nLab supergravity

Idea

A quantum field theory of supergravity is similar to the theory of gravity, but where (in first order formulation) the latter is given by an action functional (the Einstein-Hilbert action functional) on the space of connections (over spacetime) with values in the Poincare Lie algebra $\mathfrak{iso}(n,1)$, supergravity is defined by an extension of this to an action functional on the space of connections with values in the super Poincare Lie algebra $\mathfrak{siso}(n,1)$. One says that supergravity is the theory of local (Poincaré) supersymmetry in the same sense that ordinary gravity is the theory of “local Poincaré-symmetry”. These are gauge theories for the Poincare Lie algebra and the super Poincare Lie algebra, respectively, in that the field (physics) is a Cartan connection for the inclusion $o(n,1) \hookrightarrow \mathfrak{siso}(n,1)$:

if we write $\mathfrak{siso}(n,1)$ as a semidirect product of the translation Lie algebra $\mathbb{R}^{(n,1)}$, the special orthogonal Lie algebra $\mathfrak{so}(n,1)$ and a spin group representation $\Gamma$, then locally a connection is a Lie algebra valued 1-form

$$A : T X \to \mathfrak{siso}(n,1)$$

that decomposes into three components, $A = (E, \Omega, \Psi)$:

• a $\mathbb{R}^{n,1}$-valued 1-form $E$ – the vielbein

  (this encodes the pseudo-Riemannian metric and hence the field of gravity);

• a $\mathfrak{so}(n,1)$-valued 1-form $\Omega$ – called the spin connection;

• a $\Gamma$-valued 1-form $\Psi$ – called the gravitino field.

Typically in fact the field content of supergravity is larger, in that a field $A$ is really an ∞-Lie algebra-valued differential form with values in an ∞-Lie algebra such as the supergravity Lie 3-algebra (DAuriaFreCastellani) $\mathfrak{sugra}(10,1)$. Specifically such a field

$$A : T X \to \mathfrak{sugra}(10,1)$$

has one more component

• a 3-form $C$ – the supergravity C-field.

The gauge transformations on the space of such connections that are parameterized by the elements of $\Gamma$ are called supersymmetries.

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

The condition of gauge invariance of an action functional on $\mathfrak{siso}$-connections is considerably more restrictive than for one on $\mathfrak{iso}$-connections. For instance there is, under mild assumptions, a unique maximally supersymmetric supergravity extension of the ordinary Einstein-Hilbert action on a 4-dimensional manifold. This in turn is obtained from the unique (under mild assumptions) maximally supersymmetric supergravity action functional on a (10,1)-dimensional spacetime by thinking of the 4-dimensional action function as being a dimensional reduction of the 11-dimensional one.

This uniqueness (under mild conditions) is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize gravity as a quantum field theory. Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully renormalizable. This couldn’t be shown computationally – until recently: triggered by new insights recently there there has been lots of renewed activity on the renormalizability of maximal supergravity.

Physical motivations

The idea of supergravity (SuGra) is theoretically suggestive: Where ordinary gravity is Cartan geometry for the Spin subgroups of Poincaré group, the latter have natural (higher) supergroup extensions (supersymmetry, SuSy), and supergravity is essentially the super Cartan geometry for the Spin subgroups of these (higher) super Poincaré groups (more on this below).

Less widely appreciated may be that discussion of supergravity is also physically motivated (beyond the widely discussed application of holography to solid state physics and to quark-gluon plasma physics):

• In fundamental physics, the high energy local SuSy embodied by SuGra remains phenomenologically viable, in contrast to the low energy global SuSy that has notoriously not been seen in experiment. In particular, SuGra is known to improve models of inflationary cosmology, by naturally furnishing not only the required scalar fields themselves but also explaining their observationally favored plateau potentials (the $\alpha$-attractor mechanism) and their stability against quantum corrections (an $\eta$-problem resolution).

• In particle physics, despite common perception, supersymmetry is experimentally seen in the form of hadron supersymmetry in the “flavor sector” instead of the commonly considered “color sector” of the standard model. Here it naturally fits into holographic models realizing quantum chromodynamics on supergravitational branes.

• In solid state physics, topological order in quantum materials is a 21st-century holy grail, plausibly necessary for stabilizing future utility-scale quantum computing hardware. To date, a single class of candidate systems has been experimentally realized with certainty: fractional quantum Hall (FQH) liquids.

  Remarkably, it has been established that the gapped excitations of certain non-abelian FQH liquids fall into multiplets comprising a (massive graviton-like) spin-2 magneto-roton and a (massive gravitino-like) spin-3/2 neutral fermion mode which are partners under an emergent supersymmetry, suggestive of an effective (2D massive) supergravity description.

  Furthermore, the effective symmetry of FQH excitations is “special” diffeomorphism invariance ($W_\infty$-symmetry). In combination with SuSy this matches the characteristic symmetry on super $p$-brane probes of supergravity backgrounds, suggesting that the effective field theory of FQH liquids may naturally be understood via geometric engineering of strongly coupled quantum systems on branes probing supergravity orbifolds.

