Types of quantum field thories
supergravity in .
for the moment see the respective section at D'Auria-Fre formulation of supergravity
where is the th Pontryagin class.
Concerning the integrality of
Notice that . So
is divisble by 6.
Assume that is further divisble by 2 (see the relevant discussion at M5-brane).
Then the above becomes
and hence then is divisible at least by 24.
and so when is further divisible by 2 we have that is divisible by 48. Hence 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 is the curvature 3-form of the supergravity C-field and 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.
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|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|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
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
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
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.
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 ; 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 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 subgroup.
Some of the above is taken from this TP.SE thread.
A classification of symmetric solutions is discussed in
See also the relevant references at M5-brane.
Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review (arXiv:hep-th/0307152)