nLab D=11 N=1 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

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

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

$$\lambda \coloneqq \frac{1}{2}p_1 \,.$$

Concerning the integrality of the I8-term

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

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

• the M2-brane

• the M5-brane

• the M-wave

• the Kaluza-Klein monopole

(e.g. EHKNT 07)

Related concepts

• supergravity

• M-theory, F-theory

• D=12 supergravity, D=14 supersymmetry

• 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

• Horava-Witten theory

• string theory FAQ – Does string theory predict supersymmetry?

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

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	$\,$	$\,$
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-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
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
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-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-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

References

General

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

• Werner Nahm, Supersymmetries and their Representations, Nucl.Phys. B135 (1978) 149 (spire, pdf)

The theory was then actually constructed in

• Eugene Cremmer, Bernard Julia, Joël Scherk, Supergravity in theory in 11 dimensions, Phys. Lett. 76B (1978) 409 (doi:10.1016/0370-2693(78)90894-8)

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

• Mike Duff, chapter I of: The World in Eleven Dimensions: Supergravity, Supermembranes and M-theory, IoP 1999 (publisher)

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, Nuclear Physics B201 (1982) 101-140

of which a textbook account is in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, chapters III.8 and V.4-V.11 in Supergravity and Superstrings - A Geometric Perspective, World Scientific, 1991

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 Tsimpis 04

More recent textbook accounts include

• Antoine Van Proeyen, Daniel Freedman, section 10 of: Supergravity, Cambridge University Press, 2012 (pdf, doi:10.1017/CBO9781139026833)

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

• Paul Howe, Weyl Superspace, Physics Letters B Volume 415, Issue 2, 11 December 1997, Pages 149–155 (arXiv:hep-th/9707184)

following

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

Much computational detail is displayed in

• André Miemiec, Igor Schnakenburg, Basics of M-Theory, Fortsch. Phys. 54 (2006) 5-72 (arXiv:hep-th/0509137, doi:10.1002/prop.200510256)

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.

• Mike Duff, Kellogg Stelle, Multi-membrane solutions of $D = 11$ supergravity, Phys. Lett. B 253, 113 (1991) (spire:299386, doi:10.1016/0370-2693(91)91371-2)

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.

See also

• Don Page, Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity, Phys. Rev. D 28, 2976 (1983) (spire:14480 doi:10.1103/PhysRevD.28.2976)

• Ergin Sezgin, 11D Supergravity on $AdS_4 \times S^7$ versus $AdS_7 \times S^4$ (arXiv:2003.01135)

A classification of symmetric solutions is discussed in

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

• Linus Wulff, All symmetric space solutions of eleven-dimensional supergravity (arXiv:1611.06139)

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

• Jan Gutowski, George Papadopoulos, Static M-horizons (arXiv:1106.3085)

See also

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

Resolution of scalar field-dressed Schwarzschild black holes in D=11 supergravity:

• Ibrahima Bah, Pierre Heidmann, Geometric Resolution of Schwarzschild Horizon [arXiv:2303.10186]

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):

• Kasper Peeters, Pierre Vanhove, Anders Westerberg, Supersymmetric $R^4$ actions and quantum corrections to superspace torsion constraints (arXiv:hep-th/0010182)

• H. Lu, Christopher Pope, Kellogg Stelle, Paul Townsend, Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory, JHEP 0410:019, 2004 (arXiv:hep-th/0312002)

  (specifically for M-theory on G2-manifolds)

• H. Lu, Christopher Pope, Kellogg Stelle, Paul Townsend, String and M-theory Deformations of Manifolds with Special Holonomy, JHEP 0507:075, 2005 (arXiv:hep-th/0410176)

  (specifically for M-theory on G2-manifolds)

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

• Paul Howe, $R^4$ terms in supergravity and M-theory (arXiv:hep-th/0408177)

• Martin Cederwall, Ulf Gran, Bengt Nilsson, Dimitrios Tsimpis, Supersymmetric Corrections to Eleven-Dimensional Supergravity, JHEP0505:052, 2005 (arXiv:hep-th/0409107)

• Yoshifumi Hyakutake, Sachiko Ogushi, $R^4$ Corrections to Eleven Dimensional Supergravity via Supersymmetry, Phys.Rev. D74 (2006) 025022 (arXiv:hep-th/0508204)

