# nLab D=11 N=1 supergravity

Contents

### 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

$\int_X \left( \frac{1}{6} \left( C \wedge G \wedge G - C \wedge \frac{1}{8} \left( p_2 + (\frac{1}{2}p_1)^2 \right) \right) \right)$

where $p_i$ is the $i$th Pontryagin class.

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

Concerning the integrality of

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

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

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

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

Notice that $1440 = 6 \times 8 \times 30$. So

$p_2 - (\frac{1}{2}p_2)^2 = 6 ( (\frac{1}{2}p_1)^2 - 30 \times 8 I)$

is divisible by 6.

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

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

Then the above becomes

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

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

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

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.

### Higher curvature corrections

Possible higher curvature corrections to 11-dimensional supergravity are discussed in the references listed below.

The first correction is an $R^4$-term at order $\ell^3_{P}$ (11d Planck length). In Tsimpis 04 it is shown that part of this is a topological term (total derivative) which relates to the shifted C-field flux quantization.

For effects of higher curvature corrections in a Starobinsky model of cosmic inflation see there.

### 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

$\propto \int_{X_11} G_4 \wedge G_7$

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).

(hm, check)

### BPS states

The basic BPS states 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
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(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$\,$
A-brane$\,$
B-brane$\,$
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
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

### General

That there is a maximal dimension $d = 11$ 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 Tsimpis 04

More recent textbook accounts include

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

following

Much computational detail is displayed in

In terms of pure spinors:

• Max Guillen, Pure spinors and $D=11$ supergravity (arXiv:2006.06014)

### 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 compactification with geometry $AdS_4 \times S^7$ and 4-form flux proportional to the volume form on $AdS_4$.

• 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 geometry 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 $S^7$; 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 $AdS_4$ 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 $SO(7)$ 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

• Kellogg Stelle, section 3 of BPS Branes in Supergravity (arXiv:hep-th/9803116)

• Francois Englert, Laurent Houart, Axel Kleinschmidt, Hermann Nicolai, Nassiba Tabti, An $E_9$ multiplet of BPS states, JHEP 0705:065,2007 (arXiv:hep-th/0703285)

• Andrew Callister, Douglas Smith, Topological BPS charges in 10 and 11-dimensional supergravity, Phys. Rev. D78:065042,2008 (arXiv:0712.3235)

• Andrew Callister, Douglas Smith, Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA, Phys.Rev.D80:125035,2009 (arXiv:0907.3614)

• Andrew Callister, Topological BPS charges in 10- and 11-dimensional supergravity, thesis 2010 (spire)

• A. A. Golubtsova, V.D. Ivashchuk, BPS branes in 10 and 11 dimensional supergravity, talk at DIAS 2013 (pdf slides)

• Cristine N. Ferreira, BPS solution for eleven-dimensional supergravity with a conical defect configuration (arXiv:1312.0578)

Discussion of black hole horizons includes

• Teng Fei, Bin Guo, Duong H. Phong, A Geometric Construction of Solutions to 11D Supergravity (arXiv:1805.07506)

### Higher curvature corrections to $D=11$ supergravity

Discussion of higher curvature corrections to 11-dimensional supergravity (i.e. in M-theory):

• Arkady Tseytlin, $R^4$ terms in 11 dimensions and conformal anomaly of (2,0) theory, Nucl. Phys. B584: 233-250, 2000 (arXiv:hep-th/0005072)

#### Via 11d superspace cohomology

Via 11d superspace-cohomology (or mostly):

#### Via superparticle scattering in 11d

Via analysis of would-be superparticle scattering amplitudes on D=11 supergravity backgrounds:

#### Via lifting 10d stringy corrections

From lifting alpha'-corrections in type IIA string theory through the duality between M-theory and type IIA string theory:

#### Via the ABJM M2-brane model

From the ABJM model for the M2-brane:

• Damon J. Binder, Shai Chester, Silviu S. Pufu, Absence of $D^4 R^4$ in M-Theory From ABJM (arXiv:1808.10554)

Discussion in view of the Starobinsky model of cosmic inflation is in

and in view of de Sitter spacetime vacua:

### Scattering amplitudes and Effective action

Computation of Feynman amplitudes/scattering amplitudes and effective action in 11d supergravity:

### Truncations and compactifications

Kaluza-Klein compactifications of supergravity and its consistent truncations:

Discussion of Freund-Rubin compactifications:

### Topology and anomaly cancellation

Discussion of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles. (For more see at M5-brane – anomaly cancellation).