• In mathematical metaphysics, the brane bouquet shows that out of the superpoint — the classifying space for fermion fields per se — grows a hierarchy of (the local structures of) ever-higher dimensional supergravity theories and their super $p$-brane content, culminating in the local structure of 11D supergravity with its M2/M5-branes. Therefore, super-gravity and branes are fundamentally hard-coded into the BIOS of mathematical physics. The surprising emergence of SuGra in the effective theories of hadrons and of FQH liquids, noted above, may be an instance of this universality.

As super Cartan geometry

Ordinary Einstein gravity has a natural formulation in terms of Cartan geometry for the inclusion of the Lorentz Lie algebra into the Poincaré Lie algebra $\mathfrak{o}(d-1,1) \hookrightarrow \mathfrak{Iso}(\mathbb{R}^{d-1,1})$. In this first order formulation of gravity a field configuration is a Cartan connection with such coefficients.

This perspective directly generalizes to supergeometry and yields the superspace formulation of theories of supegravity – super Cartan geometry.

After picking a dimension $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the Poincaré Lie algebra, then a choice of “number of supersymmetries” is a choice of real spin representation $N$. Then the direct sum

$$\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}) \coloneqq \underbrace{\mathfrak{Iso}(\mathbb{R}^{d-1,1})}_{even} \oplus \underbrace{N}_{odd}$$

regarded as a super vector space with $N$ in odd degree becomes a super Lie algebra by letting the $[even,odd]$ bracket be given by the defining action and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing – the super Poincaré Lie algebra. This still canonical contains the Lorentz Lie algebra $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the quotient

$$\mathbb{R}^{d-1,1|N} \coloneqq \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1})$$

is called super Minkowski spacetime (equipped with its super translation Lie algebra structure).

From this, a super-Cartan geometry is defined in direct analogy to the Cartan formulation of Riemannian geometry

(higher-)Cartan geometry	$\mathfrak{g}$	$\mathfrak{h}$	$\mathfrak{g}/\mathfrak{h}$
Einstein gravity	$\mathfrak{Iso}(\mathbb{R}^{d-1,1})$	$\mathfrak{o}(d-1,1)$	$\mathbb{R}^{d-1,1}$
supergravity	$\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$	$\mathfrak{o}(d-1,1)$	$\mathbb{R}^{d-1,1\vert N}$
11-dimensional supergravity	$\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert N=1})$	$\mathfrak{o}(d-1,1)$	$\widehat{\mathbb{R}}^{10,1\vert N=1}$

Indeed, all the traditional literature on supergravity (e.g. (Castellani-D’Auria-Fré 91)) is phrased, more or less explicitly, in terms of Cartan connections for the inclusion of the Lorentz group into the super Poincaré group this way, this being the formalization of what physicists mean when saying that they pass to “local supersymmetry”.

One subtlety to take care of is that this makes spacetime a super-spacetime locally modeled on super Minkowski spacetime. But the resulting theory is supposed to be a field theory on an ordinary spacetime locally modeled on ordinary Minkowski spacetime. This is enforced by a further constraint on the super-Cartan connection which forces it to be determined by the bosonic manifold underlying the given supermanifold. This constraint is variously known as the superspace constraints or as rheonomy .

The other subtlety to take care of is that a key aspect of higher dimensional supergravity theories is that their field content necessarily includes, in addition to the graviton and the gravitino, higher differential n-form fields, notably the 2-form B-field of 10-dimensional type II supergravity and heterotic supergravity as well as the 3-form C-field of 11-dimensional supergravity.

This means that these higher dimensional supergravity theories are not in fact entirely described by super-Cartan geometry – but by super-higher Cartan geometry.

This follows a key insight due to (D’Auria-Fré-Regge 80, D’Auria-Fré 82) – the D'Auria-Fre formulation of supergravity – that the “tensor multiplet” fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general Chevalley-Eilenberg algebras.

Namely, we may add to the above CE-algebra

• a single generator $c_3$ of degree $(3,even)$

and extend the differential to that by the formula

$$d_{CE} \, c_3 = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \,.$$

This still squares to zero due to the remarkable property of 11d super Minkowski spacetime by which $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ is a representative of an exception super-Lie algebra cohomology class. (The collection of all these exceptional classes constitutes what is known as the brane scan).

In the textbook (Castellani-D’Auria-Fré 91) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.

But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of Lie algebras but of strong homotopy Lie algebras, hence of L-infinity algebras, in fact of Lie (p+1)-algebras for $(p+1)$ the degree of the relevant differential form field.

Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane})$

of the supergravity Lie 3-algebra $\mathfrak{m}2\mathfrak{brane}$.

Now a morphism

$$\Omega^\bullet(U) \stackrel{}{\longleftarrow} CE(\mathfrak{m}2\mathfrak{brane}) \;\colon\; A$$

encodes graviton and gravitino fields as above, but in addition it encodes a 3-form

$$C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U)$$

whose curvature

$$G_4 = \mathbf{d}C_3 + \frac{1}{2}\bar \Psi \Gamma^{a b} \wedge \Psi \wedge E_a \wedge E_b$$

satisfies a modified Bianchi identity, crucial for the theory of 11-dimensional supergravity (D’Auria-Fré 82).

