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Take ${\eta }^{\alpha \beta }=\mathrm{diag}(-1,1)$ with coordinate zero equal to $\tau$ and coordinate one to $\sigma$. In the supplied representation setThey satisfy the Clifford algebra $\{{\rho }^{\alpha },{\rho }^{\beta }\}=2{\eta }^{\alpha \beta }1$. For anticommuting spinor components and an ordinary matrix $B$, relabelling the two summed indices givesNow $C^T=-C$, whereas $C{\rho }^0=-1$ and $C{\rho }^1=\mathrm{diag}(-1,1)$ are symmetric. Therefore the Majorana Grassmann bilinear interchange yieldsThese signs require fermionic, Grassmann-odd fields; commuting spinors would have the opposite interchange signs.

In flat conformal gauge, suppress the common factor $(4\pi {\alpha }^{\prime }{)}^{-1}$ and write the combined density as $\mathcal{L}=-{\partial }_{\alpha }X\cdot {\partial }^{\alpha }X+\bar{\psi }\cdot {\rho }^{\alpha }{\partial }_{\alpha }\psi$. The supersymmetry variation including its odd parameter is an even derivation. From $({\rho }^{\alpha }{)}^{T}C=-C{\rho }^{\alpha }$ we obtainAlso $(C{\rho }^{\alpha }{\rho }^{\beta }{)}^T=-C{\rho }^{\beta }{\rho }^{\alpha }$, so interchanging odd spinors gives $\bar{\chi }{\rho }^{\alpha }{\rho }^{\beta }ϵ=\overline{ϵ}{\rho }^{\beta }{\rho }^{\alpha }\chi$. The two density variations, with target contractions understood, are thereforeIn the second term the symmetric second derivative annihilates the antisymmetric part of ${\rho }^{\beta }{\rho }^{\alpha }$. Using $2{\eta }^{\alpha \beta }-{\rho }^{\beta }{\rho }^{\alpha }={\rho }^{\alpha }{\rho }^{\beta }$ now gives the exact off-shell identityThus the action is invariant when this boundary flux vanishes and the transformation preserves the field boundary conditions, for example with suitable falloff or compatible supersymmetric endpoint conditions. On a spatial circle, periodic fields admit the constant parameter; antiperiodic fermions with periodic bosons require the local superconformal treatment, since a nonzero constant parameter would make the boson variation antiperiodic. The constant parameter and ordinary derivatives describe rigid supersymmetry in flat gauge; a locally supersymmetric curved-worldsheet formulation includes the worldsheet gravitino. The result is the rigid worldsheet supersymmetry boundary term without using field equations.

For the commutator on $X$, keep the order of the odd parameters:For the fermion, the order is equally important. Moving ${ϵ}_2$ through both odd factors in $({\overline{ϵ}}_1{\partial }_{\alpha }\psi ){ϵ}_2$ incurs two minus signs. ConsequentlyTo check the spinor matrix explicitly, set ${ϵ}_1=(a,b{)}^T$, ${ϵ}_2=(c,d{)}^T$, with all four components anticommuting and monomials ordered $a,b,c,d$. ThenDirect multiplication gives ${\rho }^{\alpha }M+M{\rho }^{\alpha }=2V^{\alpha }1$. HenceVariation of the fermion action gives the Dirac equation ${\rho }^{\alpha }{\partial }_{\alpha }\psi =0$. On this equation the second term vanishes, provingThis is on-shell closure of rigid worldsheet supersymmetry; the action's invariance was off shell, but its fermionic transformation algebra closes in this field content only on shell.

The associated superconformal current is proportional to $\psi \cdot \partial X$, of conformal weight $3/2$. Its fermionic modes $G_r$ extend the stress-tensor generators to the N=1 super-Virasoro algebra:The modes are half-integral in the Neveu–Schwarz sector and integral in the Ramond sector. Their anticommutator giving a Virasoro generator is the conformal-mode counterpart of the translation commutator just derived. Ten coordinate bosons and ten Majorana fermions give matter central charge $c=10+10/2=15$, canceled by the usual reparametrization and superconformal ghosts in the critical RNS string.

