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:

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

$Z\,$ 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