11-dimensional supergravity



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N=1N=1 supergravity in d=11d = 11.

for the moment see the respective section at D'Auria-Fre formulation of supergravity

The action functional


Kinetic terms


The higher Chern-Simons term

under construction

X(16(CGGC18(p 2+(12p 1) 2))) \int_X \left( \frac{1}{6} \left( C \wedge G \wedge G - C \wedge \frac{1}{8} \left( p_2 + (\frac{1}{2}p_1)^2 \right) \right) \right)

where p ip_i is the iith Pontryagin class.

λ:=12p 1. \lambda := \frac{1}{2}p_1 \,.

Concerning the integrality of

I 8:=148(p 2+(λ) 2) I_8 := \frac{1}{48}(p_2 + (\lambda)^2)

on a spin manifold XX. (Witten96, p.9)

First, the index of a Dirac operator on XX is

I=11440(7(12p 1) 2p 2). I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z} \,.

Notice that 1440=6x8x301440 = 6 x 8 x 30. So

p 2(12p 2) 2=6((12p 1) 230x8I) p_2 - (\frac{1}{2}p_2)^2 = 6 ( (\frac{1}{2}p_1)^2 - 30 x 8 I)

is divisble by 6.

Assume that (12p 1)(\frac{1}{2}p_1) is further divisble by 2 (see the relevant discussion at M5-brane).

(12p 1)=2x. (\frac{1}{2}p_1) = 2 x \,.

Then the above becomes

p 2(12p 2) 2=24(x 230x2I) p_2 - (\frac{1}{2}p_2)^2 = 24 ( x^2 - 30 x 2 I)

and hence then p 2+(12p 1) 2p_2 + (\frac{1}{2}p_1)^2 is divisible at least by 24.

But moreover, on a Spin manifold the first fractional Pontryagin class 12p 1\frac{1}{2}p_1 is the Wu class ν 4\nu_4 (see there). By definition this means that

x 2=x(12p 1)mod2 x^2 = x (\frac{1}{2}p_1) \; mod \; 2

and so when (12p 1) 2(\frac{1}{2}p_1)^2 is further divisible by 2 we have that p 2(12p 1) 2p_2 - (\frac{1}{2}p_1)^2 is divisible by 48. Hence I 8I_8 is integral.

The hidden deformation

There is in fact a hidden 1-parameter deformation of the Lagrangian of 11d sugra. Mathematically this was maybe first noticed in (D’Auria-Fre 82) around equation (4.25). This shows that there is a topological term which may be expressed as

X 11G 4G 7 \propto \int_{X_11} G_4 \wedge G_7

where G 4G_4 is the curvature 3-form of the supergravity C-field and G 7G_7 that of the magnetically dual C6-field. However, (D’Auria-Fre 82) consider only topologically trivial (trivial instanton sector) configurations of the supergravity C-field, and since on them this term is a total derivative, the authors “drop” it.

The term then re-appears in the literatur in (Bandos-Berkovits-Sorokin 97, equation (4.13)). And it seems that this is the same term later also redicovered around equation (4.2) in (Tsimpis 04).

BPS states

The basic BPS spates? of 11d SuGra are

(e.g. EHKNT 07)

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



That there is a maximal dimension d=11d = 11 in which supergravity may exist was found in

The theory was then actually constructed in

Formulation in terms of supergeometry (“superspace formulation”) is in

  • Eugene Cremmer, S. Ferrara, Formulation of Eleven-Dimensional Supergravity in Superspace, Phys.Lett. B91 (1980) 61

  • Lars Brink, Paul Howe, Eleven-Dimensional Supergravity on the Mass-Shell in Superspace, Phys.Lett. B91 (1980) 384

The history as of 1990s with an eye towards the development to M-theory is survey in

The description of 11d supergravity in terms of the D'Auria-Fre formulation of supergravity originates in

of which a textbook account is in

The topological deformation (almost) noticed in equation (4.25) of D’Auria-Fre 82 later reappears in (4.13) of

and around (4.2) of

  • Dimitrios Tsimpis, 11D supergravity at 𝒪(l 3)\mathcal{O}(l^3), JHEP0410:046,2004 (arXiv:hep-th/0407271)

More recent textbook accounts include

Discussion of the equivalence of the 11d SuGra equations of motion with the supergravity torsion constraints is in


  • A. Candiello, K. Lechner, Duality in Supergravity Theories, Nucl.Phys. B412 (1994) 479-501 (arXiv:hep-th/9309143)

Much computational detail is displayed in

  • A. Miemiec, I. Schnakenburg, Basics of M-Theory, Fortsch.Phys. 54 (2006) 5-72 (arXiv:hep-th/0509137)

Classical solutions and BPS states

Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the branes and duality paradigm and the AdS/CFT correspondence.

One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund-Rubin background with geometry AdS 4×S 7AdS_4 \times S^7 and 4-form flux proportional to the volume form on AdS 4AdS_4.

  • Peter Freund, Mark Rubin, Dynamics of Dimensional Reduction Phys.Lett. B97 (1980) 233-235 (inSpire)

The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon limit of coincident M2-branes.

Shortly after the original Freund-Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the S 7S^7; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in AdS 4AdS_4 and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the SO(8) symmetry of the sphere to an SO(7)SO(7) subgroup.

  • Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity Phys.Lett. B119 (1982) 339 (inSPIRE)

Some of the above is taken from this TP.SE thread.

A classification of symmetric solutions is discussed in

Discussion of black branes and BPS states includes

Discussion of black hole horizons includes

Truncations and compactifications

Topology and anomaly cancellation

Discussion of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles.

See also the relevant references at M5-brane.

Description by exceptional generalized geometry

Revised on August 8, 2015 07:04:26 by Urs Schreiber (