Baum–Connes conjecture: Difference between revisions
Jump to navigation
Jump to search
en>Michael Hardy punctuation corrections required by WP:MOS |
|||
Line 1: | Line 1: | ||
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: | |||
:[[supercharge|supersymmetric charges]]: <math>Q, \bar{Q}</math> | |||
:supersymmetric central charge: <math>Z\,</math> | |||
:time [[translation (geometry)|translation]] generator: <math>H\,</math> | |||
:space translation generator: <math>P\,</math> | |||
:[[Lorentz boost|boost]] generator: <math>N\,</math> | |||
:[[(-1)F|fermionic parity]]: <math>\Gamma\,</math> | |||
:[[identity element|unit element]]: <math>I\,</math> | |||
The following relations are satisfied by the generators: | |||
:<math>\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} | |||
</math> | |||
<math>Z\,</math> is a [[center (algebra)|central]] element. | |||
The supersymmetry algebra admits a <math>\mathbb{Z}_2</math>[[graded algebra|-grading]]. The generators <math>H, P, N, Z, I\, | |||
</math> are even (degree 0), the generators <math>Q, \bar{Q}, \Gamma\,</math> are odd (degree 1). | |||
2(''H'' − ''P'') gives the left-moving momentum and 2(''H'' + ''P'') the right-moving momentum. | |||
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. | |||
==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 | |||
{{DEFAULTSORT:N = 1 Supersymmetry Algebra In 1 + 1 Dimensions}} | |||
[[Category:Supersymmetry]] | |||
[[Category:Mathematical physics]] | |||
[[Category:Lie algebras]] |
Revision as of 04:42, 2 July 2013
In 1 + 1 dimensions the N = 1 supersymmetry algebra (also known as because we have one left-moving SUSY generator and one right moving one) has the following generators:
- supersymmetric charges:
- supersymmetric central charge:
- time translation generator:
- space translation generator:
- boost generator:
- fermionic parity:
- unit element:
The following relations are satisfied by the generators:
is a central element.
The supersymmetry algebra admits a -grading. The generators 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 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