Supersymmetry algebras in 1 + 1 dimensions
A two dimensional Minkowski space, i.e. a flat space with one time and one spatial dimension, has a two-dimensional Poincaré group IO(1,1) as its symmetry group. The respective Lie algebra is called the Poincaré algebra. It is possible to extend this algebra to a supersymmetry algebra, which is a [math]\displaystyle{ \mathbb{Z}_2 }[/math]-graded Lie superalgebra. The most common ways to do this are discussed below.
N=(2,2) algebra
Let the Lie algebra of IO(1,1) be generated by the following generators:
- [math]\displaystyle{ H = P_0 }[/math] is the generator of the time translation,
- [math]\displaystyle{ P = P_1 }[/math] is the generator of the space translation,
- [math]\displaystyle{ M = M_{01} }[/math] is the generator of Lorentz boosts.
For the commutators between these generators, see Poincaré algebra.
The [math]\displaystyle{ \mathcal{N}=(2,2) }[/math] supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) [math]\displaystyle{ Q_+, \, Q_-, \, \overline{Q}_+, \, \overline{Q}_- }[/math], which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators [math]\displaystyle{ Q_+ }[/math] and [math]\displaystyle{ \overline{Q}_+ }[/math] transform as left-handed Weyl spinors, while [math]\displaystyle{ Q_- }[/math] and [math]\displaystyle{ \overline{Q}_- }[/math] transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus[1]:283
[math]\displaystyle{ \begin{align} &\begin{align} &Q_+^2 = Q_{-}^2 = \overline{Q}_+^2 = \overline{Q}_-^2 =0, \\ &\{ Q_{\pm}, \overline{Q}_{\pm} \} = H \pm P, \\ \end{align} \\ &\begin{align} &\{\overline{Q}_+, \overline{Q}_- \} = Z, && \{Q_+, Q_- \} = Z^*, \\ &\{Q_-, \overline{Q}_+ \} =\tilde{Z}, && \{Q_+, \overline{Q}_-\} = \tilde{Z}^*,\\ &{[iM, Q_{\pm}]} = \mp Q_{\pm}, && {[iM, \overline{Q}_{\pm}]} = \mp \overline{Q}_{\pm}, \end{align} \end{align} }[/math]
where all remaining commutators vanish, and [math]\displaystyle{ Z }[/math] and [math]\displaystyle{ \tilde{Z} }[/math] are complex central charges. The supercharges are related via [math]\displaystyle{ Q_{\pm}^\dagger = \overline{Q}_\pm }[/math]. [math]\displaystyle{ H }[/math], [math]\displaystyle{ P }[/math], and [math]\displaystyle{ M }[/math] are Hermitian.
Subalgebras of the N=(2,2) algebra
The N=(0,2) and N=(2,0) subalgebras
The [math]\displaystyle{ \mathcal{N} = (0,2) }[/math] subalgebra is obtained from the [math]\displaystyle{ \mathcal{N} = (2,2) }[/math] algebra by removing the generators [math]\displaystyle{ Q_- }[/math] and [math]\displaystyle{ \overline{Q}_- }[/math]. Thus its anti-commutation relations are given by[1]:289
[math]\displaystyle{ \begin{align} &Q_+^2 = \overline{Q}_+^2 = 0, \\ &\{ Q_{+}, \overline{Q}_{+} \} = H + P \\ \end{align} }[/math]
plus the commutation relations above that do not involve [math]\displaystyle{ Q_- }[/math] or [math]\displaystyle{ \overline{Q}_- }[/math]. Both generators are left-handed Weyl spinors.
Similarly, the [math]\displaystyle{ \mathcal{N} = (2,0) }[/math] subalgebra is obtained by removing [math]\displaystyle{ Q_+ }[/math] and [math]\displaystyle{ \overline{Q}_+ }[/math] and fulfills
[math]\displaystyle{ \begin{align} &Q_-^2 = \overline{Q}_-^2 = 0, \\ &\{ Q_{-}, \overline{Q}_{-} \} = H - P. \\ \end{align} }[/math]
Both supercharge generators are right-handed.
The N=(1,1) subalgebra
The [math]\displaystyle{ \mathcal{N} = (1,1) }[/math] subalgebra is generated by two generators [math]\displaystyle{ Q_+^1 }[/math] and [math]\displaystyle{ Q_-^1 }[/math] given by
[math]\displaystyle{ \begin{align} Q^1_{\pm} = e^{i \nu_{\pm}} Q_{\pm} + e^{-i \nu_{\pm}} \overline{Q}_{\pm} \end{align} }[/math]for two real numbers [math]\displaystyle{ \nu_+ }[/math]and [math]\displaystyle{ \nu_- }[/math].
By definition, both supercharges are real, i.e. [math]\displaystyle{ (Q_{\pm}^1)^\dagger = Q^1_\pm }[/math]. They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by[1]:287
[math]\displaystyle{ \begin{align} &\{ Q^1_{\pm}, Q^1_{\pm} \} = 2 (H \pm P), \\ &\{ Q^1_{+}, Q^1_{-} \} = Z^1, \end{align} }[/math]
where [math]\displaystyle{ Z^1 }[/math] is a real central charge.
The N=(0,1) and N=(1,0) subalgebras
These algebras can be obtained from the [math]\displaystyle{ \mathcal{N} = (1,1) }[/math] subalgebra by removing [math]\displaystyle{ Q_-^1 }[/math] resp. [math]\displaystyle{ Q_+^1 }[/math]from the generators.
See also
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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