Four-dimensional N=1 supersymmetry (4D $\mathcal{N}=1$ SUSY)

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.

Superspace integration uses the Berezin integral over Grassmann variables. A chiral $d^2\theta$ integral extracts an F-term, while a full $d^4\theta$ integral extracts a D-term.

A superfield is a function on superspace. Its finite Taylor expansion in Grassmann coordinates packages component fields into a supersymmetry representation.

A superfield is a field on superspace with a specified Super-Poincaré group transformation law. Being a polynomial in anticommuting coordinates alone does not specify this law or make coefficients a supermultiplet. Conversely, a suitably graded function on full superspace, acted on by coordinate pullback, is an unconstrained scalar superfield. A polynomial with spacetime-independent coefficients can be a special constant-component chiral superfield; dependence on spacetime is not itself the defining criterion. Chirality and reality impose additional conditions.

With left Grassmann derivatives, let $Q_{\alpha }={\partial }_{\alpha }-i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{Q}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Left-placed odd parameters generate $\delta \theta =ϵ$, $\delta \bar{\theta }=\overline{ϵ}$ and $\delta x^{\mu }=i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }$. A scalar superfield transforms by pullback under this superspace translation. The parameter-weighted variation is an even derivation, although each supercharge is odd.

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.

A function $S(x,\theta ,\bar{\theta })=\varphi (x)$ is an allowed special configuration of an unconstrained superfield. The subspace where all other components remain zero is not invariant: $\delta S=(i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }){\partial }_{\mu }\varphi$ produces components depending on Grassmann variables whenever $\varphi$ is nonconstant. Thus an isolated spacetime scalar field is not closed under supersymmetry. A constant gauge-singlet value is a trivial invariant exception.

A product of scalar superfields is a scalar superfield: the pullback of a product is the product of the pullbacks at the same transformed superspace point. Infinitesimally, the even supersymmetry variation obeys $\delta (ST)=(\delta S)T+S(\delta T)$. This follows from the graded Leibniz rule for the odd supercharges, after including their odd parameters.

A vector superfield is a real superfield, $V=V^{†}$, used to describe a supersymmetric gauge multiplet. In Wess-Zumino gauge its components are a gauge field, a gaugino and a real auxiliary field.

A gauge connection on superspace has both vector and spinor components. Subtracting the flat-frame torsion from the graded commutator of its covariant derivatives defines its supercurvature. The conventional four-dimensional $\mathcal{N}=1$ spinor-curvature constraints leave one real vector superfield as a prepotential. Its chiral field-strength superfield contains the ordinary gauge curvature, gaugino and real auxiliary field.

Wess-Zumino gauge uses supergauge freedom to remove the scalar and spinor components of a general vector superfield that do not belong to the physical gauge multiplet.

In four-dimensional $\mathcal{N}=1$ supersymmetry, a gauge-transformation superfield is a dimensionless chiral, Lie-algebra-valued parameter $\Lambda$. Its complex nature allows gauge transformations to preserve chirality.

A supergauge transformation acts on a charged chiral superfield and vector superfield together. In one common convention, ${\Phi }^{\prime }=e^{-2i\Lambda }\Phi$ and $e^{2V^{\prime }}=e^{-2i{\Lambda }^{†}}{e}^{2V}{e}^{2i\Lambda }$.

Eliminating the real auxiliary field $D$ gives a nonnegative D-term scalar potential. For a $U(1)$ theory with charges $q_i$ and a Fayet–Iliopoulos parameter $\xi$, one convention gives $V_D=\frac{1}{2}[g{\sum }_i{q}_i|{\varphi }_i{|}^2+\xi {]}^2$.

For one charged chiral superfield, canonical kinetic terms, no nonconstant gauge-invariant superpotential, and $g>0$, the sign of $q\xi$ selects the vacuum. If $q\xi >0$, the minimum has $\varphi =0$, positive vacuum energy, unbroken internal gauge symmetry and a gaugino Goldstino; the scalar squared mass is $gq\xi$ while its chiral fermion is massless. If $q\xi <0$, a scalar condensate cancels the auxiliary field, preserving supersymmetry while Higgsing the gauge group. These are classical statements; a consistent quantum charged spectrum must cancel its gauge anomalies.