So this collection of differential form data is no longer a Lie algebra valued differential form, it is an L-infinity algebra valued differential form, with values in the supergravity Lie 3-algebra.

The quotient

$$\widehat{\mathbb{R}}^{10,1|N=1} \coloneqq \mathfrak{g}/\mathfrak{h} = \mathfrak{m}2\mathfrak{brane} / \mathfrak{o}(\mathbb{R}^{10,1|N=1})$$

is known as extended super Minkowski spacetime.

The Lie integration of this is a smooth 3-group $G$ which receives a map from the Lorentz group.

This means that a global description of the geometry which (Castellani-D’Auria-Fré 91) discuss locally on charts has to be a higher kind of Cartan geometry which is locally modeled not just on cosets, but on the homotopy quotients of (smooth, supergeometric, …) infinity-groups – higher Cartan geometry.

Solutions with global supersymmetry

A solution to the bosonic Einstein equations of ordinary gravity – some Riemannian manifold – has a global symmetry if it has a Killing vector.

Accordingly, a configuration that solves the supergravity Euler-Lagrange equations is a global supersymmetry if it has a Killing spinor: a covariantly constant spinor.

Here the notion of covariant derivative includes the usual Levi-Civita connection, but also in general torsion components and contributions from other background gauge fields such as a Kalb-Ramond field and the RR-fields in type II supergravity or heterotic supergravity.

Of particular interest to phenomenologists around the turn of the millennium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form $M^4 \times Y^6$ for $M^4$ the locally observed Minkowski spacetime (that plays a role as the background for all available particle accelerator experiments) and a small closed 6-dimensional Riemannian manifold $Y^6$.

In the absence of further fields besides gravity, the condition that such a configuration has precisely one Killing spinor and hence precisely one global supersymmetry turns out to be precisely that $Y^6$ is a Calabi-Yau manifold. This is where all the interest into these manifolds in string theory comes from. (Notice though that nothing in the theory itself demands such a compactification. It is only the phenomenological assumption of the factorized spacetime compactification together with $N = 1$ supersymmetry that does so.)

More generally, in the presence of other background gauge fields, the Calabi-Yau condition here is deformed. One also speaks of generalized Calabi-Yau spaces. (For instance (GMPT05)).

For more see

• supersymmetry and Calabi-Yau manifolds

Properties

As a background for Green-Schwarz $\sigma$-models

The equations of motion of those theories of supergravity which qualify as target spaces for Green-Schwarz action functional sigma models (e.g. 10d heterotic supergravity for the heterotic string and 10d type II supergravity for the type II string) are supposed to be equivalent to those $\sigma$-models being well defined (the WZW-model term being well defined, hence $\kappa$-symmetry being in effect). See at Green-Schwarz action – References – Supergravity equations of motion for pointers.

Scalar moduli spaces and $U$-duality

The compact exceptional Lie groups form a series

$$E_8, E_7, E_6$$

which is usefully thought of to continue as

$$E_5 := Spin(10), E_4 := SU(5), E_3 := SU(3) \times SU(2) \,.$$

Supergravity theories are controled by the corresponding split real forms

$$E_{8(8)}, E_{7(7)}, E_{6(6)}$$

$$E_{5(5)} := Spin(5,5), E_{4(4)} := SL(5, \mathbb{R}), E_{3(3)} := SL(3, \mathbb{R}) \times SL(2, \mathbb{R}) \,.$$

For instance the scalar fields in the field supermultiplet of $3 \leq d \leq 11$-dimensional supergravity have moduli spaces parameterized by the homogeneous spaces

$$E_{n(n)}/ K_n$$

for

$$n = 11 - d \,,$$

where $K_n$ is the maximal compact subgroup of $E_{n(n)}$:

$$K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)$$

$$K_5 \simeq Spin(5) \times Spin(5), K_4 \simeq Spin(5), K_3 \simeq SU(2) \times SO(2) \,.$$

Therefore $E_{n(n)}$ acts as a global symmetry on the supergravity fields.

This is no longer quite true for their UV-completion by the corresponding compactifications of string theory (e.g. type II string theory for type II supergravity, etc.). Instead, on these a discrete subgroup

$$E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)}$$

acts as global symmetry. This is called the U-duality group of the supergravity theory (see there for more).

It has been argued that this pattern should continue in some way further to the remaining values $0 \leq d \lt 3$, with “Kac-Moody groups” corresponding to the Kac-Moody algebras

$$\mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11} \,.$$

Continuing in the other direction to $d = 10$ ($n = 1$) connects to the T-duality group $O(d,d,\mathbb{Z})$ of type II string theory.