There are two distinct parts of the hierarchy problem. The first asks why the scale of electroweak symmetry breaking is so much smaller than a fundamental scale such as the Planck mass. The second, the technical hierarchy problem, asks why that separation survives quantum corrections once it is chosen. With a fundamental Higgs boson, a mass term $m_H^2{H}^{†}H$ is allowed by the ordinary gauge symmetry, so heavy thresholds can contribute additive terms of order a heavy mass squared. This is different from the chiral protection of a fermion mass and the gauge protection of a vector mass.

For example, a cutoff description of the Standard Model gives the schematic one-loop sensitivityThe regulator-dependent quadratic term is not itself an observable, but a physical heavy particle of mass $M$ coupled to the Higgs likewise produces a matching correction of order coupling squared times $M^2/(16{\pi }^2)$. Maintaining $|m_H^2|\ll M^2$ then generically requires cancellation between large unrelated terms. This is the naturalness concern; changing the regulator does not eliminate the heavy-threshold issue.

Exact supersymmetry relates bosonic and fermionic couplings and masses. Their loop contributions have opposite signs, and the relations enforce supersymmetric cancellation of quadratic divergences. Boson-fermion degeneracy in a supermultiplet also transfers fermionic mass protection to its scalar partners. It is not sufficient merely to add arbitrary scalars: the supersymmetric coupling relations are what make the cancellation persist. Perturbative superpotential non-renormalization further protects holomorphic mass parameters, although wave-function renormalization still makes canonically normalized couplings run.

For a realistic separation between ordinary particles and their partners, supersymmetry must be broken. Soft supersymmetry breaking introduces masses and selected positive-mass-dimension interactions without reinstating the original ultraviolet quadratic divergence. A paired propagator difference behaves at high Euclidean momentum asso a residual scalar correction is of the orderThis soft scalar-mass sensitivity controls the technical problem if the relevant soft masses and thresholds are not too far above the weak scale. Very heavy partners can still require tuning through finite and logarithmic terms. Supersymmetric cancellations alone therefore do not guarantee a naturally small weak scale for an arbitrary broken spectrum.

The origin of the hierarchy needs further dynamics. In dynamical supersymmetry breaking, an asymptotically free hidden supersymmetry-breaking sector can generate an exponentially smaller scale by dimensional transmutation,If that sector actually has no supersymmetric vacuum, its breaking can be communicated to the visible sector to produce small soft terms. The exponential creates a hierarchy without specifying a tiny dimensionless coupling at the high scale, but the existence of breaking and its mediation are additional model-building requirements. In the MSSM, running Higgs soft masses can trigger electroweak symmetry breaking. The supersymmetric mu problem remains: the allowed superpotential mass $\mu H_u{H}_d$ must also be near the soft scale, and non-renormalization protects a chosen small $\mu$ without explaining its origin.

Extra-dimensional proposals use geometry to explain why observable gravitational and weak scales differ. In large extra dimensions, assume gravity propagates in $4+n$ dimensions with fundamental scale $M_{\ast }$, while ordinary fields are localized on a brane. Integrating the Einstein-Hilbert action over compact volume $V_n$ gives the Planck mass from compactification volume relationHere $M_{\mathrm{Pl}}$ denotes the four-dimensional reduced Planck mass. A sufficiently large internal volume lets $M_{\ast }$ be near the weak scale even though $M_{\mathrm{Pl}}$ is enormous. The Higgs need then be stable only up to the lower fundamental cutoff. This explains the weakness of long-distance gravity geometrically but trades the scale question for the origin and stabilization of a large compact volume. Ordinary matter is localized to avoid an unwanted low-mass tower of its own Kaluza-Klein modes; gravitational modes, departures from four-dimensional gravity and bulk energy loss impose phenomenological requirements.

The Randall–Sundrum model instead uses a warped five-dimensional interval, conventionally the doubled $S^1/{\mathbb{Z}}_2$ orbifold with fixed branes at $0$ and $L$:For a scalar confined to the distant brane, the kinetic and mass terms carry factors $e^{-2kL}$ and $e^{-4kL}$. Canonical rescaling therefore producesA moderately large $kL$ yields an exponentially small observable mass without a comparably large compact-volume factor. The inter-brane size is the radion, so one still needs a mechanism fixing its expectation value. Consistent curvature, brane tensions and ultraviolet physics are also part of the model. Warping is a redshift mechanism; it does not provide the same cancellation of arbitrary scalar loops as supersymmetry.

