Supersymmetry algebras in 1 + 1 dimensions

Let the Lie algebra of IO(1,1) be generated by the following generators:

• $H\; =\; P\_0$

is the generator of the time translation,

• $P\; =\; P\_1$

is the generator of the space translation,

• $M\; =\; M\_\{01\}$ is the generator of Lorentz boosts.

For the commutators between these generators, see Poincaré algebra.

The $\backslash mathcal\{N\}=(2,2)$ supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) $Q\_+,\; \backslash ,\; Q\_-,\; \backslash ,\; \backslash overline\{Q\}\_+,\; \backslash ,\; \backslash overline\{Q\}\_-$, which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators $Q\_+$ and $\backslash overline\{Q\}\_+$ transform as left-handed Weyl spinors, while $Q\_-$ and $\backslash overline\{Q\}\_-$ transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus{{Cite book|title=Mirror symmetry|date=2003|publisher=American Mathematical Society|others=Hori, Kentaro.|isbn=9780821829554|location=Providence, RI|oclc=52374327}}{{Rp|283}}

$\backslash begin\{align\}$

&\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 $\backslash tilde\{Z\}$

are complex central charges. The supercharges are related via $Q\_\{\backslash pm\}^\backslash dagger\; =\; \backslash overline\{Q\}\_\backslash pm$

. $H$

, $P$

, and $M$ are Hermitian.

The $\backslash mathcal\{N\}\; =\; (0,2)$ subalgebra is obtained from the $\backslash mathcal\{N\}\; =\; (2,2)$

algebra by removing the generators $Q\_-$ and $\backslash overline\{Q\}\_-$. Thus its anti-commutation relations are given by{{Rp|289}}

\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 $\backslash overline\{Q\}\_-$. Both generators are left-handed Weyl spinors.

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

\begin{align}

&Q_-^2 = \overline{Q}_-^2 = 0, \\

&\{ Q_{-}, \overline{Q}_{-} \} = H - P. \\

\end{align}

Both supercharge generators are right-handed.

The $\backslash mathcal\{N\}\; =\; (1,1)$ 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 $\backslash nu\_+$and $\backslash nu\_-$.

By definition, both supercharges are real, i.e. $(Q\_\{\backslash pm\}^1)^\backslash dagger\; =\; Q^1\_\backslash pm$

. They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by{{Rp|287}}

\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.

These algebras can be obtained from the $\backslash mathcal\{N\}\; =\; (1,1)$ subalgebra by removing $Q\_-^1$ resp. $Q\_+^1$from the generators.

• Supersymmetry
• Super-Poincaré algebra (in 1+3 dimensions)

• 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

Category:Supersymmetry

Category:Mathematical physics

Category:Lie algebras