See the references (below).

	supergravity gauge group (split real form)	T-duality group (via toroidal KK-compactification)	U-duality	maximal gauged supergravity
	$SL(2,\mathbb{R})$	1	$SL(2,\mathbb{Z})$ S-duality	D=10 type IIB supergravity
	SL$(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3)$\times$ SU(2)	SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 supergravity	E₈-equivariant elliptic cohomology
E₁₀	$E_{10(10)}$	$O(9,9;\mathbb{Z})$	$E_{10(10)}(\mathbb{Z})$
E₁₁	$E_{11(11)}$	$O(10,10;\mathbb{Z})$	$E_{11(11)}(\mathbb{Z})$

(Hull-Townsend 94, table 1, table 2)

Exceptional geometry

For the moment see the remarks/references on supergravity at exceptional geometry and exceptional generalized geometry.

Examples

The usual folklore is that for supergravity Lagrangians “of ordinary type” it turns out that

• $N = 1$ 11-dimensional supergravity

is the highest dimension possible, but see also at

• 12-dimensional supergravity

for some qualification.

Most lower dimensional theories in this class appear as KK-compactifications of this theory or are deformations of such:

• 10-dimensional supergravity

  • type II supergravity

    • type IIA supergravity

      • massive type IIA supergravity

    • type IIB supergravity

  • heterotic supergravity

• 9-dimensional supergravity

• 8-dimensional supergravity

• 7-dimensional supergravity

• 6-dimensional supergravity

• 5-dimensional supergravity

• 4-dimensional supergravity

• 3-dimensional supergravity

• 2-dimensional supergravity

In dimension $(1+0)$ supergravity coupled to sigma-model fields is the spinning particle.

In dimension $(1+1)$ supergravity coupled to sigma-model fields is the spinning string/NSR superstring.

In non-Lorenzian signature it is also possible to consider

• 12-dimensional supergravity

Phenomenology

Discussion of evidence for supergravity from experiment/phenomenology includes the following:

in (Dalianis-Farakos 15) it is argued that the Starobinsky model of cosmic inflation, which is strongly preferred by experiment, further improves after embedding into supergravity.

Related concepts

• Rarita-Schwinger field

• gauged supergravity

• magic supergravity

• topologically twisted supergravity

• gravitino condensation

• duality in physics, duality in string theory

• special geometry

• torsion constraints of supergravity

• Dragon's theorem

• D'Auria-Fre formulation of supergravity

• string theory FAQ – Does string theory predict supersymmetry?

• cosmic inflation and supersymmetry breaking via higher curvature corrections of supergravity are discussed in the context of the Starobinsky model of cosmic inflation

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D = 2n)$	type IIA	$\,$	$\,$
D(-2)-brane	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
D8-brane	$\,$	$\,$
$(D = 2n+1)$	type IIB	$\,$	$\,$
D(-1)-brane	$\,$	$\,$	$\,$
D1-brane	$\,$	$\,$	2d CFT with BH entropy
D3-brane	$\,$	$\,$	N=4 D=4 super Yang-Mills theory
D5-brane	$\,$	$\,$	$\,$
D7-brane	$\,$	$\,$	$\,$
D9-brane	$\,$	$\,$	$\,$
(p,q)-string	$\,$	$\,$	$\,$
(D25-brane)	(bosonic string theory)
NS-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G₂-manifold
topological M5-brane	$\,$	C6-field on G₂-manifold
S-brane
SM2-brane, membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

References

General

Early review:

• Peter van Nieuwenhuizen, Supergravity, Physics Reports, 68, (1981) 189-398 [doi:10.1016/0370-1573(81)90157-5]

• Mike Duff, Bengt Nilsson, Christopher Pope, Kaluza-Klein supergravity, Physics Reports 130 1–2 (1986) 1-142 [spire:229417, doi:10.1016/0370-1573(86)90163-8]

  (emphasis on Kaluza-Klein compactification)

• Yuri Manin, §5.7 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]

• Yoshiaki Tanii, Introduction to Supergravities in Diverse Dimensions, in YITP Workshop on Supersymmetry, Kyoto (1996) [arXiv:hep-th/9802138]

• Bernard de Wit, Jan Louis, Supersymmetry and Dualities in various dimensions, NATO Sci. Ser. C 520 (1999) 33-101 [arXiv:hep-th/9801132, inspire:453367]

Textbook accounts:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Steven Weinberg: Supergravity, Section 31 in: The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge University Press (2000) [ISBN:9781139632638, doi:10.1017/CBO9781139644198 spire:527189, pdf]

  “Gravity exists, so if there is any truth to supersymmetry then any realistic supersymmetry theory must eventually be enlarged to a supersymmetric theory of matter and gravitation, known as supergravity. Supersymmetry without supergravity is not an option, though it may be a good approximation at energies far below the Planck scale.”