For canonical kinetic terms, vanishing superpotential, $g>0$ and $q\xi >0$, the D-term scalar potential $V=\frac{1}{2}(\xi +gq|\varphi {|}^2{)}^2$ has its minimum at $\varphi =0$. Both real scalar modes have squared mass $gq\xi$, while the chiral Weyl fermion, photon and gaugino are massless; the gaugino is the goldstino. The scalar-to-fermion squared-mass splitting is $gq\xi$. On the other branch $q\xi <0$, $v^2=-\xi /(gq)$ cancels $D$ and preserves supersymmetry; the Higgsed vector, remaining real scalar and Dirac fermion have common squared mass $2g^2{q}^2{v}^2$. A single nonzero-charge chiral fermion has a gauge anomaly, so a quantum realization requires an anomaly-cancelling completion; these are the stated classical vacuum spectra.

For canonical Kähler potential, set $a=|{\varphi }_0{|}^2$, $b=|{\varphi }_{+}{|}^2$, $c=|{\varphi }_{-}{|}^2$. The supplied supergravity F-term potential is $|\lambda {|}^2{e}^{{\kappa }^2(a+b+c)}[ab+ac+bc+3{\kappa }^{2}abc+{\kappa }^{4}abc(a+b+c)]$. Adding $g(b-c-\zeta {)}^2$ with $g>0$, $\lambda \ne 0$ and $\zeta >0$ gives a supersymmetric vacuum at $a=c=0$, $b=\zeta$. The branch $b=c=0$ has positive energy $g{\zeta }^2$, D-term breaking and a flat direction of a scalar potential in ${\varphi }_0$. Its charged squared masses are $|\lambda {|}^{2}a{e}^{{\kappa }^{2}a}\mp 2g\zeta$, so it is a local minimum valley only when both are positive. For negative $\zeta$, exchange the charged fields. These are branches rather than two isolated minima; a conventional constant Fayet–Iliopoulos term additionally needs a consistent supergravity gauge completion.

A Fayet–Iliopoulos term is a gauge-invariant term linear in the auxiliary $D$ field of an Abelian vector multiplet. Supersymmetry is broken only if no charged-scalar configuration can make the shifted D-term vanish simultaneously with every F-term.

In the conventional two-derivative supergravity action described by a Kähler potential, superpotential and gauge kinetic function, a constant Fayet–Iliopoulos term cannot be added independently of the gauging. It makes the Abelian gauging an R-symmetry gauging, and the superpotential must have the corresponding gauge transformation. Thus the rigid Abelian constant is subject to extra local consistency conditions. This statement concerns the conventional matter-coupled action, not distinct nonstandard FI constructions.

In an exactly supersymmetric Abelian theory a one-loop auxiliary-field tadpole can generate a cutoff-dependent Fayet–Iliopoulos term proportional to the trace of the matter charges. Gauge invariance with arbitrary background spurions forbids a general coupling-dependent full-superspace coefficient multiplying the vector superfield. In the supersymmetry-preserving Wilsonian setting the additive perturbative correction is exhausted by the one-loop term; it vanishes for a traceless charge generator. Canonical vector normalization and soft breaking require separate treatment. For a simple non-Abelian group the term itself is forbidden.

A constant linear auxiliary-field term ${\xi }_a{D}^a$ is gauge-invariant only when $\xi$ is an invariant vector in the dual adjoint representation. Equivalently, it annihilates all Lie brackets. A simple non-Abelian Lie algebra equals its commutator algebra and has no such nonzero vector. Abelian factors do admit the term. Identifying a Fayet–Iliopoulos parameter in a simple non-Abelian theory therefore requires setting it to zero, rather than treating it as an arbitrary coupling.

For a non-Abelian vector superfield, the chiral field-strength superfield isIt obeys ${\bar{D}}_{\dot{\alpha }}{W}_{\alpha }=0$ and transforms covariantly under a supergauge transformation.

For a real Abelian vector superfield, $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$ is chiral. The supercovariant derivative algebra with left derivatives gives $D^{\alpha }{\bar{D}}^2{D}_{\alpha }={\bar{D}}_{\dot{\alpha }}{D}^2{\bar{D}}^{\dot{\alpha }}$, proving the displayed reality condition. Its component equations contain the ordinary gauge-theory Bianchi identity, not the Maxwell equations of motion. For a non-Abelian gauge group, replace these by gauge-covariant derivatives and compare both field strengths in the same gauge frame; the identity follows from the graded Jacobi identity of the superconnection.

For an Abelian vector superfield, the chiral field-strength superfield is $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. Three barred supersymmetric covariant derivatives vanish, proving ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. This is a chiral spinor superfield with a gaugino as its lowest component. Phase and component signs depend on the stated Wess-Zumino gauge convention; the reality of $V$ alone does not fix those phase conventions.

The four-dimensional supersymmetric Yang-Mills action is the chiral superspace integral of $\mathrm{Tr}(W^{\alpha }{W}_{\alpha })$ plus its Hermitian conjugate. Its component fields are a Yang-Mills gauge field, a gaugino and an auxiliary field.