Finally, protecting a Higgs mass is not the same as solving the vacuum-energy hierarchy. Exact global supersymmetry has zero-energy supersymmetric vacua, but broken supersymmetry generically leaves vacuum energy set by its breaking dynamics. In supergravity, positive auxiliary-field contributions coexist with a negative superpotential term, and obtaining a very small cosmological constant requires further structure or cancellation. Neither soft supersymmetry nor the two geometric mechanisms automatically resolves that problem. The mechanisms can also coexist in one higher-dimensional supersymmetric theory.

The Noether gauging procedure starts by allowing a constant symmetry parameter to depend on position. For an internal symmetry written as $\delta {\varphi }^i={ϵ}^A{T}_A{\varphi }^i$, integration by parts identifies the Noether current:up to a boundary term. Introduce a gauge field $A_{\mu }^A$ and add $-g{A}_{\mu }^A{j}_A^{\mu }$, with the inhomogeneous gauge transformation $\delta A_{\mu }^A=g^{-1}{\partial }_{\mu }{ϵ}^A+\cdots$. This cancels the displayed variation. The current coupling has its own variation, so one iterates: add the needed higher-order interactions and corrections to the gauge transformations, and require closure of the local symmetry algebra. For an ordinary gauge theory, this reorganizes derivatives into gauge covariant derivatives and fixes the nonlinear gauge field strength. Gauging is therefore more than replacing a constant parameter by a function.

For rigid supersymmetry, the corresponding Noether current is the spinor-valued supercurrent $S^{\mu }$. A local Majorana spinor parameter gives a variation proportional to $({\partial }_{\mu }\overline{ϵ})S^{\mu }$. A gravitino ${\Psi }_{\mu }$, with an inhomogeneous transformation $\delta {\Psi }_{\mu }\sim {\kappa }^{-1}{\partial }_{\mu }ϵ$, cancels it through a coupling proportional to $\kappa {\bar{\Psi }}_{\mu }{S}^{\mu }$. The commutator of two rigid supersymmetry transformations is a translation. With local parameters, the translation parameter becomes position dependent, so the completion must also gauge translations: it introduces the vierbein, diffeomorphism invariance of general relativity, and local Lorentz transformations. The gravitino and graviton thus belong to the same supergravity multiplet.

For pure minimal four-dimensional supergravity, start with the free massless Fierz-Pauli action and the free Rarita-Schwinger field kinetic term. The Noether gauging procedure replaces the former by the Einstein-Hilbert action and couples the latter to the vierbein and spin connection. In a canonical normalization the leading transformations can be writtenAn overall sign or phase changes with the curvature and Majorana spinor conventions, but the relative normalization must be used consistently. To see why these terms fit together, first work to quadratic order in fermions, so the spin connection is torsion free. Varying the Rarita-Schwinger field kinetic term under $\delta {\Psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$ produces an antisymmetric double spinor covariant derivative. The spinor curvature identity and the Einstein tensor contraction of spin curvature give, in the matching curvature convention,The resulting Einstein tensor term cancels the variation of the Einstein-Hilbert action under $\delta e_{\mu }{}^a$. This is the essential first nontrivial cancellation in the construction.

At higher fermion order, the variations of the vierbein and spin connection produce further terms. Treating the spin connection as independent makes its Euler-Lagrange field equation algebraic. Its solution is a gravitino-induced torsion connection. With ${\Psi }_a=e_a{}^{\mu }{\Psi }_{\mu }$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, a conventional canonical normalization givesEquivalently, with $T^a=d{e}^a+{\tilde{\omega }}^a{}_b\wedge e^b$, the torsion tensor obeysThe Majorana Grassmann bilinear interchange makes the right-hand side antisymmetric in $\mu ,\nu$. Eliminating the spin connection generates the four-fermion completion. In the 1.5-order formalism one substitutes its solution but varies the other fields holding the connection fixed: the omitted chain-rule term is $(\delta S/\delta \omega )\delta \tilde{\omega }=0$. The remaining cubic fermion variations cancel by the Clifford algebra and Fierz rearrangements. This produces the displayed supergravity action in its compact connection-dependent form. Calling this formulation on shell means that no independent auxiliary fields have been retained and that closure of the supersymmetry algebra uses the propagating Euler-Lagrange field equations; it does not mean that action invariance is obtained only by setting every variation to zero on a solution.

