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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 $1440 = 6 x 8 x 30$. So
is divisble by 6.
Assume that $(\frac{1}{2}p_1)$ is further divisble 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.
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).
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).
11d supergravity was originally found in
Formulation in terms of supergeometry is in
E. 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 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
More recent textbook accounts include
Discussion of the equivalence of the 11d SuGra equations of motion with the supergravity torsion constraints is in
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 $AdS_4 \times S^7$ 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 limit 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.
A classification of symmetric solutions is discussed in
Discussin of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles.
See also the relevant references at M5-brane.
Danile 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)