nLab 3-brane in 6d

Idea

There is supposed to be a $(p=3)$-brane in 6-dimensional super-spacetime given by the Green-Schwarz action functional induced by the exceptional super Lie algebra $(3+2)$-cocycle on $\mathfrak{siso}(5,1)$ (Hughes-Liu-Polchinski 86).

This is thought to be the intersection locus of two M5-branes (Papadopoulos & Townsend 1996, Tseytlin 1996, Howe, Lambert & West 1998, p. 2, Kachru, Oz & Yin 1998).

Since this brane has codimension 2, it is a defect brane.

Related concepts

• D3-brane, self-dual string

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)

The Brane molecule

Furthermore, there exists a more general classification of possible supermembranes in spacetime with $S$ spatial dimensions and $T$ time dimensions, appearing in (Blencowe-Duff 88). In this sense, the brane scan is but the $T=1$ branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See D=12 supergravity and bosonic M-theory.

It’s generalization featuring L-infinity-algebra extensions is the brane bouquet.

References

• Miles Blencowe, Mike Duff, Supermembranes and the Signature of Space-time, Nucl. Phys. B310 (1988) 387-404 (spire:262142, 10.1016/0550-3213(88)90155-1, pdf)

References

The original construction is in

• James Hughes, Jun Liu, Joseph Polchinski, Supermembranes, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374 (spire)

Discussion building on that includes

• Martin Rocek, Arkady Tseytlin, Partial breaking of global D=4 supersymmetry, constrained superfields, and 3-brane actions, Phys.Rev.D59:106001, 1999 (arXiv:hep-th/9811232)

The relevant cocycle for discussion as a Green-Schwarz sigma-model is given in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, (III.7.18) of Supergravity and Superstrings - A Geometric Perspective, World Scientific, 1991

  (in D'Auria-Fré formulation of supergravity)

Discussion of the 3-brane in 6d explicitly as a black brane in an M5-brane/NS5-brane worldvolume is due to

• Paul Howe, Neil Lambert, Peter West, The Threebrane Soliton of the M-Fivebrane, Phys. Lett. B419 (1998) 79-83 (arXiv:hep-th/9710033)

and the understanding of this configuration as resulting from two intersecting M5-branes is due to

• George Papadopoulos, Paul Townsend, Intersecting M-branes, Phys. Lett. B380 (1996) 273 (arXiv:hep-th/9603087)

• Arkady Tseytlin, Harmonic superpositions of M-branes, Nucl. Phys. B475 (1996) 149 (arXiv:hep-th/9604035, doi:10.1016/0550-3213(96)00328-8)

• Hironori Mori, M-theory Perspectives on Codimension-2 Defects, Osaka (2016) [inspire:1519095]

• Hironori Mori, Yuji Sugimoto, Surface Operators from M-strings, Phys. Rev. D 95 026001 (2017) [arXiv:1608.02849, doi:10.1103/PhysRevD.95.026001]

with a matrix model-description in:

• Shamit Kachru, Yaron Oz, Zheng Yin, Matrix Description of Intersecting M5 Branes JHEP 9811:004, (1998) (arXiv:hep-th/9803050)

For more on this see

• Joaquim Gomis, David Mateos, Joan Simón, Paul Townsend, Brane-Intersection Dynamics from Branes in Brane Backgrounds, Phys. Lett. B430 (1998) 231-236 (arXiv:hep-th/9803040)

See also

• S. Bellucci, N. Kozyrev, S. Krivonos, A Sutulin, Component on-shell actions of supersymmetric 3-branes: I. 3-brane in $D = 6$, Class. Quantum Grav. 32 (2015) 035025 (doi:10.1088/0264-9381/32/3/035025)

The relation to D=4 N=2 super Yang-Mills theory is discussed in

• Paul Howe, Neil Lambert, Peter West, Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills, Phys. Lett. B418 (1998) 85-90 (arXiv:hep-th/9710034)

• Neil Lambert, Peter West, Gauge Fields and M-Fivebrane Dynamics, Nucl. Phys. B524 (1998) 141-158 (arXiv:hep-th/9712040)

• Neil Lambert, Peter West, $N=2$ Superfields and the M-Fivebrane, Phys. Lett. B424 (1998) 281-287 (arXiv:hep-th/9801104)

• Neil Lambert, Peter West, Monopole Dynamics from the M-Fivebrane, Nucl. Phys. B556 (1999) 177-196 (arXiv:hep-th/9811025)

and via F-theory in

• Robert de Mello Koch, Alastair Paulin-Campbell, Joao P. Rodrigues, Monopole Dynamics in $\mathcal{N}=2$ super Yang-Mills Theory From a Threebrane Probe, Nucl. Phys. B559 (1999) 143-164 (arXiv:hep-th/9903207)

As M5-probe branes in an AdS7-CFT6 background (i.e. in the near horizon limit of black M5-branes):

• Varun Gupta, Holographic M5 branes in $AdS_7 \times S^4$, J. High Energ. Phys. 2021 32 (2021) [arXiv:2109.08551, doi:10.1007/JHEP12(2021)032]

• Varun Gupta, More Holographic M5 branes in $AdS_7 \times S^4$ [arXiv:2301.02528]