In the antisymmetrized kinetic term, ${\Psi }_{\sigma }$ is a spinor-valued one-form and the derivative is conventionally writtenThis is the Lorentz-covariant exterior-derivative convention for that term. If one instead uses the full affine covariant derivative of a vector-spinor, a coordinate-index connection is also present and the antisymmetrization must include its torsion tensor correction. The crucial physical distinction from pure metric general relativity is the spin connection contains gravitino-induced contorsion, rather than being the torsion-free Levi-Civita connection. It reduces to the ordinary spinor covariant derivative when the gravitino vanishes.

Use the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$ and unit-weight antisymmetrization. The duality identity between ${\gamma }_5{\gamma }_{\nu }$ and three antisymmetrized gamma matrices turns the equation intoThe overall nonzero factor in this identity depends on the orientation and ${\gamma }_5$ convention and does not change its solutions. This Rarita-Schwinger field equation has the gauge symmetrysince ${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\partial }_{\rho }ϵ=0$. It is the free-field limit of local supersymmetry, so it is suitable for the gravitino.

To check which spin it propagates, put $A={\gamma }^{\mu }{\Psi }_{\mu }$ and $B={\partial }^{\mu }{\Psi }_{\mu }$. Expanding the three gamma matrices using the Clifford algebra givesin four dimensions. Thus the Euler-Lagrange field equation requires $B=/\partial A$. Under the gauge transformation, $A\mapsto A+/\partial ϵ$; locally one may choose gamma-trace gauge fixing, $A=0$, by solving $/\partial ϵ=-A$. The Euler-Lagrange field equation then yieldsThese are the massless Dirac equation, transversality and gamma-trace constraints that remove the lower-spin part of a vector-spinor.

For a more explicit massless Rarita-Schwinger polarization count, take a positive-frequency plane wave with null momentum along the third axis, in signature $(-,+,+,+)$. Transversality relates its temporal and longitudinal components. Residual gauge transformations with $/pϵ=0$ remove both. Write the surviving components as $u_1,u_2$; their gamma-trace condition reads ${\gamma }^1{u}_1+{\gamma }^2{u}_2=0$, henceThe massless Dirac equation leaves two complex positive-frequency amplitudes in $u_1$. If $i{\gamma }^1{\gamma }^2{u}_1=s{u}_1$, $s=\pm 1$, then $(u_1,u_2)$ is proportional to $(1,-is)$ in the transverse vector index. With spatial rotation generators $J_3^{\mathrm{vec}}{}_{12}=-i$ and $J_3^{\mathrm{spin}}=-i{\gamma }^1{\gamma }^2/2$, the vector has helicity $-s$ and the Majorana spinor has helicity $-s/2$, giving total helicity $-3s/2$. The Majorana spinor condition fixes the negative-frequency coefficients from these amplitudes. Therefore a massless gravitino has spin $3/2$, with the two physical helicities $\pm 3/2$. A complex vector-spinor would also have independent antiparticle modes.

The requested count is instead the off-shell component count of minimal supergravity, before imposing any Euler-Lagrange field equations. A four-dimensional Majorana spinor has four real components. Its vector index supplies four copies, and local supersymmetry supplies four real gauge functions:One must not additionally subtract transversality and the massless Dirac equation in this count: those are on shell conditions. The gamma-trace gauge fixing already uses the same four gauge functions subtracted above. For the graviton, a symmetric metric tensor has ten real components, with four diffeomorphism gauge functions. Equivalently, a vierbein has sixteen real components, minus six local Lorentz transformation and four diffeomorphism functions:The gravitino and graviton each have two propagating states on shell, but their off-shell counts disagree by six.