• Daniel Freedman, Antoine Van Proeyen: Supergravity, Cambridge University Press (2012) [doi:10.1017/CBO9781139026833]

• Tomás Ortín: Gravity and Strings, Cambridge University Press (2010, 2015) [doi:10.1017/CBO9780511616563, doi:10.1017/CBO9781139019750]

• Pietro Fré, Ch 6 in: Black Holes, Cosmology and Introduction to Supergravity, volume 2 of: Gravity, a Geometrical Course, Springer (2013) [doi:10.1007/978-94-007-5443-0]

• Michel Rausch de Traubenberg, Mauricio Valenzuela, A Supergravity Primer – From Geometrical Principles to the Final Lagrangian, World Scientific (2020) [doi:10.1142/11557]

Collection of original articles:

• Abdus Salam, Ergin Sezgin (eds.): Supergravities in Diverse Dimensions, Elsevier & World Scientific (1990) [doi:10.1142/0277]

Survey:

• Florian Domingo, Michel Rausch de Traubenberg, Supergravity: Application in Particle Physics, in: Handbook of Quantum Gravity, Springer (2023) [arXiv:2209.12541 hep-th, doi:10.1007/978-981-19-3079-9]

• Ergin Sezgin, Survey of supergravities, in: Handbook of Quantum Gravity, Springer (2023) [arXiv:2312.06754 hep-th, doi:10.1007/978-981-19-3079-9]

Lecture notes:

• P. Binetruy, G. Girardi, R. Grimm, Supergravity couplings: a geometric formulation, Phys.Rept.343:255-462,2001 (arXiv:hep-th/0005225)

• Friedemann Brandt, Lectures on supergravity, Fortsch. Phys. 50 (2002) 1126-1172 [arXiv:hep-th/0204035, doi:10.1002/1521-3978(200210)50:10/11%3C1126::AID-PROP1126%3E3.0.CO;2-B]

• Bernard de Wit, Supergravity (arXiv:hep-th/0212245)

• Antoine Van Proeyen, Structure of supergravity theories (arXiv:hep-th/0301005)

• Joachim Gomis, Three lectures on Supergravity (pdf slides)

On Chern-Simons gravity-style action functionals for supergravity:

• Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities [arXiv:0502193]

Further surveys:

• Michael Duff, The status of local supersymmetry, in: From Quarks to Black Holes: Progress in understanding the logic of Nature, World Scientific (2005) 60-116 [arXiv:hep-th/0403160, doi:10.1142/9789812701794_0004]

A fair bit of detail on supersymmetry and on supergravity is in

• P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. R. Morrison, E. Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

The original article that introduced the D'Auria-Fre formulation of supergravity:

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140

The underlying super Cartan geometry is made fully explicit in:

• Konstantin Eder, Super Cartan geometry and the super Ashtekar connection (arXiv:2010.09630)

A compendium, of relevant action functionals and equations of motion is in

• M. J. D. Hamilton, The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation (arXiv:1607.00327)

On black hole-solutions:

• Riccardo D'Auria, Pietro Fre, BPS black holes in supergravity, (hep-th/9812160)

• Antonio Gallerati, Constructing black hole solutions in supergravity theories (arXiv:1905.04104)

On pure spinor-techniques:

• Martin Cederwall, Pure spinors in classical and quantum supergravity, in Handbook of Quantum Gravity (2023) [arXiv:2210.06141]

On higher curvature corrections:

• Mehmet Ozkan, Yi Pang, Ergin Sezgin, Higher Derivative Supergravities in Diverse Dimensions [arXiv:2401.08945]

An S-matrix perspective:

• Brando Bellazzini, Alex Pomarol, Marcello Romano, Francesco Sciotti: (Super) Gravity from Positivity [arXiv:2507.12535]

• Tony Gherghetta, Wenqi Ke: Supergravity from the Bottom Up [arXiv:2507.12538]

On renormalization of supergravity:

• S. Deser, J.H. Kay, K.S. Stelle, Renormalizability Properties of Supergravity, Phys Rev Lett 38, 527 (1977) (reproduced as arXiv:1506.03757)

U-duality

On U-duality of supergravity compactifications:

Some basic facts:

• Jacques Distler, Split real forms (blog post).

The $E_{7(7)}$-symmetry was first discussed in

• Bernard de Wit, Hermann Nicolai, $D = 11$ Supergravity With Local $SU(8)$ Invariance, Nucl. Phys.

  B 274, 363 (1986)

and $E_{8(8)}$ in

• Hermann Nicolai, $D = 11$ Supergravity with Local $SO(16)$ Invariance , Phys. Lett. B 187, 316 (1987).

• K. Koepsell, Hermann Nicolai, Henning Samtleben, An exceptional geometry for $d = 11$ supergravity?, Class. Quant. Grav. 17, 3689 (2000) (arXiv:hep-th/0006034).

The discrete quantum subgroups were discussed in

• Chris Hull, Paul Townsend, Unity of Superstring Dualities Nucl.Phys.B438:109-137 (1995) (arXiv:hep-th/9410167)

which also introduced the term “U-duality”.

