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| In 1 + 1 dimensions the ''N'' = 1 [[supersymmetry]] [[algebra over a field|algebra]] (also known as <math>\mathcal{N}=(1,1)</math> because we have one left-moving SUSY generator and one right moving one) has the following [[generating set|generator]]s:
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| :[[supercharge|supersymmetric charges]]: <math>Q, \bar{Q}</math> | |
| :supersymmetric central charge: <math>Z\,</math>
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| :time [[translation (geometry)|translation]] generator: <math>H\,</math>
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| :space translation generator: <math>P\,</math>
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| :[[Lorentz boost|boost]] generator: <math>N\,</math>
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| :[[(-1)F|fermionic parity]]: <math>\Gamma\,</math>
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| :[[identity element|unit element]]: <math>I\,</math>
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| The following relations are satisfied by the generators:
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| :<math>\begin{align}
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| & \{ \Gamma,\Gamma \} =2I && \{ \Gamma, Q \} =0 && \{ \Gamma, \bar{Q} \} =0\\
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| &\{ Q,\bar{Q} \}=2Z && \{ Q, Q \}=2(H+P) && \{ \bar{Q}, \bar{Q} \} =2(H-P) \\
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| & [N,Q]=\frac{1}{2} Q && [N,\bar{Q} ]=-\frac{1}{2} \bar{Q} && [N-[1-q,\Gamma]=0 \\
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| & [N,H+P]=H+P && [N,H-P]=-(H-P) &&
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| \end{align}
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| </math>
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| <math>Z\,</math> is a [[center (algebra)|central]] element.
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| The supersymmetry algebra admits a <math>\mathbb{Z}_2</math>[[graded algebra|-grading]]. The generators <math>H, P, N, Z, I\,
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| </math> are even (degree 0), the generators <math>Q, \bar{Q}, \Gamma\,</math> are odd (degree 1).
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| 2(''H'' − ''P'') gives the left-moving momentum and 2(''H'' + ''P'') the right-moving momentum.
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| Basic [[representation theory|representation]]s of this algebra are the '''vacuum''', '''kink''' and '''boson-fermion''' representations, which are relevant e.g. to the supersymmetric (quantum) [[sine-Gordon]] model.
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| ==References==
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| * K. Schoutens, Supersymmetry and factorized scattering, Nucl.Phys. B344, 665–695, 1990
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| *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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| {{DEFAULTSORT:N = 1 Supersymmetry Algebra In 1 + 1 Dimensions}}
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| [[Category:Supersymmetry]]
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| [[Category:Mathematical physics]]
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| [[Category:Lie algebras]]
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