N=1 supersymmetry algebra in 1+1 dimensions

From Wikipedia, the free encyclopedia

In 1+1 dimensions the N=1 supersymmetry algebra 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,\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).


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.

[edit] 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