Review and further discusssion is in

• Shun’ya Mizoguchi, Germar Schroeder, On Discrete U-duality in M-theory, Class.Quant.Grav. 17 (2000) 835-870 (arXiv:hep-th/9909150)

A careful discussion of the topology of the U-duality groups is in

• Arjan Keurentjes, The topology of U-duality (sub-)groups (arXiv:hep-th/0309106)

• Arjan Keurentjes, U-duality (sub-)groups and their topology (arXiv:hep-th/0312134)

A discussion in the context of generalized complex geometry / exceptional generalized complex geometry is in

• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)

• Nicholas Houston, Supergravity and Generalized Geometry Thesis (2010) (pdf)

The case of “$E_{10}$” is discussed in

• Thibault Damour, Marc Henneaux, Hermann Nicolai, $E(10)$ and a ‘small tension expansion’ of M

  theory_, Phys. Rev. Lett. 89, 221601 (2002) (arXiv:hep-th/0207267);

• Axel Kleinschmidt, Hermann Nicolai, $E(10)$ and $SO(9,9)$ invariant supergravity, JHEP 0407,

  041 (2004) (arXiv:hep-th/0407101)

and that of “$E_{11}$” in

• Peter West, $E_{11}$ and M-theory, Class. Quant. Grav. 18, 4443 (2001) (arXiv:hep-th/0104081).

General discussion of the Kac-Moody groups arising in this context is for instance in

• Philipp Fleig, Axel Kleinschmidt, Eisenstein series for infinite-dimensional U-duality groups (arXiv:1204.3043)

Non-relativistic supergravity

On non-relativistic supergravity:

• Roel Andringa, Eric A. Bergshoeff, Jan Rosseel, Ergin Sezgin: Newton-Cartan Supergravity, Classical and Quantum Gravity 30 20 (2013) [doi:10.1088/0264-9381/30/20/205005, arXiv:1305.6737]

• Eric Bergshoeff, Jan Rosseel, Thomas Zojer: Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095, doi:10.1088/0264-9381/32/20/205003]

• Eric A. Bergshoeff, Jan Rosseel: Non-Lorentzian Supergravity, in Handbook of Quantum Gravity, Springer (2024) [arXiv:2211.02604, doi:10.1007/978-981-99-7681-2_52]

• Patrick Concha, Lucrezia Ravera: Non-Lorentzian Supergravity and Kinematical Superalgebras, JHEP 03 (2025) 032 [arXiv:2412.07665]

On non-relativistic SuGra as an effective field theory for excitation modes of FQH systems (cf. supersymmetry in fractional quantum Hall systems):

• Dung Xuan Nguyen, Kartik Prabhu, Ajit C. Balram, Andrey Gromov: Supergravity model of the Haldane-Rezayi fractional quantum Hall state, Phys. Rev. B 107 (2023) 125119 [doi:10.1103/PhysRevB.107.125119, arXiv:2212.00686]

• Yi-Hsien Du: Chiral Graviton Theory of Fractional Quantum Hall States [arXiv:2509.04408]

  (generalization to non-abelian Read-Rezayi states)

History

The original idea of supergravity is attributed to:

• Dmitry Volkov, Vladimir P. Akulov: Possible universal neutrino interaction, JETP Lett. 16 (1972) 438–440, Pisma Zh. Eksp. Teor. Fiz. 16 (1972) 621–624 [inSpire:76156, pdf, pdf]

  which introduces anticommuting gauge parameters and ends with the sentence: “We note that if we introduce also gauge fields corresponding to the transformations (3), then, as a consequence of the Higgs effect (Higgs (1966)), a massive gauge field with spin 3/2 arises*

followed up by the first model of supergravity (in nonlinear realization) constructed in:

• Dmitry Volkov, Vyacheslav Soroka: Higgs effect for Goldstone particles with spin $1/2$, JETP Lett. 18 (1973) 312–314, Pisma Zh. Eksp. Teor. Fiz. 18 (1973) 529–532[inSpire:84359, pdf, pdf]

However, the term “supergravity” was coined later by Freedman, Nieuwenhuizen, Ferrara 76, whose work on supergravity is regarded as foundational for the subject.

This early history is discussed in:

• Steven Duplij: Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn’t it?, East Eur. J. Phys. v3 (2019) 81–82 [arXiv:1910.03259]

• Sergei M. Kuzenko, Local supersymmetry: Variations on a theme by Volkov and Soroka (arXiv:2110.12835)

Supergravity, in the guise of 4d supergravity, was first found (constructed) in

• Daniel Freedman, Peter van Nieuwenhuizen, Sergio Ferrara: Progress toward a theory of supergravity, Phys. Rev. D 13 (1976) 3214 [doi.org/10.1103/PhysRevD.13.3214]

Accounts of the early history of supergravity::

• Sergio Ferrara: Supergravity and the quest for a unified theory [arxiv:hep-th/9405065]

• Dmitry Volkov, Supergravity before 1976,

  International Conference on the History of Original Ideas and Basic Discoveries in Particle Physics, Erice, 1994 (Springer, Chapter or arXiv:hep-th/9410024)

• R. Arnowitt, Ali Chamseddine, Pran Nath, The Development of Supergravity Grand Unification: Circa 1982-85 (arXiv:1206.3175)

• Dmitry Volkov, Supergravity before and after 1976. The story of goldstonions, in Concise Encyclopedia of Supersymmetry, S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004 or arXiv:hep-th/9404153

• David Appell, When supergravity was born, 2012 (pdf)

• Peter van Nieuwenhuizen, Aspects of supergravity, 2014 (pdf)