An auxiliary field has an algebraic Euler-Lagrange field equation and no propagating kinetic term. In old-minimal supergravity, a complex scalar $M$ contributes two real bosonic components and a real vector $b_{\mu }$ contributes four, so the bosonic count becomes $6+2+4=12$, matching the gravitino. Their supersymmetry transformations also absorb the terms that would otherwise require the propagating Euler-Lagrange field equations to close the local algebra. Thus auxiliary fields allow off-shell supersymmetry closure and matching component counts without adding physical particles. Eliminating them recovers an on shell formulation.

For a single chiral superfield, the supergravity F-term potential, with $\kappa =m_p^{-1}$ restored, isThe canonical Kähler potential has $K_{z\bar{z}}=K^{z\bar{z}}=1$, so the Kähler covariant derivative of a superpotential isConsequently the Polonyi model has the scalar potentialThe negative term is essential; retaining only the square of the supergravity auxiliary field would give the global F-term scalar potential rather than the supergravity F-term potential.

Set $x=\kappa z=a+iy$ and $b=\kappa \beta =2-\sqrt{3}$. In this notation $V=|\mu {|}^2{m}_p^2{e}^{|x{|}^2}U$, whereAt $a_0=\sqrt{3}-1$, $y=0$, one has $a_0+b=1$ and $1+a_0(a_0+b)=\sqrt{3}$, so $U=0$. To prove that this is a local minimum in both real directions, rather than checking only the real axis, put $u=a-a_0$. Direct expansion gives the stable zero-energy Polonyi vacuum identityEvery coefficient outside the square is strictly positive, and the exponential factor is positive. Thus $V\ge 0$ everywhere and equality requires $u=y=0$, provided $\mu \ne 0$. This proves the stronger conclusiona unique global minimum of this Polonyi model. If $\mu =0$, the scalar potential vanishes identically and this uniqueness and supersymmetry breaking conclusion do not hold.

For an independent local stability check, the Hessian matrix of $U$ at this global minimum is diagonal:Since $U$ and its first derivatives vanish there, differentiating the exponential introduces no additional Hessian matrix terms at the global minimum. For canonically normalized real fluctuations $z=z_0+(s+it)/\sqrt{2}$, the squared scalar masses are thereforewhich are both positive.

Although the vacuum energy is zero, the supergravity auxiliary field is not. In the convention $F^z=-e^{{\kappa }^{2}K/2}\overline{D_{z}W}$, its vacuum expectation value obeysThe matter fermion transformation contains a term proportional to $F^zϵ$, so no nonzero constant supersymmetry parameter leaves this vacuum invariant. Hence supersymmetry is spontaneously broken even though the vacuum energy vanishes: the positive auxiliary field contribution cancels the negative gravitational term. The matter fermion supplies the goldstino, which is absorbed by the gravitino in the super-Higgs mechanism.

Using the gravitino mass from a superpotential, and $z_0+\beta =m_p$, givesAn auxiliary field $F^z$ has mass dimension two. Defining the supersymmetry breaking scale by ${\Lambda }_{\mathrm{SUSY}}=\sqrt{|F^z|}$, one obtainsThus the gravitino mass is small compared with the supersymmetry breaking scale when that scale is sub-Planckian, as in the usual hierarchy $|\mu |\ll m_p$. This smallness is gravitational suppression; it is not a conclusion valid for arbitrary choices of the mass parameter.

Use the scalar components of the three chiral superfields, and set $a=|{\varphi }_0{|}^2$, $b=|{\varphi }_{+}{|}^2$, $c=|{\varphi }_{-}{|}^2$ and $\Sigma =a+b+c$. We assume the physical sign $g>0$ and a nonzero coupling $\lambda$. The coefficient $g$ below is exactly the coefficient in the supplied D-term scalar potential; it should not be silently replaced by a differently normalized squared gauge coupling. The canonical Kähler potential has identity Kähler metric, and the three Kähler covariant derivatives of a superpotential areSubstitution in the supergravity F-term potential givesIn particular, the subtraction cancels only half of the six ${\kappa }^{2}abc$ terms contributed by the three squares. The full scalar potential isIts nonnegativity follows term by term from $a,b,c\ge 0$ and $g>0$. This is a special property of this cubic superpotential, not of a general supergravity F-term potential.

