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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
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
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.
Dan 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)