N = 1 supersymmetry algebra in 1 + 1 dimensions

In 1 + 1 dimensions the N = 1 supersymmetry algebra (also known as ${\mathcal {N}}=(1,1)$ because we have one left-moving SUSY generator and one right moving one) has the following generators:

supersymmetric charges:

Q,{\bar {Q}}

supersymmetric central charge:

Z\,

time translation generator:

H\,

space translation generator:

P\,

boost generator:

N\,

fermionic parity:

\Gamma \,

unit element:

I\,

The following relations are satisfied by the generators:

{\begin{aligned}&\{\Gamma ,\Gamma \}=2I&&\{\Gamma ,Q\}=0&&\{\Gamma ,{\bar {Q}}\}=0\\&\{Q,{\bar {Q}}\}=2Z&&\{Q,Q\}=2(H+P)&&\{{\bar {Q}},{\bar {Q}}\}=2(H-P)\\&[N,Q]={\frac {1}{2}}Q&&[N,{\bar {Q}}]=-{\frac {1}{2}}{\bar {Q}}&&[N,\Gamma ]=0\\&[N,H+P]=H+P&&[N,H-P]=-(H-P)&&\end{aligned}}

$Z\,$ is a central element.

The supersymmetry algebra admits a $\mathbb {Z} _{2}$ -grading. The generators $H,P,N,Z,I\,$ are even (degree 0), the generators $Q,{\bar {Q}},\Gamma \,$ are odd (degree 1).

2(H − P) gives the left-moving momentum and 2(H + P) the right-moving momentum.

Basic representations of this algebra are the vacuum, kink and boson-fermion representations, which are relevant e.g. to the supersymmetric (quantum) sine-Gordon model.

References

K. Schoutens, Supersymmetry and factorized scattering, Nucl.Phys. B344, 665–695, 1990
T.J. Hollowood, E. Mavrikis, The N = 1 supersymmetric bootstrap and Lie algebras, Nucl. Phys. B484, 631–652, 1997, arXiv:hep-th/9606116

This article is issued from Wikipedia - version of the 7/5/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.