User:EduardoW/Supersymmetry algebra in 1 + 1 dimensions

A two dimensional Minkowski space, i.e. a flat space with one time and one spacial dimension, has a the 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 $$\mathbb{Z}_2$$-graded Lie superalgebra. The most common ways to do this are discussed below.

== $$\mathcal{N} = (2,2) $$ algebra == Let the Lie algebra of IO(1,1) be generated by the following generators: $$ is the generator of the time translation, $$ is the generator of the space translation, For the commutators between these generators, see Poincaré algebra.
 * $$H = P_0
 * $$P = P_1
 * $$M = M_{01}$$ is the generator of Lorentz boosts.

The $$\mathcal{N}=(2,2)$$ supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) $$Q_+, \, Q_-, \, \overline{Q}_+, \, \overline{Q}_-$$, which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators $$Q_+$$ and $$\overline{Q}_+$$ transform as left-handed Weyl spinors, while $$Q_-$$ and $$\overline{Q}_-$$ transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus

$$\begin{align} &\begin{align} &Q_+^2 = Q_{-}^2 = \overline{Q}_+^2 = \overline{Q}_-^2 =0, \\ &\{ Q_{\pm}, \overline{Q}_{\pm} \} = H \pm P, \\ \end{align} \\ &\begin{align} &\{\overline{Q}_+, \overline{Q}_- \} = Z, && \{Q_+, Q_- \} = Z^*, \\ &\{Q_-, \overline{Q}_+ \} =\tilde{Z}, && \{Q_+, \overline{Q}_-\} = \tilde{Z}^*,\\ &{[iM, Q_{\pm}]} = \mp Q_{\pm}, && {[iM, \overline{Q}_{\pm}]} = \mp \overline{Q}_{\pm}, \end{align} \end{align} $$

where all remaining commutators vanish, and $$Z $$ and $$\tilde{Z} $$ are complex central charges. The supercharges are related via $$Q_{\pm}^\dagger = \overline{Q}_\pm $$. $$H $$, $$P $$ and $$M$$ are Hermitian.

== Subalgebras of the $$\mathcal{N} = (2,2) $$ algebra ==

The $$\mathcal{N} = (0,2)$$ and $$\mathcal{N} = (2,0)$$ subalgebras
The $$\mathcal{N} = (0,2)$$ subalgebra is obtained from the $$\mathcal{N} = (2,2) $$ algebra by removing the generators $$Q_-$$ and $$\overline{Q}_-$$. Thus its anti-commutation relations are given by

$$ \begin{align} &Q_+^2 = \overline{Q}_+^2 = 0, \\ &\{ Q_{+}, \overline{Q}_{+} \} = H + P \\ \end{align} $$

plus the commutation relations above that do not involve $$Q_-$$ or $$\overline{Q}_-$$. Both generators are left-handed Weyl spinors.

Similarly, the $$\mathcal{N} = (2,0)$$ subalgebra is obtained by removing $$Q_+$$ and $$\overline{Q}_+$$ and fulfills

$$ \begin{align} &Q_-^2 = \overline{Q}_-^2 = 0, \\ &\{ Q_{-}, \overline{Q}_{-} \} = H - P. \\ \end{align} $$

Both supercharge generators are right-handed.

The $$\mathcal{N} = (1,1)$$ subalgebra
The $$\mathcal{N} = (1,1)$$ subalgebra is generated by two generators $$Q_+^1$$ and $$Q_-^1$$ given by

$$ \begin{align} Q^1_{\pm} = e^{i \nu_{\pm}} Q_{\pm} + e^{-i \nu_{\pm}} \overline{Q}_{\pm} \end{align} $$for two real numbers $$\nu_+$$and $$\nu_-$$.

By definition, both supercharges are real, i.e. $$(Q_{\pm}^1)^\dagger = Q^1_\pm $$. They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by

$$ \begin{align} &\{ Q^1_{\pm}, Q^1_{\pm} \} = 2 (H \pm P), \\ &\{ Q^1_{+}, Q^1_{-} \} = Z^1, \end{align} $$

where $$Z^1$$ is a real central charge.

The $$\mathcal{N} = (0,1)$$ and $$\mathcal{N} = (1,0)$$ subalgebras
These algebras can be obtained from the $$\mathcal{N} = (1,1)$$ subalgebra by removing $$Q_-^1$$ resp. $$Q_+^1$$from the generators.