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.
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):
$\stackrel{d}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
11 | M2 | M5 | ||||||||
10 | D0 | F1, D1 | D2 | D3 | D4 | NS5, D5 | D6 | D7 | D8 | D9 |
9 | $\ast$ | |||||||||
8 | $\ast$ | |||||||||
7 | M2${}_{top}$ | |||||||||
6 | F1${}_{little}$, S1${}_{sd}$ | S3 | ||||||||
5 | $\ast$ | |||||||||
4 | $\ast$ | * | ||||||||
3 | * |
(The first colums follow the exceptional spinors table.)
The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):
$\stackrel{d}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
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).
Giulio Bonelli, Alessandro Tanzini, Maxim Zabzine, On topological M-theory (arXiv:hep-th/0509175)
Ling Bao, Topological membranes, Nucl. Phys. Proc. Suppl. 171 (2007) 259-260 (inspire)
In the context of exceptional generalized geometry: