# nLab 11-dimensional supergravity

### Context

#### Gravity

gravity, supergravity

# Contents

## Idea

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

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

## The action functional

(…)

(…)

### The higher Chern-Simons term

under construction

${\int }_{X}\left(\frac{1}{6}\left(C\wedge G\wedge G-C\wedge \frac{1}{8}\left({p}_{2}+\left(\frac{1}{2}{p}_{1}{\right)}^{2}\right)\right)\right)$\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.

$\lambda :=\frac{1}{2}{p}_{1}\phantom{\rule{thinmathspace}{0ex}}.$\lambda := \frac{1}{2}p_1 \,.

Concerning the integrality of

${I}_{8}:=\frac{1}{48}\left({p}_{2}+\left(\lambda {\right)}^{2}\right)$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

$I=\frac{1}{1440}\left(7\left(\frac{1}{2}{p}_{1}{\right)}^{2}-{p}_{2}\right)\in ℤ\phantom{\rule{thinmathspace}{0ex}}.$I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z} \,.

Notice that $1440=6x8x30$. So

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

$\left(\frac{1}{2}{p}_{1}\right)=2x\phantom{\rule{thinmathspace}{0ex}}.$(\frac{1}{2}p_1) = 2 x \,.

Then the above becomes

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

and hence then ${p}_{2}+\left(\frac{1}{2}{p}_{1}{\right)}^{2}$ 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

${x}^{2}=x\left(\frac{1}{2}{p}_{1}\right)\phantom{\rule{thickmathspace}{0ex}}\mathrm{mod}\phantom{\rule{thickmathspace}{0ex}}2$x^2 = x (\frac{1}{2}p_1) \; mod \; 2

and so when $\left(\frac{1}{2}{p}_{1}{\right)}^{2}$ is further divisible by 2 we have that ${p}_{2}-\left(\frac{1}{2}{p}_{1}{\right)}^{2}$ 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).

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
$\left(D=2n\right)$type IIA$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D0-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$BFSS matrix model
D2-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D4-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D8-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D1-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$2d CFT with BH entropy
D3-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$N=4 D=4 super Yang-Mills theory
D5-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D7-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D9-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
NS-branetype I, II, heteroticcircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
string$\phantom{\rule{thinmathspace}{0ex}}$B2-field2d SCFT
NS5-brane$\phantom{\rule{thinmathspace}{0ex}}$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
M2-brane$\phantom{\rule{thinmathspace}{0ex}}$C3-fieldABJM theory, BLG model
M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field6d (2,0)-superconformal QFT
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\phantom{\rule{thinmathspace}{0ex}}$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

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 $𝒪\left({l}^{3}\right)$, 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 ${\mathrm{AdS}}_{4}×{S}^{7}$ 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 $\mathrm{SO}\left(7\right)$ 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

• 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.