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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.

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 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 – 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 𝔬(d1,1)ℑ𝔰𝔬( d1,1)\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 persepctive directly generalizes to supergeometry and yields the superspace formulation of theories of supegravity – super Cartan geometry.

After picking a dimension dd\in \mathbb{N} and writing ℑ𝔰𝔬( d1,1)\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 NN. Then the direct sum

ℑ𝔰𝔬( d1,1|N)ℑ𝔰𝔬( d1,1) evenN odd \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 NN in odd degree becomes a super Lie algebra by letting the [even,odd][even,odd] bracket to be given by the defining action and by letting the [odd,odd][odd,odd] bracket be given by a canonically induced bilinear and 𝔬\mathfrak{o}-equivariant pairing – the super Poincaré Lie algebra. This still canonical contains the Lorentz Lie algebra 𝔬( d1,1)\mathfrak{o}(\mathbb{R}^{d-1,1}) and the quotient

d1,1|Nℑ𝔰𝔬( d1,1|N)/𝔬( d1,1) \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 geometry𝔤\mathfrak{g}𝔥\mathfrak{h}𝔤/𝔥\mathfrak{g}/\mathfrak{h}
Einstein gravityℑ𝔰𝔬( d1,1)\mathfrak{Iso}(\mathbb{R}^{d-1,1})𝔬(d1,1)\mathfrak{o}(d-1,1) d1,1\mathbb{R}^{d-1,1}
supergravityℑ𝔰𝔬( d1,1|N)\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})𝔬(d1,1)\mathfrak{o}(d-1,1) d1,1|N\mathbb{R}^{d-1,1\vert N}
11-dimensional supergravityℑ𝔰𝔬(^ 10,1|N=1)\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert N=1})𝔬(d1,1)\mathfrak{o}(d-1,1)^ 10,1|N=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 determing 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-fom 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 – by 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-algabra

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

and extend the differential to that by the formula

d CEc 3=12ψ¯Γ abψe ae b. 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 12ψ¯Γ abψe ae bCE 4(ℑ𝔰𝔬(10,1|N=1))\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 algebra, hence of L-infinity algebras, in fact of Lie (p+1)-algebras for (p+1)(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 3c_3 as the CE-algebra CE(𝔪2𝔟𝔯𝔞𝔫𝔢)CE(\mathfrak{m}2\mathfrak{brane})

of the supergravity Lie 3-algebra 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane}.

Now a morphism

Ω (U)CE(𝔪2𝔟𝔯𝔞𝔫𝔢):A \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 3A(c 3)Ω (3,even)(U) C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U)

whose curvature

G 4=dC 3+12Ψ¯Γ abΨE aE b 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

^ 10,1|N=1𝔤/𝔥=𝔪2𝔟𝔯𝔞𝔫𝔢/𝔬( 10,1|N=1) \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 GG 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-groupshigher 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×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.


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.

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



An early survey is

Textbook accounts include

Lecture notes include

Furrther surveys 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


Discussion of implications of supergravity for phenomenology/cosmology includes

  • Ioannis Dalianis, Fotis Farakos, On the initial conditions for inflation with plateau potentials: the R+R 2R + R^2 (super)gravity case (arXiv:1502.01246)

Revised on March 18, 2015 20:07:27 by Urs Schreiber (