nLab
11-dimensional supergravity

Idea

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

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

$${\int }_X\left(\frac{1}{6}(C\wedge G\wedge G-C\wedge \frac{1}{8}(p_2+(\frac{1}{2}{p}_1{)}^2))\right)$$\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 $p_i$ is the $i$th Pontryagin class.

$$\lambda :=\frac{1}{2}{p}_1.$$\lambda := \frac{1}{2}p_1 \,.

Concerning the integrality of

$$I_8:=\frac{1}{48}(p_2+(\lambda {)}^2)$$I_8 := \frac{1}{48}(p_2 + (\lambda)^2)

on a spin manifold $X$. (Witten96, p.9)

First, the index of a Dirac operator on $X$ is

$$I=\frac{1}{1440}(7(\frac{1}{2}{p}_1{)}^2-p_2)\in \mathbb{Z}.$$I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z} \,.

Notice that $1440=6x8x30$. So

$$p_2-(\frac{1}{2}{p}_2{)}^2=6((\frac{1}{2}{p}_1{)}^2-30x8I)$$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 $(\frac{1}{2}{p}_1)$ is further divisble by 2 (see the relevant discussion at M5-brane).

$$(\frac{1}{2}{p}_1)=2x.$$(\frac{1}{2}p_1) = 2 x \,.

Then the above becomes

$$p_2-(\frac{1}{2}{p}_2{)}^2=24(x^2-30x2I)$$p_2 - (\frac{1}{2}p_2)^2 = 24 ( x^2 - 30 x 2 I)

and hence then $p_2+(\frac{1}{2}{p}_1{)}^2$ is divisible at least by 24.

But moreover, on a Spin manifold the first fractional Pontryagin class $\frac{1}{2}{p}_1$ is the Wu class ${\nu }_4$ (see there). By definition this means that

$$x^2=x(\frac{1}{2}{p}_1)\mathrm{mod}2$$x^2 = x (\frac{1}{2}p_1) \; mod \; 2

and so when $(\frac{1}{2}{p}_1{)}^2$ is further divisible by 2 we have that $p_2-(\frac{1}{2}{p}_1{)}^2$ is divisible by 48. Hence $I_8$ 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

$$\propto {\int }_{X_{11}}{G}_4\wedge G_7$$\propto \int_{X_11} G_4 \wedge G_7

where $G_4$ is the curvature 3-form of the supergravity C-field and $G_7$ 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).

Related concepts

• supergravity

• 10-dimensional type II supergravity, heterotic supergravity

• 7-dimensional supergravity

• 5-dimensional supergravity

• 4-dimensional supergravity

  • M-theory on G2-manifolds, G2-MSSM

  • Freund-Rubin compactification

• 3-dimensional supergravity

• supergravity C-field, supergravity Lie 3-algebra, supergravity Lie 6-algebra

• Hořava-Witten theory

• M-theory

Table of branes appearing in supergravity/string theory

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	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
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	$$	C6-field	6d (2,0)-superconformal QFT
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

References

General

11d supergravity was originally found in

• E. Cremmer, Bernard Julia, J. Scherk, Supergravity in theory in 11 dimensions Phys. Lett. 76B (1978) 409

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

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup

of which a textbook account is in

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

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

• Igor Bandos, Nathan Berkovits, Dmitri Sorokin, Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes, Nucl. Phys. B522 (1998) 214-233 (arXiv:hep-th/9711055)

and around (4.2) of

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

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 ${\mathrm{AdS}}_4\times S^7$ and 4-form flux proportional to the volume form on ${\mathrm{AdS}}_4$.

• 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 $S^7$; 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 ${\mathrm{AdS}}_4$ 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 $\mathrm{SO}(7)$ 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

• José Figueroa-O’Farrill, Symmetric M-Theory Backgrounds (arXiv:1112.4967)

Truncations and compactifications

• Hermann Nicolai, Krzysztof Pilch, Consistent truncation of $d=11$ supergravity on ${\mathrm{AdS}}_{4}x{S}^7$ (arXiv:1112.6131)

Topology and anomaly cancellation

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

• Edward Witten, On Flux Quantization In M-Theory And The Effective Action (arXiv:hep-th/9609122)

See also the relevant references at M5-brane.

• Dan Freed, Two nontrivial index theorems in odd dimensions (arXiv:dg-ga/9601005)

• Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review (arXiv:hep-th/0307152)

Description by exceptional generalized geometry

• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)