta$, the rigid odd transformation $\delta \theta =ϵ$, $\delta X^m=-i\overline{ϵ}{\gamma }^m\theta$ leaves ${\Pi }^m$ invariant. Its commutator gives an ordinary translation.

With left Grassmann derivatives, define ${\mathcal{P}}_{\mu }=-i{\partial }_{\mu }$, ${\mathcal{Q}}_{\alpha }=-i{\partial }_{{\theta }^{\alpha }}-{\sigma }_{\alpha \dot{\gamma }}^{\mu }{\bar{\theta }}^{\dot{\gamma }}{\partial }_{\mu }$ and ${\bar{\mathcal{Q}}}_{\dot{\beta }}=i{\partial }_{{\bar{\theta }}^{\dot{\beta }}}+{\theta }^{\gamma }{\sigma }_{\gamma \dot{\beta }}^{\mu }{\partial }_{\mu }$. The anticommutators of the derivative parts and of the coordinate parts vanish. Each mixed derivative/coordinate pair contributes $-i{\sigma }^{\mu }{\partial }_{\mu }$ by the graded product rule, giving the displayed mixed anticommutator. Equal-chirality anticommutators vanish, and all these operators commute with translations. Under Lorentz transformations their spinor indices transform in conjugate Weyl spinor representations, completing the minimal Super-Poincaré algebra. A convention for left derivatives is essential to the signs.

In the minimal Super-Poincaré algebra, Lorentz covariance allows $[P^{\mu },Q_{\alpha }]=A{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{Q}}^{\dot{\beta }}$. The graded Jacobi identity with $P^{\mu },Q_{\alpha },Q_{\beta }$ forces $A=0$. Thus each supercharge preserves four-momentum and its mass Casimir, giving supersymmetric mass degeneracy in an unbroken physical supermultiplet.

For $\{Q_{\alpha },Q_{\beta }^{†}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$, summing diagonal components gives $4H$. The expectation of each anticommutator is $\Vert Q_{\alpha }v{\Vert }^2+\Vert Q_{\alpha }^{†}v{\Vert }^2$, so $H$ is a positive semidefinite operator. A vacuum has zero energy exactly when every supercharge annihilates it. Spontaneously broken global supersymmetry with an existing vacuum therefore gives positive vacuum energy density. This statement fixes the energy zero through the algebra and does not apply to the general supergravity F-term potential.

For a normalized state in a unitary representation of the ordinary four-dimensional Super-Poincaré algebra, define $M_{ab}=⟨\{Q_a,Q_b\}⟩/2$ in a real Hermitian charge basis. Then $\Vert u\cdot Q\psi {\Vert }^2=u^{T}Mu$, so its annihilator is precisely the kernel of a linear map represented by the positive matrix $M$. In a suitable real basis its eigenvalues are the two eigenvalues of $⟨P^0⟩I+⟨\mathbf{P}⟩\cdot \mathbf{\sigma }$, each repeated twice. A nonzero kernel therefore has dimension two at nonzero null momentum expectation or four at zero expectation. Complex non-Hermitian lowering operators and algebras with domain-wall extensions are not covered by this argument.

A central charge in four-dimensional extended supersymmetry is a Lorentz scalar even generator appearing in $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. It commutes with the displayed translation, Lorentz algebra, and supercharge generators. An added R-symmetry automorphism can act on the central charges.

Use $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. Unitary skew-diagonalization of an antisymmetric matrix gives two-by-two blocks with entries $Z_r=2z_r$. At rest, the paired supercharges have anticommutators $2(M\pm |z_r|)$. The squared norms of an operator and its adjoint sum to the expectation of their anticommutator, so positive norm requires $M\ge |z_r|$ for every block. A convention with $2Z$ in the algebra instead writes $M\ge \max |Z_r|$.

A BPS state saturates a BPS bound in supersymmetry. The zero-norm combinations of supercharges and their adjoints annihilate its entire irreducible supermultiplet. Thus part of the supersymmetry is preserved. For a positive-mass representation, its massive supermultiplet is shortened; massless supermultiplets require the separate null-momentum construction. Charge conjugation can require a separate conjugate multiplet in the physical spectrum.

A wall of marginal stability is a locus in the parameter or vacuum space where the complex central charges in supersymmetry of possible decay products align in phase. The triangle inequality then becomes an equality, $|Z_1+Z_2|=|Z_1|+|Z_2|$, permitting a BPS state to reach the threshold for decay into other BPS states. Away from such alignment, the strict triangle inequality can prohibit the decay. Supersymmetry protects the mass-charge relation of an existing short state; it does not guarantee that the state exists or remains stable everywhere in parameter space.

