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:

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

$Z\,$ 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