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$N=1$ supergravity in $d = 11$.
for the moment see the respective section at D'Auria-Fre formulation of supergravity
(…)
(…)
under construction
where $p_i$ is the $i$th Pontryagin class.
Concerning the integrality of
on a spin manifold $X$. (Witten96, p.9)
First, the index of a Dirac operator on $X$ is
Notice that $1440 = 6 \times 8 \times 30$. So
is divisible by 6.
Assume that $(\frac{1}{2}p_1)$ is further divisible by 2 (see the relevant discussion at M5-brane).
Then the above becomes
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
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.
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 related to the flux quantization?-condition of the supergravity C-field.
For effects of higher curvature corrections in a Starobinsky model of cosmic inflation see there.
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
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)
The basic BPS states of 11d SuGra are
(e.g. EHKNT 07)
10-dimensional type II supergravity, heterotic supergravity
supergravity C-field, supergravity Lie 3-algebra, supergravity Lie 6-algebra
string theory FAQ – Does string theory predict supersymmetry?
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
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:
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 $AdS_4$.
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.
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
See also
Discussion of higher curvature corrections:
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)
Dimitrios Tsimpis, 11D supergravity at $\mathcal{O}(l^3)$, 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)
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)
and from the ABJM model:
See also:
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)
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)
Kaluza-Klein compactifications of supergravity and its consistent truncations:
Discussion of Freund-Rubin compactifications:
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)
Last revised on February 9, 2021 at 10:28:11. See the history of this page for a list of all contributions to it.