superalgebra and (synthetic ) supergeometry
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).
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).
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 | * | * | ||||||||
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) |
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
Last revised on January 15, 2019 at 09:03:32. See the history of this page for a list of all contributions to it.