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 ${\mathbb{Z}}_2$-graded Lie superalgebra. The most common ways to do this are discussed below.

Template:Nowrap algebra

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

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

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

Subalgebras of the Template:Nowrap algebra

The Template:Nowrap and Template:Nowrap subalgebras

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

$\begin{array}{cc}& Q_{+}^2={\bar{Q}}_{+}^2=0,\\ & \{Q_{+},{\bar{Q}}_{+}\}=H+P\\ \end{array}$

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

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

$\begin{array}{cc}& Q_{-}^2={\bar{Q}}_{-}^2=0,\\ & \{Q_{-},{\bar{Q}}_{-}\}=H-P.\\ \end{array}$

Both supercharge generators are right-handed.

The Template:Nowrap subalgebra

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

$\begin{array}{c}{Q}_{\pm }^1=e^{i{\nu }_{\pm }}{Q}_{\pm }+e^{-i{\nu }_{\pm }}{\bar{Q}}_{\pm }\end{array}$for two real numbers ${\nu }_{+}$and ${\nu }_{-}$.

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

$\begin{array}{cc}& \{Q_{\pm }^1,Q_{\pm }^1\}=2(H\pm P),\\ & \{Q_{+}^1,Q_{-}^1\}=Z^1,\end{array}$

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

The Template:Nowrap and Template:Nowrap subalgebras

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

See also

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

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