Contents

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 $\mathrm{𝔦𝔰𝔬}(n,1)$, supergravity is defined by an extension of this to an action functional on the space connections with values in the super Poincare Lie algebra $\mathrm{𝔰𝔦𝔰𝔬}(n,1)$. 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:

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

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

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

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

• a $\mathrm{𝔰𝔬}(n,1)$-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) $\mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1)$. Specifically such a field

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.

The condition of gauge invariance of an action functional on $\mathrm{𝔰𝔦𝔰𝔬}$-connections is considerably more restrictive than for one on $\mathrm{𝔦𝔰𝔬}$-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.

As a gauge theory

The non-spinorial part of action functionals of supergravity theories are typically given in first order formulation as functional on a space of connections with values in the Poincare Lie algebra $\mathrm{𝔦𝔰𝔬}(n,1)$. Including the fermionic fields, this becomes connections with values in the super Poincare Lie algebra $\mathrm{𝔰𝔦𝔰𝔬}(10,1)$.

This might suggest that supergravity is to be thought of as a gauge theory. There are indeed various action functionals of Chern-Simons theory-form for supergrvaity theories (Zanelli). These yield theories whose bosonic action functional is the Einstein-Hilbert action in certain conctraction limits.

More generally (DAuriaFreCastellani) have shown that at least some versions, such as the maximal 11-dimensional supergravity are naturally understood as higher gauge theories whose fields are ∞-Lie algebra-valued forms with values in ∞-Lie algebras such as the supergravity Lie 3-algebra. This is described in detail at D'Auria-Fre formulation of supergravity.

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 millenium (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 tbe 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

Scalar moduli spaces and $U$-duality

The compact exceptional Lie groups form a series

which is usefully thought of to continue as

Supergravity theories are controled by the corresponding split real forms

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

for

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

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

acts as global symmetry. This is called the U-duality group of the supergravity theory.

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

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).

Exceptional geometry

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

Examples

For supergravity Lagrangians “of ordinary type” it turns out that

• $N=1$ 11-dimensional supergravity

is the highest dimensional possible. All lower dimensional theories in this class appear as compactifications of this theory or otherwise derived from it, such as

• 10-dimensional type II supergravity, heterotic supergravity

• 7-dimensional supergravity

• 5-dimensional supergravity

• 4-dimensional supergravity

• 3-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.

Related concepts

• gauged supergravity

• duality in physics, duality in string theory

• special geometry

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	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$$	$$
D8-brane	$$	$$
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
D9-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D SuGra/M-theory	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT
M9-brane			heterotic string theory
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

References

General

A modern reference for the diverse flavours of supergravity theories is

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

Introductory lecture notes are in

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

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 is

• R. D’Auria, P. Fré Geometric supergravity in $D=11$ and its hidden supergroup Geometric Supergravity in D=11 and its hidden supergroup

The standard textbook monograph on supergravity and string theory using these tools is

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre, Supergravity and Superstrings - A Geometric Perspective

U-duality

Some basic facts are recalled in

• 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 $\mathrm{SU}(8)$ Invariance, Nucl. Phys. B 274, 363 (1986)

and $E_{8(8)}$ in

• Hermann Nicolai, $D=11$ Supergravity with Local $\mathrm{SO}(16)$ 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”.

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 $\mathrm{SO}(9,9)$ 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)

Gauged supergravity

• Natxo Alonso-Alberca; and Tomáas Ortín, Gauged/Massive supergravities in diverse dimensions (pdf)

Chern-Simons supergravity

A survey of the Chern-Simons gravity-style action functionals for supergravity is in

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

History

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

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

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