nLab super 2-brane in 4d

Idea

By the brane scan there exists a classical (pre-quantum) Green-Schwarz super p-brane sigma model for $p =2$ (supermembrane) with target space being $d = 4$ dimensional super Minkowski spacetime with N=1 supersymmetry, or more generally a superspacetime which is a super Cartan geometry solving the equations of motion of d=4 N=1 supergravity. (The literature traditionally knows this as the “4d supermembrane”.)

Its double dimensional reduction is the super 1-brane in 3d (see there) for $N=(1,1)$ 3d supergravity.

Related entries

• super 1-brane in 3d

• super 1-brane in 4d

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					*
8				*
7			M2${}_{top}$
6		F1${}_{little}$, S1${}_{sd}$		S3
5			*
4		*	*
3		*

(The first columns 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)

References

The action functional is apparently originally due to

• A. Achúcarro, Jerome Gauntlett, K. Itoha, Paul Townsend, World-volume supersymmetry from spacetime supersymmetry of the four-dimensional supermembrane, Nuclear Physics B Volume 314, Issue 1, 20 February 1989, Pages 129–157 (doi:10.1016/0550-3213(89)90115-6)

Further dicussion of the coupling to the d=4 N=1 supergravity background includes

• Burt Ovrut, Daniel Waldram, Membranes and Three-form Supergravity, Nucl.Phys. B506 (1997) 236-266 (arXiv:hep-th/9704045)

• M. P. Garcia del Moral, J. M. Pena, A. Restuccia, $N=1$ 4D Supermembrane from $11D$, JHEP0807:039,2008 (arXiv:0709.4632)

• M. Huebscher, P. Meessen, T. Ortin, Domain walls and instantons in $N=1$, $d=4$ supergravity, JHEP 1006:001,2010 (arXiv:0912.3672)

• Igor Bandos, Carlos Meliveo, Supermembrane interaction with dynamical $D=4$ $N=1$ supergravity. Superfield Lagrangian description and spacetime equations of motion (arXiv:1205.5885)

• Carlos Meliveo, On Supermembrane in $D=4$, multiple M0-brane in $D=11$ and Supersymmetric Higher Spin Theories (arXiv:1312.3512)

See also

• Igor Bandos, Fotis Farakos, Stefano Lanza, Luca Martucci, Dmitri Sorokin, Three-forms, dualities and membranes in four-dimensional supergravity, JHEP07(2018)028 (arXiv:1803.01405)

Discussion of coupling of the super 2-brane in 4d to D=4 N=1 super Yang-Mills theory:

• Igor Bandos, Stefano Lanza, Dmitri Sorokin, Supermembranes and domain walls in $\mathcal{N}=1$, $D=4$ SYM (arXiv:1905.02743)

On membranes in D=4 supergravity (super 2-brane in 4d):

• Igor Bandos, Fotis Farakos Stefano Lanza, L. Martucci, Dmitri Sorokin, Higher Forms and Membranes in 4D Supergravities, in: Proceedings of Higher Structures in M-Theory 2018, Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 (arXiv:1903.02841, doi:10.1002/prop.201910020)

Discussion of strongly coupled 4d physics akin to 11d M-theory with its M2-brane includes

• Edward Witten, Strong coupling and the cosmological constant, Mod.Phys.Lett.A10:2153-2156,1995 (arXiv:hep-th/9506101)

(on possible solutions to the cosmological constant problem).