The D-term vacuum branches of a cubic superpotential must be described as branches: there are not two isolated field-space minima. First suppose $\zeta >0$. Vanishing of $V_F$ requires $ab=ac=bc=0$, and vanishing of the D-term scalar potential requires $b-c=\zeta$. Together these conditions give the vacuum expectation valuesThe phase of ${\varphi }_{+}$ lies on a gauge orbit, and one may choose it positive and real. At this global minimum, all $D_{i}W$ and the real auxiliary field of the vector multiplet vanish. It is a supersymmetric vacuum, even though the internal gauge symmetry is spontaneously broken. For $\zeta <0$, interchange ${\varphi }_{+}$ and ${\varphi }_{-}$: the nonzero vacuum expectation value has magnitude $\sqrt{-\zeta }$. For $\zeta =0$, the supersymmetric vacuum is the entire family ${\varphi }_{+}={\varphi }_{-}=0$, with ${\varphi }_0$ arbitrary.

The other branch has ${\varphi }_{+}={\varphi }_{-}=0$ and arbitrary ${\varphi }_0$. Here $W=D_{i}W=0$ but the real auxiliary field is nonzero for $\zeta \ne 0$, andThus this branch has pure D-term supersymmetry breaking and a flat direction of a scalar potential in the neutral scalar. It is not stable at every neutral-field value. At a fixed $a=|{\varphi }_0{|}^2$, expansion in the charged scalars yieldsHence the two charged squared masses and the stability condition areAbove this threshold, both charged directions have positive mass squared. Continuity makes them uniformly positive in a sufficiently small neighborhood of the chosen neutral-field value, so the stationary branch is a genuine, non-strict local minimum valley. Its neutral direction stays flat. Below the threshold, a charged mass squared is negative and the branch is a saddle point. At equality it is a boundary of the stable valley, not a local minimum in the full field space: changing $a$ slightly downward and switching on the unstable charged field lowers $V$. For example, for $\zeta >0$, put $a=a_c-ϵ$, $b={ϵ}^2$, $c=0$. Since $M(a)=|\lambda {|}^{2}a{e}^{{\kappa }^{2}a}$ has $M^{\prime }(a_c)>0$, one finds $V-g{\zeta }^2=-M^{\prime }(a_c){ϵ}^3+O({ϵ}^4)<0$.

There are no further local-minimum branches under these generic assumptions. If $b+c>0$, the expression for $V_F$ is strictly increasing in $a$, so a stationary configuration with a charged condensate must have $a=0$. At $a=0$, an interior critical point with $b,c>0$ is impossible, becauseOn the boundary $c=0$, the relevant scalar potential is $g(b-\zeta {)}^2$; on $b=0$ it is $g(c+\zeta {)}^2$. Their nonnegative-domain minima give exactly the supersymmetric vacuum described above, or the origin of the charged-field branch. Thus, for nonzero $\zeta$, the two types are the zero-energy supersymmetric gauge orbit and the positive-energy D-breaking minimum valley above the threshold. If $\zeta =0$, the two types coalesce into the flat supersymmetric vacuum; the literal two-minimum assertion needs this parameter qualification.

In the global supersymmetry limit $\kappa \to 0$, the F-term scalar potential becomes $|\lambda {|}^2(ab+ac+bc)$ and the D-term scalar potential retains its form. The same two types of branches occur, with the threshold $|\lambda {|}^{2}a>2g|\zeta |$. Supergravity instead introduces the exponential and additional positive terms. The goldstino of global supersymmetry breaking is a physical fermion; in local supersymmetry it participates in the super-Higgs mechanism. A positive vacuum energy also gravitates in supergravity. In particular, the positive-energy branch here has $W=0$ and hence vanishing gravitino mass from a superpotential; it is not the tuned zero-energy broken vacuum of Question 3.