A chiral superfield obeys ${\bar{D}}_{\dot{\alpha }}\Phi =0$. In chiral coordinates $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$ it expands as

A chiral spinor superfield carries a Spinor representation of the Lorentz group index and satisfies ${\bar{D}}_{\dot{\alpha }}{\Psi }_{\beta }=0$. The chiral field-strength superfield is a fermionic example: its lowest component is the Weyl spinor gaugino. Its spinor index distinguishes it from a scalar chiral superfield, even though both obey the same chirality constraint.

A nonsingular holomorphic function of chiral superfields remains chiral because ${\bar{D}}_{\dot{\alpha }}f(\Phi )={\sum }_i{f}_i(\Phi ){\bar{D}}_{\dot{\alpha }}{\Phi }_i=0$. Dependence on conjugate superfields generally destroys the constraint. This explains the holomorphic form of a superpotential.

For $X=x+\sqrt{2}\theta G+{\theta }^{2}F$, the nilpotency condition gives $x^2=xG=0$ and $2xF=GG$. On the branch with invertible commuting part of $F$, $x=GG/(2F)$, while the goldstino $G$ and auxiliary field $F$ are independent. Products of three identical two-component odd spinor entries vanish, verifying the remaining constraints. The scalar is therefore composite. The branch hypothesis matters: $X=0$ also obeys the constraint and need not break supersymmetry.

With $X^2=0$ and invertible $F_X$, the additional chiral superfield $Y=y+\sqrt{2}\theta \chi +{\theta }^2{F}_Y$ satisfying $XY=0$ has $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$. Its Weyl spinor $\chi$ and auxiliary field $F_Y$ remain independent. This eliminates the independent scalar. It does not force $Y^2=0$.

On the invertible-$F_X$ branch, use $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$ and the two-component Grassmann algebra identity $(G\chi {)}^2=-\frac{1}{2}(GG)(\chi \chi )$. Then $y^2=-(GG)(\chi \chi )/(2F_X^2)$, $y^3=y^2\chi =0$ and $y^2{F}_Y-y\chi \chi =0$. These are the components of $Y^3=0$. Generally $y^2$ is nonzero, so cubic nilpotency must not be replaced by quadratic nilpotency. The allowed analytic holomorphic monomials are $1,X,Y,Y^2$.

An antichiral superfield obeys $D_{\alpha }{\Phi }^{†}=0$ and is the Hermitian conjugate of a chiral superfield. It depends naturally on ${\bar{y}}^{\mu }=x^{\mu }-i\theta {\sigma }^{\mu }\bar{\theta }$ and $\bar{\theta }$.

Supersymmetric covariant derivatives anticommute with the supersymmetry generators. Their antichiral member defines a chiral superfield through ${\bar{D}}_{\dot{\alpha }}\Phi =0$.

In flat four-dimensional N=1 supersymmetry, let ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$, contracting the two spinor indices with the antisymmetric epsilon invariant tensor. The supercovariant derivative algebra with left derivatives has $\{{\bar{D}}_{\dot{\alpha }},{\bar{D}}_{\dot{\beta }}\}=0$. There are only two dotted components, so every product of three barred derivatives repeats one component and vanishes. Therefore ${\bar{D}}_{\dot{\alpha }}{\bar{D}}^{2}X=0$ for any superfield $X$: ${\bar{D}}^{2}X$ is a chiral superfield. This map is called chiral projection, although ${\bar{D}}^2$ is not an idempotent projector.

If $\Phi$ is a chiral superfield, holomorphic closure of chiral superfields also makes $W^{\prime }(\Phi )$ chiral. Thus an equation ${\bar{D}}^2{\Phi }^{†}=4W^{\prime }(\Phi )$ is consistent with the chiral constraint: applying any ${\bar{D}}_{\dot{\alpha }}$ gives zero on both sides. Both sides are scalar superfields, and the squared supersymmetric covariant derivative commutes with the supercharges, so the equation is Lorentz invariant and supersymmetry covariant.

For left Grassmann derivatives and $D_{\alpha }={\partial }_{\alpha }+i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{\mu }$, ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}{\partial }_{\mu }$, the coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ gives $D_{\alpha }={\partial }_{\alpha }{|}_y+2i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{y^{\mu }}$ and ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}{|}_y$. The odd chain rule gives ${\bar{\partial }}_{\dot{\alpha }}(\theta {\sigma }^{\mu }\bar{\theta })=-(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}$. A chiral superfield is consequently independent of $\bar{\theta }$ at fixed $y$.