• Yoshifumi Hyakutake, Sachiko Ogushi, Higher Derivative Corrections to Eleven Dimensional Supergravity via Local Supersymmetry, JHEP0602:068, 2006 (arXiv:hep-th/0601092)

• Anirban Basu, Constraining gravitational interactions in the M theory effective action, Classical and Quantum Gravity, Volume 31, Number 16, 2014 (arXiv:1308.2564)

• Bertrand Souères, Dimitrios Tsimpis, The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity, Phys. Rev. D 95, 026013 (2017) (arXiv:1612.02021)

• Bertrand Souères, Supergravities in Superspace, Lyon 2018 (tel:[phone], pdf)

Via superparticle scattering in 11d

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

• Kasper Peeters, Jan Plefka, Steffen Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 0508 (2005) 095 (arXiv:hep-th/0507178)

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:

• James Liu, Ruben Minasian, Higher-derivative couplings in string theory: dualities and the B-field, Nucl. Phys. B 2013 (arXiv:1304.3137)

Via type IIB supergravity:

• James Liu, Ruben Minasian, Raffaele Savelli, Andreas Schachner, Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles $[$arXiv:2205.11530$]$

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)

See also

See also

• Mohammad Garousi, Minimal gauge invariant couplings at order $\ell^6_p$ in M-theory (arXiv:2102.00639)

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

• Katrin Becker, Melanie Becker, Supersymmetry Breaking, M-Theory and Fluxes, JHEP 0107:038,2001 (arXiv:hep-th/0107044)

• Kazuho Hiraga, Yoshifumi Hyakutake, Inflationary Cosmology via Quantum Corrections in M-theory (arXiv:1809.04724)

• Kazuho Hiraga, Yoshifumi Hyakutake, Scalar Cosmological Perturbations in M-theory with Higher Derivative Corrections (arxiv:1910.12483)

and in view of de Sitter spacetime vacua:

• Johan Blåbäck, Ulf Danielsson, Giuseppe Dibitetto, Suvendu Giri, Constructing stable de Sitter in M-theory from higher curvature corrections (arXiv:1902.04053)

Scattering amplitudes and Effective action

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

• Stanley Deser, Domenico Seminara, Counterterms/M-theory Corrections to D=11 Supergravity, Phys.Rev.Lett.82:2435-2438, 1999 (arXiv:hep-th/9812136)

• Stanley Deser, Domenico Seminara, Tree Amplitudes and Two-loop Counterterms in D=11 Supergravity, Phys.Rev.D62:084010, 2000 (arXiv:hep-th/0002241)

• L. Anguelova, P. A. Grassi, P. Vanhove, Covariant One-Loop Amplitudes in $D=11$, Nucl. Phys. B702 (2004) 269-306 (arXiv:hep-th/0408171)

• Kasper Peeters, Jan Plefka, Steffen Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 0508 (2005) 095 (arXiv:hep-th/0507178)

• Hamid R. Bakhtiarizadeh, Gauge field corrections to eleven dimensional supergravity via dimensional reduction (arXiv:1711.11313)

Truncations and compactifications

Kaluza-Klein compactifications of supergravity and its consistent truncations:

• Mike Duff, Bengt Nilsson, Christopher Pope, Kaluza-Klein supergravity, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 (spire:229417, doi:10.1016/0370-1573(86)90163-8)

Discussion of Freund-Rubin compactifications:

• Hermann Nicolai, Krzysztof Pilch, Consistent truncation of $d = 11$ supergravity on $AdS_4 \times S^7$, JHEP 03 (2012) 099 (arXiv:1112.6131)

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).

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

• Edward Witten, Five-Brane Effective Action In M-Theory, J.Geom.Phys.22:103-133, 1997 (arXiv:hep-th/9610234)

• Dan Freed, Jeff Harvey, Ruben Minasian, Greg Moore, Gravitational Anomaly Cancellation for M-Theory Fivebranes, Adv.Theor.Math.Phys.2:601-618, 1998 (arXiv:hep-th/9803205)

• Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review, Nucl.Phys. B675 (2003) 416-446 (arXiv:hep-th/0307152)

• Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class $\mathcal{S}$ Anomalies from M-theory Inflow (arXiv:1812.04016)

• Daniel 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)

Review of U-duality and exceptional generalized geometry in KK-compactification of D=11 supergravity:

• Henning Samtleben, 11D Supergravity and Hidden Symmetries, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2303.12682]