There is a further gauge-consistency qualification. In conventional two-derivative matter-coupled supergravity, a constant Fayet–Iliopoulos term gauges an R-symmetry and requires a corresponding nontrivial transformation of the superpotential. The apparent charges $0,+1,-1$ make the supplied cubic superpotential neutral, so a nonzero independent constant $\zeta$ does not by itself specify a consistent conventional supergravity action. The computed branches are those of the supplied scalar potential; a full embedding needs compatible charges or additional fields. This restriction is explained in Van Proeyen's derivation of the superpotential gauge-covariance condition.

More generally, the supergravity moment-map constraint on D-term breaking follows directly from gauge covariance. If $k^i{\partial }_{i}W=-{\kappa }^{2}rW$ and the moment map is $\mathcal{P}=i(k^i{K}_i-r)$, then at a point with $W\ne 0$,Consequently all chiral auxiliary fields vanishing implies the D-term also vanishes there. This relation has no analogue forcing that implication in an arbitrary global supersymmetry model. Its $W\ne 0$ hypothesis matters: the formal D-breaking branch above has $W=0$, so it does not contradict the identity. These conventional gauge-completion restrictions are additional to the elementary minimization calculation.

Use left Grassmann derivatives and define the supersymmetric covariant derivatives $D_{\pm }={\partial }_{{\theta }^{\pm }}-i{\theta }^{\pm }{\partial }_{\pm }$. Multiplication in the indicated order gives exactly the printed differential operator:Thus the equation is $i{D}_{-}{D}_{+}\Phi =W(\Phi )$ in two-dimensional N=(1,1) superspace.

Under a Lorentz boost, take $x^{\pm }\mapsto e^{\pm \omega }{x}^{\pm }$ and ${\theta }^{\pm }\mapsto e^{\pm \omega /2}{\theta }^{\pm }$. The derivatives ${\partial }_{\pm }$ have boost weights $\mp 1$, and $D_{\pm }$ have weights $\mp 1/2$. Consequently $D_{-}{D}_{+}$ has weight zero, and is a Lorentz scalar. Both $\Phi$ and the ordinary function $W(\Phi )$ are scalar superfields, so the equation is Lorentz invariant.

For supersymmetry, $\{{\partial }_{\theta },\theta \}=1$ and anticommutation of distinct odd coordinates giveMoving either supercharge through the two odd derivatives introduces two minus signs, hence $[Q_t,D_{-}{D}_{+}]=0$. If $\delta \Phi ={ϵ}^{+}{Q}_{+}\Phi +{ϵ}^{-}{Q}_{-}\Phi$, with odd constant parameters, the chain rule for an even scalar superfield givesTherefore a solution is mapped to a solution, proving supersymmetry invariance.

To compute the supersymmetric Liouville equation, order the odd coordinates as ${\theta }^{-}{\theta }^{+}$ and anticommute the fermionic component fields with both coordinates. In particular, ${\partial }_{{\theta }^{+}}({\theta }^{-}{\theta }^{+})=-{\theta }^{-}$. Direct application of the operator givesLet $N=\Phi -\varphi$. Only the two products of its degree-one terms survive in $N^2$; for example, $(i{\theta }^{-}{\psi }_{+})(i{\theta }^{+}{\psi }_{-})={\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$. Hence $N^2=2{\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$ and $N^3=0$. The exponential function expands exactly asEquating the constant terms fixes the auxiliary field, $\boxed{F=e^{\varphi }}$. The linear and quadratic terms then give the requested component field equations of motion:The signs here follow from the printed operator, the printed component expansion, and left Grassmann derivatives; no convention change has been made in eliminating the auxiliary field.

The barred notation denotes the antichiral supersymmetric covariant derivative. Write barred spinor indices as dotted indices, and use the stated plus-sign convention with left Grassmann derivatives:Their supercovariant derivative algebra with left derivatives gives $\{{\bar{D}}_{\dot{\alpha }},{\bar{D}}_{\dot{\beta }}\}=0$ and $\{D_{\alpha },{\bar{D}}_{\dot{\beta }}\}=2i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\partial }_{\mu }$. The constraint $\boxed{{\bar{D}}_{\dot{\alpha }}\Phi =0}$ defines a chiral superfield, also called an L-superfield.