A supersymmetry generator is an odd Weyl spinor generator of the Super-Poincaré algebra. Acting on a physical state reverses fermion parity while preserving its four-momentum.

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.

Let $[J_i,q_a]=(R_i{)}_{ab}{q}_b$, with numerical coefficient matrices and $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. The Jacobi identity then gives $(R_j{R}_i-R_i{R}_j)q=i{ϵ}_{ijk}{R}_{k}q$, so component coefficient matrices obey the opposite bracket to state-representation matrices. Thus $R_i=-{\sigma }_i/2$ is consistent for an upper-column spinor, whereas $+{\sigma }_i/2$ in that same component prescription is not. Inverse adjoint actions and dual row conventions must be compared with their index contractions intact.

Under standard assumptions on a nontrivial analytic relativistic S-matrix, the Coleman–Mandula theorem says that every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra.

The Haag–Łopuszański–Sohnius theorem allows a Z2-graded Lie superalgebra and shows that supersymmetry, including possible central charges and R-symmetries, is the nontrivial fermionic extension compatible with an interacting relativistic S-matrix.

The superspin Casimir classifies massive supermultiplets analogously to the Pauli-Lubanski Casimir for Poincare representations. One construction contracts the tensor $C_{\mu \nu }=B_{\mu }{P}_{\nu }-B_{\nu }{P}_{\mu }$, where $B_{\mu }$ is the supersymmetric correction of the Pauli-Lubanski pseudovector.

A supermultiplet is an irreducible representation of the Super-Poincaré algebra. Conserved supercharges pair its bosonic and fermionic states at equal four-momentum.

A four-dimensional N=1 chiral multiplet consists off shell of a complex scalar $\varphi$, a Weyl fermion ${\psi }_{\alpha }$, and a complex auxiliary scalar $F$. A chiral superfield packages their supersymmetry transformations. Eliminating the auxiliary scalar leaves two bosonic and two fermionic physical polarizations; an allowed supersymmetric mass gives equal scalar and fermion masses.

A massive supermultiplet is a unitary irreducible representation of the Super-Poincaré algebra with positive timelike momentum. At rest, its nonzero supercharges become fermionic creation operators and fermionic annihilation operators. Without central charges in supersymmetry, four-dimensional extended supersymmetry has $2\mathcal{N}$ complex oscillators; a spin-$j$ Clifford vacuum gives $(2j+1)2^{2\mathcal{N}}$ states.

For a massive four-dimensional N=1 supermultiplet, superspin $j$ is the spin of its Clifford vacuum, the state annihilated by the two lowering supercharges. The creation operators produce spins $j,j,j+1/2,j-1/2$, omitting the last when $j=0$. The corresponding superspin Casimir has an eigenvalue proportional to $j(j+1)$; this representation label is distinct from the spin of any one constituent.

At rest the two complex supercharges have anticommutators $2m{\delta }_{ab}$. A normalized scalar Clifford vacuum generates four states at occupation levels $0,1,2$: two spin-zero bosonic states and one spin-half pair. Products of three creation charges vanish. The one-creation normalization is $(2m{)}^{-1/2}$ and the two-creation normalization is $(2m{)}^{-1}$. The resulting on-shell content is a complex scalar and a massive fermion with equal masses.

For $\mathcal{N}$ supersymmetries and $k$ saturated two-by-two central-charge blocks, $2k$ complex oscillators act trivially. A spin-$j$ Clifford vacuum therefore gives $(2j+1)2^{2\mathcal{N}-2k}$ states and preserves $4k$ of the $4\mathcal{N}$ real supercharges. With $\mathcal{N}=2$ and a scalar vacuum, the shortened multiplet has two boson states and two fermion states, before any CPT completion of a supermultiplet.

A supersymmetric vector multiplet contains a gauge boson and its gaugino partners, with additional scalar fields for extended supersymmetry. Its gauge group representation is the adjoint group representation.

A four-dimensional massive $\mathcal{N}=1$ vector multiplet contains one spin-one particle, two spin-one-half particles and one real spin-zero particle. Its on-shell state counts are $3+1=4$ bosonic and $2+2=4$ fermionic. These masses agree in an unbroken supersymmetric vacuum. The supersymmetric Higgs mechanism assembles it from a massless vector multiplet and one chiral superfield per broken gauge generator.

The massive N=1 vector multiplet contains spins $1,1/2,1/2,0$. Its massless helicity content splits into a CPT-complete vector multiplet with helicities $(1,1/2)$ and $(-1/2,-1)$, and a CPT-complete chiral multiplet with helicities $(1/2,0)$ and $(0,-1/2)$. The longitudinal vector polarization becomes one of the two scalar polarizations; the other comes from the original real scalar. This is the representation-theoretic inverse of the supersymmetric Higgs mechanism.

