Supergravity (SUGRA)

Supergravity is a theory with local supersymmetry and necessarily contains gravity.

The scalar fields of chiral matter induce a composite phase connection proportional to $i(K_i{\partial }_{\mu }{z}^i-K_{\bar{i}}{\partial }_{\mu }{\bar{z}}^{\bar{i}})$. For canonical Kähler potential it is proportional to the scalar phase current. The chiral matter fermions and gravitino have appropriate Kähler weights; expanding their connection terms produces scalar-current–fermion-bilinear couplings. Overall signs and coefficients depend on spinor and Planck-scale conventions.

Eleven-dimensional supergravity contains a metric, a three-form potential and a gravitino. Its bosonic on-shell counts are $11(11-3)/2=44$ for the metric and $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$ for the three-form, totaling $128$. Flat toroidal zero-mode reduction gives type IIA supergravity in ten dimensions and four-dimensional N=8 supergravity in four dimensions without losing these bosonic polarizations.

// Target: supersymmetry.bigb

A product compactification supported by a four-form proportional to one factor's volume form. For the maximally supersymmetric membrane throat, the other factor is a round $S^7$ and the radii obey $R_{S^7}=2R_{\mathrm{AdS}}$. The flux links the curvature scales through the supergravity field equations.

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

// Target: geometry-and-topology.bigb

Circle reduction splits the metric into a ten-dimensional graviton, one vector and one scalar, and the three-form into a three-form and two-form. Seven-torus reduction gives a four-dimensional graviton, $7+21=28$ vectors, $28+35$ direct scalars and seven two-forms. The two-form scalar duality turns the latter into seven more scalars, giving $70$ in all. A remaining four-dimensional three-form has no local polarization. These are zero-mode counts without fluxes, orbifold projections or moduli freezing.

For circle-independent fields, the metric gives the type-IIA string-frame metric, dilaton and RR one-form. Decomposing $A_3=C_3+B_2\wedge dy$ gives the RR three-form and Kalb–Ramond field. With $H_3=d{B}_2$, $F_2=d{C}_1$, the horizontal four-form is ${\tilde{F}}_4=d{C}_3-C_1\wedge H_3$. The Einstein term reduces, modulo a total derivative, to $\sqrt{-g}[e^{-2\Phi }(R+4(\nabla \Phi {)}^2)-\frac{1}{4}{F}_{2,\mu \nu }{F}_2^{\mu \nu }]$. The eleven-dimensional form norm gives $\sqrt{-g}[e^{-2\Phi }|H_3{|}^2+|{\tilde{F}}_4{|}^2]$, where $|F_k{|}^2=F^2/k!$. Ordinary reduction generates massless type IIA supergravity, not Romans mass.

Put $A_3=C_3+B_2\wedge dy$ and $F_4=G_4+H_3\wedge dy$, where $G_4=d{C}_3$. The circle component of $A_3\wedge F_4\wedge F_4$ is $[B_2\wedge G_4\wedge G_4+2C_3\wedge G_4\wedge H_3]\wedge dy$. Since $d(C_3\wedge B_2\wedge G_4)=B_2\wedge G_4\wedge G_4-C_3\wedge H_3\wedge G_4$, its integral equals three copies of $B_2\wedge G_4\wedge G_4$ up to a boundary term. The coefficient therefore becomes $-1/2$ after circle integration. Boundary terms and global potential patching must be retained when the spacetime has boundary or nontrivial flux topology.

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

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

A parallel commuting Dirac current is causal. If timelike, the geometry locally splits as a time line times a three-dimensional Ricci-flat metric; three-dimensional Ricci flatness implies flatness. If null, the stabilizer of a parallel spinor in $\mathrm{Spin}(1,3)$ contains only null rotations, and the local vacuum metric is $d{s}^2=-2dudv+d{x}^2+d{y}^2+H(u,x,y)d{u}^2$ with $H_{xx}+H_{yy}=0$. Constant spinors obeying ${\gamma }^{+}ϵ=0$ are parallel. These are local statements; global quotients must preserve a compatible spin structure.

