Contents

# Contents

## Idea

A membrane sigma-model topological quantum field theory that is roughly related to topological M-theory as the M2-brane is related to M-theory and to the topological string (A-model/B-model) as the M2-brane is related to the string and to the topological M5-brane as the M2-brane is related to the M5-brane.

The target space of the topological membrane is a G2-manifold, the action functional is governed by the higher holonomy of the compatible supergravity C-field over the membrane worldvolume.

## Definition

According to (Bao-Bengtsson-Cederwall-Nillson 05, equation (2.14)) the topological $(p=2)$-brane is the super 2-brane which exists in $D = 7$ according to the brane scan, which says that the super Poincare Lie algebra in $D = 7$ carries an exceptional Lie algebra cocycle of degree $(2+2)$, hence admits the Green-Schwarz action functional for a super 2-brane.

The brane scan.

The Green-Schwarz type super $p$-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

$\stackrel{d}{=}$$p =$123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9$\ast$
8$\ast$
7M2${}_{top}$
6F1${}_{little}$, S1${}_{sd}$S3
5$\ast$
4**
3*

(The first columns follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

$\stackrel{d}{=}$$p =$123456789
11$\Psi^2 E^2$ on sIso(10,1)$\Psi^2 E^5 + \Psi^2 E^2 C_3$ on m2brane
10$\Psi^2 E^1$ on sIso(9,1)$B_2^2 + B_2 \Psi^2 + \Psi^2 E^2$ on StringIIA$\cdots$ on StringIIB$B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4$ on StringIIA$\Psi^2 E^5$ on sIso(9,1)$B_2^4 + \cdots + \Psi^2 E^6$ on StringIIA$\cdots$ on StringIIB$B_2^5 + \cdots + \Psi^2 E^8$ in StringIIA$\cdots$ on StringIIB
9$\Psi^2 E^4$ on sIso(8,1)
8$\Psi^2 E^3$ on sIso(7,1)
7$\Psi^2 E^2$ on sIso(6,1)
6$\Psi^2 E^1$ on sIso(5,1)$\Psi^2 E^3$ on sIso(5,1)
5$\Psi^2 E^2$ on sIso(4,1)
4$\Psi^2 E^1$ on sIso(3,1)$\Psi^2 E^2$ on sIso(3,1)
3$\Psi^2 E^1$ on sIso(2,1)

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$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

In the context of exceptional generalized geometry:

Last revised on June 9, 2017 at 13:50:55. See the history of this page for a list of all contributions to it.