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.

N=(2,2) 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 ${\mathcal {N}}=(2,2)$ 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

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

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 N=(2,2) algebra

The N=(0,2) and 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

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

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

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

Both supercharge generators are right-handed.

The N=(1,1) subalgebra

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

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

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

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

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

The N=(0,1) and 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.

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