The CPT theorem reverses helicity and conjugates internal charges. If a supermultiplet is not mapped into itself, a physical spectrum must also include its conjugate supermultiplet; this can double the irreducible state count.

A massless supermultiplet is a positive-energy unitary irreducible representation of the Super-Poincaré algebra with null four-momentum. For finite helicity states in four dimensions, half the supercharges act trivially and the others form a fermionic Fock space.

For a four-dimensional $\mathcal{N}=1$ finite-helicity massless supermultiplet, one complex oscillator changes helicity by one-half. A lower-helicity labelling convention assigns $\kappa$ to the pair $(\kappa ,\kappa +1/2)$. A symmetric midpoint convention instead uses $\kappa +1/4$; report the helicity pair when comparing conventions. CPT completion of a supermultiplet pairs $\kappa$ with $-\kappa -1/2$.

For a four-dimensional massless supermultiplet with $\mathcal{N}$ independent supercharges and highest helicity $\lambda$, level $k$ has helicity $\lambda -k/2$ and multiplicity $\left(\genfrac{}{}{0ex}{}{\mathcal{N}}{k}\right)$. The total is $2^{\mathcal{N}}$ states and the helicity range has width $\mathcal{N}/2$, before any necessary CPT completion of a supermultiplet.

In four-dimensional supersymmetry, a finite-helicity massless supermultiplet has helicity interval of width $\mathcal{N}/2$. If every state obeys $|h|\le s$, that interval fits inside one of width $2s$, giving $\mathcal{N}\le 4s$. The bounds $4$ for $s=1$ and $8$ for $s=2$ are attained by the four-dimensional N=4 super Yang-Mills theory vector multiplet and the four-dimensional N=8 supergravity supergravity multiplet. Spin bounds alone do not prove that an arbitrary action is renormalizable.

Every positive-energy supermultiplet contains equally many bosonic and fermionic states. A supertrace of a positive supercharge anticommutator proves the equality, provided supersymmetry is exact and the representation is finite-dimensional at fixed momentum.

For a finite physical supermultiplet at fixed positive energy, fermion parity anticommutes with each odd supercharge. Trace cyclicity makes $\mathrm{Tr}[(-1{)}^F\{Q,Q^{†}\}]=0$. Summing the Super-Poincaré algebra diagonal entries replaces the anticommutator by $4E$, giving equal bosonic and fermionic state counts. A zero-energy vacuum is an exception to the division by $E$, so it need not have a partner. The argument requires an invariant fixed-momentum representation, which is the step lost under explicit supersymmetry breaking.

An odd involution $A$ reverses fermion parity and is its own inverse, so it identifies the bosonic and fermionic subspaces. They have equal dimension in a finite-dimensional representation. For a supercharge $q$ with $q^2=0$ and $\{q,q^{†}\}=cI$, $c>0$, use $A=(q+q^{†})/\sqrt{c}$. This gives a direct proof of boson-fermion degeneracy in a supermultiplet at positive energy. A zero-energy supersymmetric vacuum need not have an accompanying fermionic vacuum.

In an unbroken supersymmetric vacuum, physical boson and fermion partners have equal masses because supercharges commute with four-momentum. For one canonical chiral superfield, fluctuations about $W^{\prime }(v)=0$ have mass $|W^{\prime \prime }(v)|$ for both the complex scalar field and Weyl spinor.

For one canonical chiral field at $W^{\prime }(v)=0$, the quadratic scalar potential is $|W^{\prime \prime }(v){|}^2|\delta \phi {|}^2$. Writing $\delta \phi =(a+ib)/\sqrt{2}$ gives two canonically normalized real scalars with squared mass $|W^{\prime \prime }(v){|}^2$. The chiral-superfield fermion mass matrix has the same physical mass. A complex bilinear coefficient or its algebraic square is not itself a signed physical squared mass.

Superspace extends spacetime coordinates $x^{\mu }$ by anticommuting spinor coordinates ${\theta }^{\alpha }$ and ${\bar{\theta }}^{\dot{\alpha }}$.

The displayed one-forms give an invertible triangular change from the coordinate coframe on flat Majorana superspace. A rigid supertranslation has $dϵ=0$, and the variation of $d{X}^m$ cancels that of $i\bar{\theta }{\gamma }^{m}d\theta$. Their Maurer-Cartan equation is $d{\Pi }^m=id\bar{\theta }{\gamma }^{m}d\theta$. The bosonic frame transforms as a Lorentz vector and the odd frame as a Majorana spinor.

