In 11-dimensional supergravity the brane electrically charged under the supergravity C-field is the M2-brane/membrane. The dual under electric-magnetic duality is the M5-brane.
As a Green-Schwarz sigma-model: BLNPST 97
As a black brane solution of 11-dimensional supergravity the M5-brane is given (Gueven 92) by the spacetime $\mathbb{R}^{5,1} \times (\mathbb{R}^5-\{0\})$ with pseudo-Riemannian metric given by
for $H = 1 + \frac{1}{r}$ and $r$ the distance in $\mathbb{R}^5$ from the origin, and with field strength of the supergravity C-field being
This is a $1/2$-BPS state of 11-dimensional supergravity.
The near horizon geometry of this spacetime is AdS7$\times$S4. For more on this see at AdS-CFT.
More generally for lower BPS black M5-branes, the near horizon geometry is $AdS_7 \times S^4/G$, where $G$ is a finite subgroup of SU(2) (ADE subgroup) acting by left multiplication on the quaternions $\mathbb{H}$ in the canonical way, under the identitfication $S^4 \simeq S(\mathbb{R}^5) \simeq S(\mathbb{R}\oplus \mathbb{H})$ (MFF 12, section 8.3).
1/2 BPS black branes in supergravity: D-branes, F1-brane, NS5-brane, M2-brane, M5-brane
(table taken from Blumenhagen-Lüst-Theisen “Basic concepts of string theory”)
the worldvolume theory of the M5-brane is the 6d (2,0)-superconformal QFT.
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($\sim$ gerbe), presumably with structure 2-group the automorphism 2-group $AUT(G)$ of some Lie group $G$.
For instance in the proposal of (SSW11) one sees in equation (2.1) almost the data of an $\mathfrak{aut}(\mathfrak{g})$-Lie 2-algebra valued forms.
The M5-brane admits two solitonic excitations ($p$-branes within branes)
$p = 1$: the self-dual string
$p = 3$: the 3-brane in 6d (see there for more)
On dimensional reduction of 11-dimensional supergravity on a circle the M5-brane turns into the NS5-brane and the D4-brane of type II string theory.
The compactification of the 5-brane on a Riemann surface yields as worldvolume theory N=2 D=4 super Yang-Mills theory. See at N=2 D=4 SYM – Construction by compactification of 5-branes.
The AdS/CFT correspondence for the 5-brane is $AdS_7/CFT_6$ 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 $\hat G$ be the circle 3-bundle with connection on a 7-dimensional manifold $X$ with boundary the M5-brane, thought of as the compactification of the supergravity C-field from 11-dimensional supergravity down to 7-dimensional supergravity.
As discussed there, the 7-dimensional Chern-Simons theory action functional on these 3-connections is
where
$\exp(i \int_X (-))$ is the higher holonomy / fiber integration in ordinary differential cohomology from $X$ to the point
of the Beilinson-Deligne cup product 7-connection $\hat G_4 \cup \hat G_4$.
The space of states of this 7d theory on the M5 worldvolume $\partial X$ would be the space of conformal blocks of the 6d (2,0)-supersymmetric QFT on the worldvolume.
Except, that it turns out that the first Chern class of the corresponding prequantum line bundle is twice that required from geometric quantization.
Therefore the above action functional is not yet the correct one, but only a fractional version of it is. However, the class $G_4 \cup G_4$ in integral cohomology has in general no reason to be divisible by 2.
This is related to the fact that as a quadratic form on the ordinary differential cohomology group $\hat H^4(X)$, the above is not a quadratic refinement of
but of twice that. In (Witten 1996) it was argued, and later clarified in (Hopkins-Singer), that instead the action functional should be replaced by a proper quadratic refinement.
This is accomplished by shifting the center of the quadratic form by a lift $\lambda \in H^4(X, \mathbb{Z})$ of the degree-4 Wu class $\nu_4 \in H^4(X, \mathbb{Z}/2)$ from 0 to $\frac{1}{2}\lambda$.
(For that to make sense in integral cohomology, either the Wu class $\lambda$ happens to be divisible by 2 on $X$, 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 $X$ is such that $\lambda$ is divisbible by 2.)
Since $X$, being a spacetime for supergravity, admits (and is thought to be equipped with) a spin structure, by the discussion at Wu class it follows that $\lambda$ is the first fractional Pontryagin class $\frac{1}{2}p_1$
By the very definition of Wu class, it follows that for any $\hat \alpha \in \hat H^4(X)$ the combination
is divisible by 2.
Therefore define then the modified quadratic form
(see differential string structure for the definition of the differential refinement $\hat \mathbf{\lambda} = \frac{1}{2}\hat \mathbf{p}_1$), 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.
Notice that where the equations of motion of the original action functional are $\hat a = 0$, those of this shifted one are $\hat a = - \frac{1}{2}\hat \mathbf{\lambda}$. One may therefor calls $-\frac{1}{2}\lambda$ here a background charge for the 7-d Chern-Simons theory.