To solve the constraint, put $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$. For a left Grassmann derivative, ${\partial }_{{\bar{\theta }}^{\dot{\alpha }}}(\theta {\sigma }^{\mu }\bar{\theta })=-{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }$. Therefore ${\bar{D}}_{\dot{\alpha }}{y}^{\mu }=0$ and ${\bar{D}}_{\dot{\alpha }}{\theta }^{\beta }=0$, so $\bar{D}$ becomes ordinary differentiation with respect to $\bar{\theta }$ at fixed $y$. Its general solution is the chiral-superfield component expansionHere $A$ is a complex scalar, $\chi$ a Weyl spinor, and $F$ a complex auxiliary field. Before imposing field equations of motion this is four real bosonic and four real fermionic components. Chirality means independence of $\bar{\theta }$ at fixed $y$, rather than at fixed $x$. The differential supercharges $Q_{\alpha }={\partial }_{{\theta }^{\alpha }}-i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{Q}}_{\dot{\alpha }}={\partial }_{{\bar{\theta }}^{\dot{\alpha }}}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$ anticommute with both covariant derivatives, so a supersymmetry transformation preserves the chiral constraint.

Here L-superfield means a left-chiral superfield: ${\bar{D}}_{\dot{\alpha }}\Phi =0$, and its complex conjugate is an antichiral superfield, $D_{\alpha }{\Phi }^{\ast }=0$. Interpret the two barred derivatives as the contracted scalar ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$. The key point is chiral projection by squared supercovariant derivatives. Since the two components of $\bar{D}$ anticommute, any product of three barred derivatives repeats a component and is zero. Thusfor every superfield $X$, even when $X$ is antichiral. In particular, ${\bar{D}}^2{\Phi }^{\ast }$ is chiral.

The right side is chiral too. Because $\Phi$ is even, the graded derivation rule gives ${\bar{D}}_{\dot{\alpha }}({\Phi }^2)=({\bar{D}}_{\dot{\alpha }}\Phi )\Phi +\Phi ({\bar{D}}_{\dot{\alpha }}\Phi )=0$. Constant $m$ and $\lambda$ do not contribute derivatives. Consequentlyso applying the chiral constraint to the equation imposes no additional incompatible condition. This is the holomorphic closure of chiral superfields; a generic nonholomorphic right side would not satisfy it.

Both sides have zero Grassmann parity and are Lorentz scalars: the two dotted derivative indices are contracted with the invariant epsilon tensor, and $\Phi$ is a scalar superfield. Moreover, each covariant derivative anticommutes with every supercharge, so its squared contraction commutes with the supercharges. Hence the equation transforms covariantly under supersymmetry. It is the chiral field equation of motion of the Wess–Zumino model with superpotential $\mathcal{W}(\Phi )=m{\Phi }^2/2+\lambda {\Phi }^3/3$, for which ${\mathcal{W}}^{\prime }=m\Phi +\lambda {\Phi }^2$, using the normalization in the PDF. With mass dimensions $[\Phi ]=1$, $[\bar{D}]=1/2$, $[m]=1$ and $[\lambda ]=0$, each side has dimension two. No component field equations of motion are needed for this consistency argument.

Work through terms quadratic in the fermions, with Grassmann-odd Majorana spinors ${\psi }_{\mu }$ and $ϵ$. At this order the spin connection can be taken torsion-free: eliminating its gravitino-induced torsion changes the action only at four-fermion order. The Christoffel term needed to turn the printed $D_{\nu }{\psi }_{\sigma }$ into a full vector-spinor covariant derivative drops out after contraction with ${\gamma }^{\mu \nu \sigma }$, since the connection is symmetric in $\nu ,\sigma$.

First establish the Einstein tensor contraction of spin curvature. Expanding the Clifford product and using the first Bianchi identity, its five-gamma and antisymmetric three-gamma curvature contributions vanish, leavingSince $[{\nabla }_{\nu },{\nabla }_{\rho }]=R_{\nu \rho ab}{\gamma }^{ab}/4$, it follows thatThe factor $1/2$ arises from antisymmetrizing the two derivatives, not from a choice of Einstein-Hilbert normalization.

For the Einstein-Hilbert action, the metric variation isVariation with respect to the covariant metric, including the determinant variation and discarding the usual boundary divergence, givesNo gravitational field equation has been assumed here.