• Vyacheslav Soroka, The Sources of Supergravity in The Supersymmetric World. The Beginnings of the Theory, G. Kane and M. Shifman (Eds.), World Scientific, 2000 or arXiv:hep-th/0203171

• Sergio Ferrara, A. Sagnotti, Supergravity at 40: Reflections and Perspectives, Based in part on the talk delivered by S. F. at the “Infeld Colloquium and Discrete”, in Warsaw, on December 1 2016, and on a joint CERN Courier article. Dedicated to John Schwarz on the occasion of his 75-th birthday (arXiv:1702.00743)

• Stanley Deser, A brief history of Supergravity: the first three weeks (arXiv:1704.05886)

• Bernard L. Julia: An exceptional story: Symmetries and dualities between Maximal supergravity and General relativity, in: Half a Century of Supergravity [arXiv:2504.07576]

• Peter West: The discovery of supergravity and its early development [arXiv:2507.23634]

• Antoine Van Proeyen: Matter Couplings in Supergravity. The first 10 years [arXiv:2503.12510]

• Pran Nath: 50 Years of SUSY and SUGRA, circa 1974-2024, and Future Prospects [arXiv:2603.04664]

See also:

• supersymmetry – History.

Related

Further physics monographs on supergravity include

• I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace, googB

• Julius Wess, Jonathan Bagger, Supersymmetry and supergravity, 1992

• Steven Weinberg, Quantum theory of fields, volume III: supersymmetry

• Concise Encyclopedia of Supersymmetry, S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004, SUSY story narrated by its founders

The Cauchy problem for classical solutions of simple supergravity has been discussed in

• Yvonne Choquet-Bruhat, The Cauchy Problem in Classical Supergravity, Letters in Mathematical Physics 7 (1983) 459-467. 0377

A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional supergravity is volume II, starting on page 1091 in

• P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. R. Morrison and E. Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

Discussion of solutions with $N = 1$ global supersymmetry left and their relation to Calabi-Yau compactifications are for instance in

• Mariana Graña, Ruben Minasian, Michela Petrini, Alessandro Tomasiello, Generalized structures of $N=1$ vacua, arXiv:hep-th/0505212

Phenomenology

On supergravity phenomenology:

Dark matter

Discussion of the gravitino as a dark matter candidate:

• John Ellis, Keith Olive, Supersymmetric Dark Matter Candidates (arXiv:1001.3651)

A proposal for super-heavy gravitinos as dark matter, by embedding D=4 N=8 supergravity into E10-U-duality-invariant M-theory:

• Krzysztof A. Meissner, Hermann Nicolai: Standard Model Fermions and Infinite-Dimensional R-Symmetries, Phys. Rev. Lett. 121 091601 (2018) [arXiv:1804.09606, doi:10.1103/PhysRevLett.121.091601]

• Krzysztof A. Meissner, Hermann Nicolai, Planck Mass Charged Gravitino Dark Matter, Phys. Rev. D 100, 035001 (2019) (arXiv:1809.01441)

following the proposal towards the end of

• Murray Gell-Mann, introductory talk at Shelter Island II, 1983 (pdf)

  in: Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

Further discussion:

• Krzysztof A. Meissner, Hermann Nicolai, Supermassive gravitinos and giant primordial black holes (arXiv:2007.11889)

• Krzysztof A. Meissner, Hermann Nicolai, Evidence for a stable supermassive gravitino with charge $2/3$? [arXiv:2303.09131]

• Adrianna Kruk, Michał Lesiuk, Krzysztof A. Meissner, Hermann Nicolai: Signatures of supermassive charged gravitinos in liquid scintillator detectors [arXiv:2407.04883]

Standard model of particle physics

On the phenomenology of supergravity coupled to the (non-supersymmetric) standard model of particle physics:

• Cliff P. Burgess, Fernando Quevedo: Who’s Afraid of the Supersymmetric Dark? The Standard Model vs Low-Energy Supergravity, Fortschr. Physik 70 7-8 (2022) 2200077 [arXiv:2110.13275, doi:10.1002/prop.202200077]

Inflationary cosmology in Supergravity

On models of cosmic inflation in D=4 supergravity and the $\alpha$-attractor mechanism & the $\eta$-problem resolution:

(See also at Starobinsky model of cosmic inflation the references on its embedding into supergravity).

