N=1 supersymmetry algebra in 1+1 dimensions

In 1+1 dimensionsthe 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: :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 thegenerators:

:

$Z,$ 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 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 andboson-fermion representations, which are relevant e.g. to thesupersymmetric (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