Contents

supersymmetry

# Contents

## 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).

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 =$123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9*
8$\ast$
7M2${}_{top}$
6F1${}_{little}$, 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 =$123456789
11$\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 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.