Minimal four-dimensional supergravity couples a graviton to one Majorana gravitino. In the normalization $\delta e_{\mu }{}^a=2\kappa \overline{ϵ}{\gamma }^a{\psi }_{\mu }$, $\delta {\psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$, its action through quadratic fermion order is $\int e[R/(2{\kappa }^2)-2{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\psi }_{\rho }]$. Rescaling the gravitino and parameter by two gives the usual canonical kinetic coefficient $-1/2$. The curvature identity ${\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\nabla }_{\rho }ϵ=G^{\mu \nu }{\gamma }_{\nu }ϵ/2$ cancels the Einstein-Hilbert variation against the Rarita-Schwinger variation. The omitted cubic-fermion variations require the quartic-fermion completion.

Old-minimal four-dimensional supergravity adds a complex scalar $M$ and a real vector $b_{\mu }$ to the vierbein and gravitino. These six real bosonic auxiliary field components have no propagating kinetic terms. They balance the off-shell component count of minimal supergravity and permit the local supersymmetry algebra to close without imposing the propagating Euler-Lagrange field equations.

A four-dimensional Majorana spinor gravitino has $4\times 4=16$ real components, minus four local supersymmetry gauge functions, leaving twelve before using any Euler-Lagrange field equation. A vierbein has sixteen real components, minus six local Lorentz transformation and four diffeomorphism gauge functions, leaving six. Equivalently, a symmetric metric tensor has $10-4=6$. The propagating graviton and massless gravitino each have two states on shell. Six bosonic auxiliary field components complete an off-shell supergravity multiplet.

With canonical Rarita-Schwinger field normalization, one convention gives the displayed torsion tensor. Its contorsion tensor is $K_{\mu ab}={\kappa }^2({\bar{\psi }}_{\mu }{\gamma }_a{\psi }_b-{\bar{\psi }}_{\mu }{\gamma }_b{\psi }_a+{\bar{\psi }}_a{\gamma }_{\mu }{\psi }_b)/4$. The algebraic spin connection equation determines $\tilde{\omega }=\omega (e)+K$; substituting it in the action produces four-fermion terms. This is an instance of Einstein-Cartan theory and is efficiently varied using the 1.5-order formalism. A rescaled gravitino changes the displayed coefficient.

An Abelian vector multiplet contains a Maxwell field and a Majorana gaugino. With the doubled gravitino normalization, add $e[-F^2/4-\bar{\lambda }/\nabla \lambda /2-(\kappa /2){\bar{\psi }}_{\mu }{F}_{ab}{\gamma }^{ab}{\gamma }^{\mu }\lambda ]$. The leading transformations are $\delta A_{\mu }=\overline{ϵ}{\gamma }_{\mu }\lambda$ and $\delta \lambda =-F_{ab}{\gamma }^{ab}ϵ/2$. Promoting the rigid parameter to a function produces the current $(F_{ab}/2){\gamma }^{ab}{\gamma }^{\mu }\lambda$ multiplying ${\nabla }_{\mu }\overline{ϵ}$; the gravitino-current term cancels it. The Clifford identity $(F_{ab}{\gamma }^{ab}){\gamma }^{\mu }(F_{cd}{\gamma }^{cd})=8T_{\mathrm{EM}}^{\mu \nu }{\gamma }_{\nu }$ cancels the Maxwell stress variation. Supercovariant field strengths and quartic fermions complete the transformations beyond this order.

A real cosmological deformation adds $3m^{2}e/{\kappa }^2+2me{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu }{\psi }_{\nu }$ and changes the gravitino variation to ${\mathcal{D}}_{\mu }ϵ/\kappa$. The modified commutator identity is ${\gamma }^{\mu \nu \rho }{\mathcal{D}}_{\nu }{\mathcal{D}}_{\rho }ϵ=(G^{\mu \nu }+\Lambda g^{\mu \nu }){\gamma }_{\nu }ϵ/2$. It fixes the negative cosmological constant in terms of the gravitino mass parameter. A positive cosmological constant is not obtained by this real unbroken minimal deformation.

