This is a -BPS state of 11-dimensional supergravity.
(table taken from Blumenhagen-Lüst-Theisen “Basic concepts of string theory”)
This worldvolume theory involves self-dual higher gauge theory of the nonabelian kind (Witten07, Witten09): the fields are supposed to be connections on a 2-bundle( gerbe), presumably with structure 2-group the automorphism 2-group of some Lie group .
The M5-brane admits two solitonic excitations (-branes within branes)
The AdS/CFT correspondence for the 5-brane is and relates the 6d (2,0)-superconformal QFT to 7-dimensional supergravity obtained by reduction of 11-dimensional supergravity on o 4-sphere to an and asymptotically 7d anti de Sitter spacetime.
The self-dual 2-connection-field (see there for more details) on the 6-dimensional worldvolume M5-brane is supposed to have a holographic description in terms of a 7-dimensional Chern-Simons theory (Witten 1996). We discuss the relevant “fractional” quadratic form on ordinary differential cohomology that defines the correct action functional.
Let be the circle 3-bundle with connection on a 7-dimensional manifold with boundary the M5-brane, thought of as the compactification of the supergravity C-field from 11-dimensional supergravity down to 7-dimensional supergravity.
is the higher holonomy / fiber integration in ordinary differential cohomology from to the point
Therefore the above action functional is not yet the correct one, but only a fractional version of it is. However, the class in integral cohomology has in general no reason to be divisible by 2.
(For that to make sense in integral cohomology, either the Wu class happens to be divisible by 2 on , or else one has to regard it itself as a twisted differential character of sorts, as explained in (Hopkins-Singer). For the moment we will assume that is such that is divisbible by 2.)
By the very definition of Wu class, it follows that for any the combination
is divisible by 2.
Therefore define then the modified quadratic form
(see differential string structure for the definition of the differential refinement ), where, note, we have included a global factor of 2, which is now possible due to the inclusion of the integral lift of the Wu class.
To express the correct action functional for the 7d Chern-Simons theory it is useful to define the shifted supergravity C-field
which the object whose equations of motion with respect to the 7d Chern-Simons theory are still .
This is the action as it appears in (Witten96, (3.6)).
In terms of twisted differential c-structures we may summarize the outcome of this reasoning as follows:
The divisibility of the action functional requires a -twisted Wu structure in -cohomology. Its lift to integral cohomology is the -twisted differential string structure known as the “Witten quantization condition” on the supergravity C-field.
This is similar to the analogous situation in type II string theory. The the Freed-Witten anomaly cancellation condition demands that the restriction of the B-field on spacetime to an oriented D-brane has to trivialize, up to torsion, relative to the integral Stiefel-Whitney class , where is the Bockstein homomorphism induced from the short exact sequence :
The analog of this for the M5-brane is discussed in (Witten00, section 5). There it is argued that there is a class
This condition reduces to the above one for the -field under double dimensional reduction on the circle.
See at M5-brane charge
|M-theory perspective via AdS7-CFT6||F-theory perspective|
|Kaluza-Klein compactification on||compactificationon elliptic fibration followed by T-duality|
|7-dimensional Chern-Simons theory|
|AdS7-CFT6 holographic duality|
|6d (2,0)-superconformal QFT on the M5-brane with conformal invariance||M5-brane worldvolume theory|
|KK-compactification on Riemann surface||double dimensional reduction on M-theory/F-theory elliptic fibration|
|N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence||D3-brane worldvolume theory with type IIB S-duality|
|topologically twisted N=2 D=4 super Yang-Mills theory|
|KK-compactification on Riemann surface|
|A-model on , Donaldson theory|
|gauge theory induced via AdS5-CFT4|
|type II string theory|
|Kaluza-Klein compactification on|
|5-dimensional Chern-Simons theory|
|AdS5-CFT4 holographic duality|
|N=4 D=4 super Yang-Mills theory|
|topologically twisted N=4 D=4 super Yang-Mills theory|
|KK-compactification on Riemann surface|
|A-model on and B-model on , geometric Langlands correspondence|
|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|
The history as of the 1990 is reviewed in
Further reviews and general accounts include
and in non-covariant form in
A comparison of the different action functionals here is in
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, On the equivalence of different formulations of the M Theory five–brane, Phys. Lett. B408 (1997) 135-141 (arXiv:hep-th/9703127)
The computation of the small fluctuations of this GS-type sigma-model around a solution embedding as the asymptotic boundary of the AdS-spacetime near-horizon geometry of a black 5-brane as above, and the proof, to low order, that the result is the 6d (2,0)-supersymmetric QFT appearing in AdS7-CFT6 duality is due to
A review with emphasis on the coupling to the M2-brane is in
Further developments include
A precise mathematical formulation of the proposal made there is given in
A discussion that embeds this argument into the larger context of AdS-CFT duality is in
See also the references at 6d (2,0)-supersymmetric QFT.
with further comments in
Chong-Sun Chu, Sheng-Lan Ko, Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors, (arXiv:1203.4224) JHEP05(2012)028
See also (Witten 11).
P. Pasti, Dmitri Sorokin and M. Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys.Lett. B398 (1997) 41.
I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, Dmitri Sorokin and M. Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys.Rev.Lett. 78 (1997) 4332.
M. Aganagic, J. Park, C. Popescu and John Schwarz, World-Volume Action of the M Theory Five-Brane, Nucl.Phys. B496 (1997) 191.
these reference are in the wrong place here, needs fixing
as well as sections 3 and 4 of
Proposals for how to implement this are for instance in
Chong-Sun Chu, A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry (arXiv:1108.5131)
A formal proposal is here.