nLab super 1-brane in 3d



String theory




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

For N=(1,1)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).


While the light-cone gauge quantization of the Green-Schwarz superstrings breaks Lorentz group-symmetry for d=4d = 4 and d=6d = 6, it preserves Lorentz symmetry not only for d=10d = 10 (the “critical string” of heterotic string theory or type II string theory) but also for d=3d = 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=3d = 3 (Mezincescu-Routh-Townsend 13).

super 2-brane in 4d

The brane scan.

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

=d\stackrel{d}{=}p=p =123456789
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
7M2 top{}_{top}
6F1 little{}_{little}, S1 sd{}_{sd}S3

(The first columns follow the exceptional spinors table.)

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

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

The Brane molecule

Furthermore, there exists a more general classification of possible supermembranes in spacetime with SS spatial dimensions and TT time dimensions, appearing in (Blencowe-Duff 88). In this sense, the brane scan is but the T=1T=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.

The brane molecule without assuming super Poincare invariance.



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

Last revised on January 15, 2019 at 14:03:32. See the history of this page for a list of all contributions to it.