The ungauged maximal nine-dimensional supergravity zero-mode multiplet has one metric, three vectors, two two-forms, one three-form and three scalars, giving $128$ bosonic states. Two Majorana gravitini and four Majorana spin-one-half fields give $128$ fermionic states. Flat-circle reductions of massless type IIA and type IIB theories, and a flat-two-torus reduction of eleven-dimensional supergravity, give this same nonchiral spectrum.

Type IIB supergravity is the chiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond zero-form, two-form and four-form potentials, and a self-dual five-form field strength. Two same-chirality Majorana-Weyl gravitini and two opposite-to-gravitino-chirality Majorana-Weyl dilatini complete its $128+128$ massless spectrum.

Massless type IIA supergravity is the nonchiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond one-form and three-form potentials, two opposite-chirality Majorana-Weyl gravitini, and two opposite-chirality Majorana-Weyl dilatini. Its massless spectrum has $128$ bosonic and $128$ fermionic polarizations. The two supercharges have opposite ten-dimensional chirality.

The zero-form field strength deforms massless type IIA supergravity. A D8-brane changes this flux across its worldvolume; it is therefore outside the elementary massless M-theory circle duality dictionary.

// Target: physics.bigb

A dilatino is a spin-one-half fermionic partner in a supergravity theory containing a dilaton. Type IIA and type IIB theories each contain two Majorana-Weyl dilatini with eight physical polarizations apiece. Their chirality assignments differ between the two theories.

When local supersymmetry is spontaneously broken, the gravitino absorbs the goldstino and becomes massive. The two goldstino states provide its longitudinal spin-$1/2$ polarizations. With only global supersymmetry breaking, the goldstino remains a physical massless particle. An ordinary supersymmetric Higgs mechanism instead makes an internal spin-one gauge boson massive and can leave supersymmetry unbroken.

The gravitino is the spin-$3/2$ gauge field of local supersymmetry and the partner of the graviton. A massless four-dimensional gravitino has two physical helicities, $\pm 3/2$; a massive one has four, adding $\pm 1/2$. The super-Higgs mechanism supplies these extra states from the goldstino.

In a no-scale sector, the Kähler metric satisfies $K_i{K}^{i\bar{j}}{K}_{\bar{j}}=3$. If the superpotential is independent of that sector, its covariant derivatives contribute $3e^K|W{|}^2$, cancelling the universal $-3e^K|W{|}^2$ in the supergravity F-term potential. Spectator sectors can still contribute to the potential. This cancellation permits zero-energy minima with broken supersymmetry and exact flat scalar directions.

For $s=S+\bar{S}>0$, $t=T+\bar{T}-|C{|}^2>0$, $K=-\log s-3\log t$ and a holomorphic superpotential independent of $T$, the exact F-term potential is $V=e^K[s^2|D_{S}W{|}^2+(t/3)|W_C{|}^2]$. Inverting the mixed $T,C$ metric gives $K^{T\bar{T}}=t(t+|C{|}^2)/3$, $K^{T\bar{C}}=t\bar{C}/3$, $K^{C\bar{T}}=tC/3$, and $K^{C\bar{C}}=t/3$. Substitution of $D_{T}W=-3W/t$ and $D_{C}W=W_C+3\bar{C}W/t$ cancels all mixed terms, leaving $3|W{|}^2+(t/3)|W_C{|}^2$; the first term cancels the supergravity negative term.

If a no-scale supergravity scalar potential has the displayed form with $y>0$ and nonnegative $A,B$ independent of the volume coordinate $y$, every point with $A+B>0$ has a direction of strictly decreasing potential. Thus a finite stationary vacuum requires $A=B=0$. If that common zero set is empty, the potential has infimum zero along $y\to \infty$ but no finite minimum. If the common zero set is nonempty, varying $y$ on it is a flat direction of a scalar potential. This distinguishes an attained zero-energy vacuum from a decompactification runaway.

