# Contents

## Idea

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.

## Properties

### Worldvolume 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.

### Branes inside the M5

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)

### Dimensional reduction

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.

### Holographic dual

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.

### Conformal blocks and 7d Chern-Simons dual

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

$\hat G_4 \mapsto \exp(i \int_X \hat G_4 \cup \hat G_4) \,,$

where

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

$(\hat G, \hat G') \mapsto \exp(i \int_X \hat G \cup \hat G') \,,$

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 $\frac{1}{2}p_1$

$(\frac{1}{2}p_1 \; mod \; 2) \; = \; \nu_4 \in H^4(X, \mathbb{Z}/2) \,.$

By the very definition of Wu class, it follows that for any $\hat \alpha \in \hat H^4(X)$ the combination

$\hat \alpha \cup \hat \alpha + \hat \alpha \cup \hat \lambda = Sq^4(\hat \alpha) - \hat \alpha \cup \hat \lambda \; =\; 0 \; mod \; 2$

is divisible by 2.

Therefore define then the modified quadratic form

$\exp(i S^\lambda) \; : \; \hat a \mapsto \exp i \int_X \frac{1}{2} \left( \hat a \cup \hat a + \hat a \cup \hat \mathbf{\lambda} \right)$

(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:

$\exp i \left( S^\lambda\left(\hat a + \hat b \right) - S^\lambda\left( \hat a \right) - S^\lambda\left( \hat b \right) + S^\lambda\left( 0 \right) \right) = \exp i \int_X ( \hat a \cup \hat b ) \,.$

To express the correct action functional for the 7d Chern-Simons theory it is useful to define the shifted supergravity C-field

$\hat a := \hat G_4 - \frac{1}{2}\hat \mathbf{\lambda} \,,$

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

$\exp(i S(\hat G_4)) = \exp(i \int_X \frac{1}{2} ( \hat G_4 \cup \hat G_4 - (\frac{1}{2}\hat \mathbf{\lambda})^2 ) ) \,.$

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.

### Resctriction of 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 $\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \mathbb{Z}_2$:

$H_3|_Q \simeq W_3 \,,$

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

$\theta \in H^3(Q, U(1))$

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

$G|_Q = \beta'(\theta) \,.$

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.

11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$
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
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
N=2 D=4 super Yang-Mills theory with S-duality invariance; AGT correspondence
$\;\;\;\;\; \downarrow$ 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

$\,$

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 $Log_G$, geometric Langlands correspondence

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-braneheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

### General

Reviews and general accounts include

### $\sigma$-Model description

Sigma model description of the (single) M5-brane are discussed for Green-Schwarz action functional-type setups in

A compariso of the different actions here is in

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

• Eric Bergshoeff, Mees de Roo, Tomas Ortin, The Eleven-dimensional Five-brane (pdf)

### Worldvolume theory

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

The double dimensional reduction to the D4-brane D=5 super Yang-Mills theory and the relation to Khovanov homology is discussed in

• Michele Nardelli, On some equations concerning Fivebranes and Knots, Wilson Loops in Chern-Simons Theory, cusp anomaly and integrability from String theory. Mathematical connections with some sectors of Number Theory (2011) (pdf)

### Relation to D4-brane

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

### Open M5-branes

Discussion of open M5-branes ending on M9-branes in a Yang monopole is in

### Sugra solutions

The anti de Sitter spacetime $AdS_7 \times S^4$ arises in 11-dimensional supergravity as the large $N$ limit of $N$ coincident M5-branes:

• 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.

### Nonabelian 2-form fields

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

• Edward Witten, Conformal Field Theory In Four And Six Dimensions, in Ulrike Tillmann, Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, London Mathematical Society Lecture Note Series (2004) (arXiv:0712.0157)

as well as sections 3 and 4 of

Proposals for how to implement this are for instance in

A formal proposal is here.

### More on the holographic description

• A. J. Nurmagambetov, I. Y. Park, On the M5 and the AdS7/CFT6 Correspondence (arXiv:hep-th/0110192)

### More on the algebraic topology

• 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)

Revised on September 2, 2013 13:17:23 by Urs Schreiber (89.204.135.109)