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Fields and quanta



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 𝔦𝔰𝔬(n,1)\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 𝔰𝔦𝔰𝔬(n,1)\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)𝔰𝔦𝔰𝔬(n,1)o(n,1) \hookrightarrow \mathfrak{siso}(n,1):

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

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

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

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

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

has one more component

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

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

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

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 𝔦𝔰𝔬(n,1)\mathfrak{iso}(n,1). Including the fermionic fields, this becomes connections with values in the super Poincare Lie algebra 𝔰𝔦𝔰𝔬(10,1)\mathfrak{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 supergravity theories (Zanelli). These yield theories whose bosonic action functional is the Einstein-Hilbert action in certain contraction 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 millennium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form M 4×Y 6M^4 \times Y^6 for M 4M^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 6Y^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 6Y^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=1N = 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


As a background for Green-Schwarz σ\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 σ\sigma-models being well defined (the WZW-model term being well defined, hence κ\kappa-symmetry being in effect). See at Green-Schwarz action – References – Supergravity equations of motion for pointers.

Scalar moduli spaces and UU-duality

The compact exceptional Lie groups form a series

E 8,E 7,E 6 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)×SU(2). 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_{8(8)}, E_{7(7)}, E_{6(6)}
E 5(5):=Spin(5,5),E 4(4):=SL(5,),E 3(3):=SL(3,)×SL(2,). 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 3d113 \leq d \leq 11-dimensional supergravity have moduli spaces parameterized by the homogeneous spaces

E n(n)/K n E_{n(n)}/ K_n


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

where K nK_n is the maximal compact subgroup of E n(n)E_{n(n)}:

K 8Spin(16),K 7SU(8),K 6Sp(4) K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)
K 5Spin(5)×Spin(5),K 4Spin(5),K 3SU(2)×SO(2). 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)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)()E n(n) 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 0d<30 \leq d \lt 3, with “Kac-Moody groups” corresponding to the Kac-Moody algebras

𝔢 9,𝔢 10,𝔢 11. \mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11} \,.

Continuing in the other direction to d=10d = 10 (n=1n = 1) connects to the T-duality group O(d,d,)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-dualitymaximal gauged supergravity
SL(2,)SL(2,\mathbb{R})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
Spin(10)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
SU(5)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()E_{11(11)}(\mathbb{Z})

(Hull-Townsend 94)

Exceptional geometry

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


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

is the highest dimension possible. All lower dimensional theories in this class appear as KK-compactifications of this theory or are deformations of such:

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

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

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

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d



Textbook accounts include

Lecture notes include

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

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


Some basic facts are recalled in

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

and E 8(8)E_{8(8)} in

The discrete quantum subgroups were discussed in

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

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 10E_{10}” is discussed in

and that of “E 11E_{11}” in

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

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


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

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

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

Revised on January 28, 2015 22:24:44 by Urs Schreiber (