For $W=C^3+A+b$, $A=a{e}^{-\alpha S}$, the potential is $V=|C^3+b+(1+\alpha s)A{|}^2/(s{t}^3)+3|C{|}^4/(s{t}^2)$. A finite zero-energy vacuum requires $C=0$ and $b+(1+\alpha s)A=0$. With $a,b\ne 0$, such a solution exists exactly when $0<|b/a|\le 2e^{-1/2}$: the magnitude equation is $|b/a|=(1+x)e^{-x/2}$, $x=\alpha s>0$, whose maximum occurs at $x=1$. At that vacuum $F^S=F^C=0$ but $F^T=e^{K/2}t\bar{W}\ne 0$. Both components of $T$ are flat, while $C$ is quartically lifted. If no root exists, the positive potential falls toward zero as $t\to \infty$ and has no finite minimum. If $a=b=0$, the $C=0$ family is supersymmetric instead.

For the cubic-matter no-scale vacuum with an exponential dilaton, put $x=S+\bar{S}$ and $A=a{e}^{-\alpha S}$. At $C=0$, zero energy requires $G(S)=b+(1+\alpha x)A=0$. On a fixed-imaginary-part slice, $s=\mathrm{Re}S$ gives $dG/ds=\alpha (1-2\alpha s)A$. At the double root $s_0=1/(2\alpha )$, the first derivative vanishes but the second is $-2{\alpha }^{2}A$, so $G=-{\alpha }^{2}A(s-s_0{)}^2+O((s-s_0{)}^3)$. Therefore $V=|G{|}^2/(x{y}^3)$ begins at fourth order. This real scalar has zero quadratic mass but is not an exact flat direction of a scalar potential. The imaginary part of $S$ is still lifted quadratically, and the matter field $C$ also has a quartic leading potential. Exact flat directions instead come from the complex volume modulus $T$ along the zero-energy locus.

The no-scale supergravity identity cancels the negative gravitino contribution, leaving $V=e^{|C{|}^2}|3C^2+\bar{C}(C^3+B){|}^2/(T+\bar{T}{)}^3$. Zero-energy minima occur at $C=0$ and, for $B\ne 0$, three phase-related nonzero solutions with $|B|=|C|(3+|C{|}^2)$. Their $T$ supergravity auxiliary field is nonzero if $B\ne 0$, while both real components of $T$ are flat on each vacuum branch. For $B=0$, the $C=0$ branch is instead supersymmetric and has zero gravitino mass from a superpotential.

For $K=-\log (S+\bar{S})-3\log (T+\bar{T})$ and $W=a{e}^{-\alpha S}+b$ with $\alpha >0$, the potential is $|b+[1+\alpha (S+\bar{S})]a{e}^{-\alpha S}{|}^2/[(S+\bar{S})(T+\bar{T}{)}^3]$. With nonzero $a,b$, a finite zero-energy vacuum exists exactly when $0<|b/a|\le 2e^{-1/2}$. To prove this, set $x=\alpha \mathrm{Re}S>0$; the magnitude equation is $|b/a|=(1+2x)e^{-x}$, whose derivative changes sign at $x=1/2$ and whose maximum is $2e^{-1/2}$. The phase fixes the imaginary part of $S$. At a zero-energy solution the $T$ auxiliary field remains nonzero, so supersymmetry is broken and both real components of $T$ are flat. If $a=b=0$, the family is instead supersymmetric.

Let $K=-3\log \Gamma$ where $\Gamma >0$ is twice differentiable and homogeneous of degree one. If the Kähler Hessian is invertible, the Euler theorem for homogeneous functions identities give ${\tau }_i{K}_{ij}=3{\Gamma }_j/\Gamma$, then $K_{ij}^{-1}{\Gamma }_j/\Gamma ={\tau }_i/3$ and ${\Gamma }_i{K}_{ij}^{-1}{\Gamma }_j/{\Gamma }^2=1/3$. Multiplication by nine gives the no-scale identity. A positive degree-one $\Gamma$ need not induce an invertible or positive Kähler metric: $\Gamma ={\tau }_1+{\tau }_2$ gives a rank-one metric for $K$. The Hessian matrix $({\Gamma }_{ij})$ itself always has the radial null vector $\tau$, whereas the Hessian matrix of $K$ can be invertible. In a larger theory the relevant full inverse metric, or a decoupled block, must obey the identity.

After eliminating the chiral auxiliary fields in Planck-unit supergravity, one convention gives $F^i=-e^{K/2}{K}^{i\bar{j}}\overline{D_{j}W}$. The upper-index inverse satisfies $K_{i\bar{j}}{K}^{k\bar{j}}={\delta }_i^k$. Nonzero auxiliary expectation values diagnose supersymmetry breaking; their common phase convention does not affect the vanishing condition.