• Dimitri V. Nanopoulos, Keith A. Olive, Mark Srednicki, K. Tamvakis: Primordial inflation in simple supergravity, Phys. Lett. B 123 (1983) 41–44 [doi:10.1016/0370-2693(83)90954-1]

• R. Holman, Pierre Ramond, G. G. Ross: Supersymmetric inflationary cosmology, Physics Letters B 137 (1984) 343 [doi:10.1016/0370-2693(84)91729-5]

• S. James Gates Jr., Sergei V. Ketov: Superstring-inspired supergravity as the universal source of inflation and quintessence, Physics Letters B 674 (2009) 59–63 [doi:10.1016/j.physletb.2009.03.005, arXiv:0901.2467]

• Masahide Yamaguchi: Supergravity based inflation models: a review, Class. Quantum Grav. 28 (2011) 103001 [doi:10.1088/0264-9381/28/10/103001, arXiv:1101.2488]

• Sergio Ferrara, Renata Kallosh, Andrei Linde, Massimo Porrati: Minimal Supergravity Models of Inflation, Phys. Rev. D 88 (2013) 085038 [doi:10.1103/PhysRevD.88.085038, arXiv:1307.7696]

• Renata Kallosh, Andrei Linde, Diederik Roest: Superconformal Inflationary $\alpha$-Attractors, J. High Energ. Phys. 2013 198 (2013) [doi:10.1007/JHEP11(2013)198, arXiv:1311.0472]

• Renata Kallosh, Andrei Linde, Diederik Roest: Large Field Inflation and Double $\alpha$-Attractors, J. High Energ. Phys. 2014 52 (2014) [doi:10.1007/JHEP08(2014)052, arXiv:1405.3646]

• Sergei V. Ketov, Takahiro Terada: Inflation in Supergravity with a Single Chiral Superfield, Physics Letters B 736 (2014) 272–277 [doi:10.1016/j.physletb.2014.07.036, arXiv:1406.0252]

• Sergio Ferrara, Renata Kallosh, Andrei Linde: Cosmology with Nilpotent Superfields, J. High Energ. Phys. 2014 143 (2014) [doi:10.1007/JHEP10(2014)143, arXiv:1408.4096]

• Renata Kallosh, Andrei Linde, Marco Scalisi: Inflation, de Sitter Landscape and Super-Higgs effect, J. High Energ. Phys. 2015 111 (2015) [doi:10.1007/JHEP03(2015)111, arXiv:1411.5671]

• John Joseph M. Carrasco, Renata Kallosh, Andrei Linde: Cosmological Attractors and Initial Conditions for Inflation, Phys. Rev. D 92 (2015) 063519 [doi:10.1103/PhysRevD.92.063519, arXiv:1506.00936]

• Takahiro Terada: Inflation in Supergravity with a Single Superfield, Phd thesis, Tokyo (2015) [arXiv:1508.05335]

• Sergio Ferrara, Renata Kallosh, Jesse Thaler: Cosmology with orthogonal nilpotent superfields, Phys. Rev. D 93 043516 (2016) [doi:10.1103/PhysRevD.93.043516, arXiv:1512.00545]

• Sergio Ferrara, Renata Kallosh: Seven-Disk Manifold, $\alpha$-attractors and $B$-modes, Phys. Rev. D 94 (2016) 126015 [doi:10.1103/PhysRevD.94.126015, arXiv:1610.04163]

  (via 11D supergravity/M-theory on $G_2$-manifolds)

• Llibert Aresté Saló, David Benisty, Eduardo I. Guendelman, Jaume de Haro: $\alpha$-attractors in Quintessential Inflation motivated by Supergravity, Phys. Rev. D 103 (2021) 123535 [doi:10.1103/PhysRevD.103.123535, arXiv:2103.07892]

• Sergei Ketov: Inflationary Cosmology from Supergravity, in Handbook of Quantum Gravity (2023) [doi:10.1007/978-981-19-3079-9_51-1, pdf]

• Ignatios Antoniadis, Emilian Dudas, Fotis Farakos, Augusto Sagnotti: Non-Linear Supergravity and Inflationary Cosmology [arXiv:2409.14943]

• Michele Cicoli, Christopher Hughes, Ahmed Rakin Kamal, Francesco Marino, Fernando Quevedo, Mario Ramos-Hamud, Gonzalo Villa: Back to the origins of brane-antibrane inflation, Eur. Phys. J. C 85 315 (2025) [doi:10.1140/epjc/s10052-025-13982-9, arXiv:2410.00097]

• Renata Kallosh, Andrei Linde: Attractors in Cosmology, section 3 of: Attractors in Supergravity, chapter 22 in: Half a Century of Supergravity Part II – Structure and Properties of Sugra (2026) 188–199 [doi:10.1017/9781009575874.025, arXiv:2503.13682]

• Renata Kallosh, Andrei Linde: Streamlined Supergravity, J. High Energ. Phys. 2026 176 (2026) [doi:10.1007/JHEP03(2026)176, arXiv:2511.15815]

• Renata Kallosh, Andrei Linde: Singular $\alpha$-attractors, Journal of Cosmology and Astroparticle Physics, 2026 (2026) [doi:10.1088/1475-7516/2026/04/075, arXiv:2512.02969]

• Gonzalo Villa: Remarks on brane-antibrane inflation, PoS 474 (2025) [doi:10.22323/1.474.0006, arXiv:2501.09074]

Discussion in D=6 supergravity with 3-branes:

• Y. Aghababaie, Cliff P. Burgess, S. L. Parameswaran, Fernando Quevedo: Towards a Naturally Small Cosmological Constant from Branes in 6D Supergravity, Nucl. Phys. B 680 (2004) 389-414 [doi:10.1016/j.nuclphysb.2003.12.015, arXiv:hep-th/0304256]