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).

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



11d supergravity was originally found in

Formulation in terms of supergeometry is in

  • E. 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 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)

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

Classical solutions

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.

  • Mike Duff, Kai Stelle?, Multimembrane solutions of D = 11 supergravity , Phys.Lett. B253 (1991) 113-118 (web)

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

Truncations and compactifications

Topology and anomaly cancellation

Discussin 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 January 24, 2015 19:58:02 by Urs Schreiber (