Four-dimensional $\mathcal{N}=8$ supergravity has eight independent Weyl spinor supercharges. Its supergravity multiplet has helicities from $2$ to $-2$, with multiplicities $1,8,28,56,70,56,28,8,1$.

A supergravity multiplet is a supermultiplet containing a graviton and its supersymmetric partners. In four-dimensional N=8 supergravity, the irreducible massless supermultiplet contains $128$ boson and $128$ fermion states.

In Planck units, chiral multiplets in four-dimensional $\mathcal{N}=1$ supergravity have

Let $k^i$ generate a holomorphic gauge transformation, with $k^i{\partial }_{i}W=-{\kappa }^{2}rW$ and moment map $\mathcal{P}=i(k^i{K}_i-r)$. Substitution of the Kähler covariant derivative of a superpotential gives the displayed identity. Consequently, a finite point with $W\ne 0$ and all $D_{i}W=0$ also has $\mathcal{P}=0$. Pure D-term breaking with vanishing chiral auxiliary fields can therefore evade this argument only where $W=0$ or outside its hypotheses. In conventional two-derivative matter-coupled supergravity, a constant Fayet–Iliopoulos term gauges an R-symmetry and constrains the transformation of $W$; a neutral superpotential and an independently assigned nonzero constant shift are not generally consistent. The gauge-covariance restriction is derived in Van Proeyen's discussion of Fayet–Iliopoulos terms and R-symmetry.

The gravitino mass parameter in four-dimensional supergravity is the displayed expression. In a zero-energy supersymmetry breaking vacuum with only F-term breaking, $K_{i\bar{j}}{F}^i\overline{F^j}=3m_{3/2}^2/{\kappa }^2$. Writing this norm as ${\Lambda }_{\mathrm{SUSY}}^4$ gives $m_{3/2}={\Lambda }_{\mathrm{SUSY}}^2/(\sqrt{3}{m}_p)$. It is small relative to ${\Lambda }_{\mathrm{SUSY}}$ when the breaking scale is below the Planck mass. The super-Higgs mechanism supplies the longitudinal gravitino states.

The Kähler covariant derivative of a superpotential transforms by the same holomorphic factor as the superpotential under a Kähler transformation. This follows by differentiating $e^{-f}W$ and cancelling the extra derivative of $f$ against the shift of $K_i$.

The Polonyi model uses a canonical Kähler potential and a superpotential linear in one chiral superfield to obtain spontaneous supersymmetry breaking.

For $m\ne 0$ and positive $\beta$, the Polonyi model has a unique zero-energy global minimum at $\beta =2-\sqrt{3}$, $⟨z⟩=\sqrt{3}-1$. Writing $u=\mathrm{Re}z-(\sqrt{3}-1)$ and $y=\mathrm{Im}z$, its potential bracket is $(u^2+y^2+\sqrt{3}u{)}^2+(2\sqrt{3}-3)u^2+(4-2\sqrt{3})y^2$, a sum of nonnegative terms. The real and imaginary curvatures are $4\sqrt{3}$ and $8-4\sqrt{3}$. The other zero-energy stationary point with $\beta =2+\sqrt{3}$ and $z=-\sqrt{3}-1$ is a saddle point. The stable vacuum has nonzero supergravity auxiliary field, so it has supersymmetry breaking even with zero cosmological constant. The $m=0$ branch does not fix these values.

For the Polonyi model with canonical Kähler potential and $W=m^2(z+\beta )$, $\beta >0$, a constant supersymmetric vacuum with $m\ne 0$ requires real $z=(-\beta \pm \sqrt{{\beta }^2-4})/2$ and exists exactly for $\beta \ge 2$. Its supergravity F-term potential is negative because $W\ne 0$. For $0<\beta <2$, the supergravity auxiliary field cannot vanish. If $m=0$, the model is flat with unbroken supersymmetry. This parameter-dependent statement is sharper than asserting that every linear superpotential breaks supersymmetry in supergravity.