nLab super 1-brane in 3d

Idea

By the brane scan there exists a classical (pre-quantum) Green-Schwarz super p-brane sigma model for $p =1$ (superstrings) with target space being $d = 3$ dimensional super Minkowski spacetime or more generally a superspacetime which is a super Cartan geometry solving the equations of motion of $N=1$ or $N=2$ 3d supergravity. (The literature traditionally knows this as the “3d superstring”.)

For $N=(1,1)$ this is the double dimensional reduction of the super 2-brane in 4d (e.g. Mezincescu-Townsend 11, p. 44).

These superstrings in 3d target spacetime play a role in the AdS3-CFT2 and CS-WZW correspondence (e.g. (YuZ-Zhang 98)) and at least after a topological twist as exhibiting 3d Chern-Simons theory as a second quantization (see at TCFT – Effective background theories).

Quantization

While the light-cone gauge quantization of the Green-Schwarz superstrings breaks Lorentz group-symmetry for $d = 4$ and $d = 6$, it preserves Lorentz symmetry not only for $d = 10$ (the “critical string” of heterotic string theory or type II string theory) but also for $d = 3$ (Mezincescu-Townsend 10, Mezincescu-Townsend 11). This quantization turns out to be equivalent to that of the RR-sector of the worldsheet supersymmetric spinning string in $d = 3$ (Mezincescu-Routh-Townsend 13).

Related concepts

super 2-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)

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.

Compare:

• 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)

• brane bouquet

References

• Ming Yu, Bo Zhang, Light-Cone Gauge Quantization of String Theories on AdS3 Space, Nucl.Phys. B551 (1999) 425-449 (arXiv:hep-th/9812216)

• Luca Mezincescu, Paul Townsend, Anyons from Strings, Phys.Rev.Lett.105:191601,2010 (arXiv:1008.2334)

• Luca Mezincescu, Paul Townsend, Quantum 3D Superstrings, PhysRevD.84.106006 (arXiv:1106.1374)

• Luca Mezincescu, Alasdair J. Routh, Paul Townsend, Equivalence of 3D Spinning String and Superstring, J. High Energ. Phys. (2013) 2013: 24 (arXiv:1305.5049)

The F-theory-lift of the 3d superstring is discussed in

• William D. Linch III, Warren Siegel, F-theory Superspace (arXiv:1501.02761)