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# N=1 supersymmetry algebra in 1+1 dimensions

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

supersymmetric charges: $Q, bar\left\{Q\right\}$
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\left\{align\right\}
& { 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 $mathbb\left\{Z\right\}_2$-grading. The generators $H, P, N, Z, I,$ are even (degree 0), the generators $Q, bar\left\{Q\right\}, Gamma,$ 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
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