This is now indeed a quadratic refinement of the intersection pairing:
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 $\hat a = 0$.
Then in terms of the original $\hat G_4$ the action functional for the holographic dual 7d Chern-Simons theory reads
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 $2(G_4 - a)$-twisted Wu structure in $\mathbb{Z}/2$-cohomology. Its lift to integral cohomology is the $2(G_4 - a)$-twisted differential string structure known as the “Witten quantization condition” on the supergravity C-field.
We discuss the conditions on the restriction of the supergravity C-field on the ambient 11-dimensional supergravity spacetime to the M5-brane.
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 $\hat H_3 \hat H^3(X)$ on spacetime $X$ to an oriented D-brane $Q \hookrightarrow X$ has to trivialize, up to torsion, relative to the integral Stiefel-Whitney class $W_3 = \beta(w_2)$, where $\beta$ is the Bockstein homomorphism induced from the short exact sequence $\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \mathbb{Z}_2$:
thus defining a twisted spin^c-structure on the D-brane.
The analog of this for the M5-brane is discussed in (Witten00, section 5). There it is argued that there is a class
on the 5-brane such that under the Bockstein homomorphism $\beta'$ induced by the short exact sequence $\mathbb{Z} \to \mathbb{R} \to U(1)$ we have for the supergravity C-field $\hat G \in \hat H^4(X)$ the condition
By the above quantization condition, this may also be thought of as witnessing a twisted string structure on the 5-brane (Sati).
This condition reduces to the above one for the $B$-field under double dimensional reduction on the circle.
See at M5-brane charge
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
$\;\;\;\;\downarrow$ topological sector | |
7-dimensional Chern-Simons theory | |
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ 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 |
$\;\;\;\;\; \downarrow$ topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | |
A-model on $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ topological sector |
5-dimensional Chern-Simons theory |
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality |
N=4 D=4 super Yang-Mills theory |
$\;\;\;\;\; \downarrow$ topological twist |
topologically twisted N=4 D=4 super Yang-Mills theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface |
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence |
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The history as of the 1990 is reviewed in
Further reviews and general accounts include
Robbert Dijkgraaf, The mathematics of fivebranes (pdf)
Hisham Sati, Geometric and topological structures related to M-branes (2010)
The M5 was first found as a black brane of 11-dimensional supergravity (the black fivebrane) in
That this metric, as well as that of every black $p$ brane for odd $p$, is completely non-singular was observed in
The orbifold structure for lower-BPS M5-branes is discussed in
Discussion in terms of E11-U-duality and current algebra is in
Hirotaka Sugawara, Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 (arXiv:1701.06894)
Shotaro Shiba, Hirotaka Sugawara, M2- and M5-branes in E11 Current Algebra Formulation of M-theory (arXiv:1709.07169)
The sigma-model description of the (single) M5-brane of Green-Schwarz action functional-type was found in covariant form in
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)
Paolo Pasti, Dmitri Sorokin and M. Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys. Lett. B398 (1997) 41.
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)
Dmitri Sorokin, Superbranes and Superembeddings (arXiv:hep-th/9906142)
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
The double dimensional reduction of the M5-brane to the D4-brane in type II string theory is discussed in
Further developments include
The original article suggesting the description of the self-dual higher gauge theory on the 5-brane holographically by a dual higher dimensional Chern-Simons theory is
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.
The double dimensional reduction to the D4-brane D=5 super Yang-Mills theory and the relation to Khovanov homology is discussed in
with further comments in
A proposal for a construction as a higher gauge theory for string 2-connections is due to
based on
The relation of the M5-brane to the D4-brane and the D=5 super Yang-Mills theory in its worldvolume theory by double dimensional reduction is discussed in the following references
Malcolm Perry, John Schwarz, Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory, Nucl. Phys. B489 (1997) 47-64 (arXiv:hep-th/9611065)
Neil Lambert, Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills, JHEP 1101:083 (2011) (arXiv:1012.2882)
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).
Discussion of open M5-branes ending on M9-branes in a Yang monopole is in
The fact that the worldvolume theory of the M5-brane should support fields that are self-dual connections on a 2-bundle ($\sim$ a gerbe) is discussed in
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 $G \times G$ (arXiv:1108.5131)
Henning Samtleben, Ergin Sezgin, Robert Wimmer, (1,0) superconformal models in six dimensions (arXiv:1108.4060)
A formal proposal is here.
Edward Witten, Duality relations among topological effects in string theory, J. High Energy Phys. 0005 (2000) 031 (arXiv:hep-th/9912086)
Hisham Sati, Geometric and topological structures related to M-branes II: Twisted String and String^c structures (arXiv:1007.5419)
Hisham Sati, Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures (arXiv:1109.4461)