The contraction of invariant vector and spinor frames makes $H_3$ invariant under the Super-Poincaré group. Odd-coordinate differentials commute in the graded exterior algebra. The four-dimensional Fierz rearrangement identity $(C{\gamma }^m{)}_{(\alpha \beta }(C{\gamma }_m{)}_{\gamma \delta )}=0$ consequently gives $d{H}_3=i(d\bar{\theta }{\gamma }^{m}d\theta )(d\bar{\theta }{\gamma }_{m}d\theta )=0$. Its three-form degree is distinct from the four-form needed in the four-dimensional supermembrane Wess-Zumino coupling.

The supervielbein is the local frame of a superspace geometry. Pulling its bosonic-frame components back along a brane embedding gives ${\Pi }_i^a={\partial }_i{Z}^{\mathcal{M}}{E}_{\mathcal{M}}{}^a$, and hence the induced worldvolume metric $h_{ij}={\Pi }_i^a{\Pi }_j^b{\eta }_{ab}$. Fermionic coordinates remain present through their dependence on $Z$.

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A local fermionic gauge symmetry of supersymmetric brane actions. Since the kappa symmetry projector has half rank, the symmetry removes half the embedding fermions. The remaining physical fermions pair with the bosonic worldvolume degrees of freedom.

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.

In light-cone coordinates $x^{\pm }$ on two-dimensional Minkowski spacetime, add two real odd Grassmann variables ${\theta }^{\pm }$. With left Grassmann derivatives, the supercharges and supersymmetric covariant derivatives can be represented asThe identity $\{{\partial }_{\theta },\theta \}=1$ gives $Q_{\pm }^2=i{\partial }_{\pm }$, $D_{\pm }^2=-i{\partial }_{\pm }$, $\{Q_{+},Q_{-}\}=\{D_{+},D_{-}\}=0$, and $\{Q_s,D_t\}=0$ for every $s,t$. A Lorentz boost acts as $x^{\pm }\mapsto e^{\pm \omega }{x}^{\pm }$, ${\theta }^{\pm }\mapsto e^{\pm \omega /2}{\theta }^{\pm }$. Thus $D_{\pm }$ have boost weights $\mp 1/2$, and $D_{-}{D}_{+}$ is a Lorentz scalar. Since it commutes with both supercharges, an equation $i{D}_{-}{D}_{+}\Phi =W(\Phi )$ for an even scalar superfield is Lorentz invariant and supersymmetry covariant.

In two-dimensional N=(1,1) superspace, one convention for the supersymmetric Liouville equation is $i{D}_{-}{D}_{+}\Phi =e^{\Phi }$. With the ordered expansion $\Phi =\varphi +i{\theta }^{-}{\psi }_{+}+i{\theta }^{+}{\psi }_{-}+i{\theta }^{-}{\theta }^{+}F$ and left Grassmann derivatives, its left side isWriting $N=\Phi -\varphi$, Grassmann parity gives $N^2=2{\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$ and $N^3=0$. HenceComparing coefficients eliminates the auxiliary field as $F=e^{\varphi }$ and gives ${\partial }_{-}{\psi }_{-}=e^{\varphi }{\psi }_{+}$, ${\partial }_{+}{\psi }_{+}=-e^{\varphi }{\psi }_{-}$, ${\partial }_{-}{\partial }_{+}\varphi =-e^{2\varphi }+i{e}^{\varphi }{\psi }_{+}{\psi }_{-}$. The fermion bilinear sign depends on the stated ordering and derivative convention.

For left Grassmann derivatives, ${ϵ}^{12}=-{ϵ}_{12}=1$, $\theta \theta ={\theta }^{\alpha }{\theta }_{\alpha }$ and $\bar{\theta }\bar{\theta }={\bar{\theta }}_{\dot{\alpha }}{\bar{\theta }}^{\dot{\alpha }}$, the antisymmetric products are ${\theta }^{\alpha }{\theta }^{\beta }=-{ϵ}^{\alpha \beta }\theta \theta /2$ and ${\bar{\theta }}^{\dot{\alpha }}{\bar{\theta }}^{\dot{\beta }}={ϵ}^{\dot{\alpha }\dot{\beta }}\bar{\theta }\bar{\theta }/2$. With contractions ${\partial }^2={\partial }^{\alpha }{\partial }_{\alpha }$ and ${\bar{\partial }}^2={\bar{\partial }}_{\dot{\alpha }}{\bar{\partial }}^{\dot{\alpha }}$, both squared derivatives give $-4$ on the matching quadratic monomial. If $\mathrm{Tr}({\sigma }^{\mu }{\bar{\sigma }}^{\nu })=2{\eta }^{\mu \nu }$, the mixed bilinear product is $+(\theta \theta )(\bar{\theta }\bar{\theta }){\eta }^{\mu \nu }/2$. Reversing the metric-relative trace convention reverses this last sign.