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 ${\displaystyle \mathbb {Z} _{2}}$-graded Lie superalgebra. The most common ways to do this are discussed below.

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

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

• ${\displaystyle H=P_{0}}$ is the generator of the time translation,
• ${\displaystyle P=P_{1}}$ is the generator of the space translation,
• ${\displaystyle M=M_{01}}$ is the generator of Lorentz boosts.

For the commutators between these generators, see Poincaré algebra.

The ${\displaystyle {\mathcal {N}}=(2,2)}$ supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) ${\displaystyle Q_{+},\,Q_{-},\,{\overline {Q}}_{+},\,{\overline {Q}}_{-}}$, which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators ${\displaystyle Q_{+}}$ and ${\displaystyle {\overline {Q}}_{+}}$ transform as left-handed Weyl spinors, while ${\displaystyle Q_{-}}$ and ${\displaystyle {\overline {Q}}_{-}}$ transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus[1]^:283

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

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

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

The ${\displaystyle {\mathcal {N}}=(0,2)}$ and ${\displaystyle {\mathcal {N}}=(2,0)}$ subalgebras

The ${\displaystyle {\mathcal {N}}=(0,2)}$ subalgebra is obtained from the ${\displaystyle {\mathcal {N}}=(2,2)}$ algebra by removing the generators ${\displaystyle Q_{-}}$ and ${\displaystyle {\overline {Q}}_{-}}$. Thus its anti-commutation relations are given by[1]^:289

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

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

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

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

Both supercharge generators are right-handed.

The ${\displaystyle {\mathcal {N}}=(1,1)}$ subalgebra

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

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

By definition, both supercharges are real, i.e. ${\displaystyle (Q_{\pm }^{1})^{\dagger }=Q_{\pm }^{1}}$. They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by[1]^:287

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

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

The ${\displaystyle {\mathcal {N}}=(0,1)}$ and ${\displaystyle {\mathcal {N}}=(1,0)}$ subalgebras

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

1. Mirror symmetry. Hori, Kentaro. Providence, RI: American Mathematical Society. 2003. ISBN 9780821829554. OCLC 52374327.CS1 maint: others (link)