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 Z 2 {\displaystyle \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 {\displaystyle H=P_{0}} is the generator of the time translation,
P = P 1 {\displaystyle P=P_{1}} is the generator of the space translation,
M = M 01 {\displaystyle M=M_{01}} is the generator of Lorentz boosts. For the commutators between these generators, see Poincaré algebra. The N = ( 2 , 2 ) {\displaystyle {\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 − , Q ¯ + , Q ¯ − {\displaystyle Q_{+},\,Q_{-},\,{\overline {Q}}_{+},\,{\overline {Q}}_{-}} , which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators Q + {\displaystyle Q_{+}} and Q ¯ + {\displaystyle {\overline {Q}}_{+}} transform as left-handed Weyl spinors, while Q − {\displaystyle Q_{-}} and Q ¯ − {\displaystyle {\overline {Q}}_{-}} transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus
Q + 2 = Q − 2 = Q ¯ + 2 = Q ¯ − 2 = 0 , { Q ± , Q ¯ ± } = H ± P , { Q ¯ + , Q ¯ − } = Z , { Q + , Q − } = Z ∗ , { Q − , Q ¯ + } = Z ~ , { Q + , Q ¯ − } = Z ~ ∗ , [ i M , Q ± ] = ∓ Q ± , [ i M , Q ¯ ± ] = ∓ Q ¯ ± , {\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 Z {\displaystyle Z} and Z ~ {\displaystyle {\tilde {Z}}} are complex central charges. The supercharges are related via Q ± † = Q ¯ ± {\displaystyle Q_{\pm }^{\dagger }={\overline {Q}}_{\pm }} . H {\displaystyle H} , P {\displaystyle P} , and M {\displaystyle M} are Hermitian.
Subalgebras of the N=(2,2) algebra
The N=(0,2) and N=(2,0) subalgebras The N = ( 0 , 2 ) {\displaystyle {\mathcal {N}}=(0,2)} subalgebra is obtained from the N = ( 2 , 2 ) {\displaystyle {\mathcal {N}}=(2,2)} algebra by removing the generators Q − {\displaystyle Q_{-}} and Q ¯ − {\displaystyle {\overline {Q}}_{-}} . Thus its anti-commutation relations are given by
Q + 2 = Q ¯ + 2 = 0 , { Q + , Q ¯ + } = H + P {\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 Q − {\displaystyle Q_{-}} or Q ¯ − {\displaystyle {\overline {Q}}_{-}} . Both generators are left-handed Weyl spinors. Similarly, the N = ( 2 , 0 ) {\displaystyle {\mathcal {N}}=(2,0)} subalgebra is obtained by removing Q + {\displaystyle Q_{+}} and Q ¯ + {\displaystyle {\overline {Q}}_{+}} and fulfills
Q − 2 = Q ¯ − 2 = 0 , { Q − , Q ¯ − } = H − P . {\displaystyle {\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 N = ( 1 , 1 ) {\displaystyle {\mathcal {N}}=(1,1)} subalgebra is generated by two generators Q + 1 {\displaystyle Q_{+}^{1}} and Q − 1 {\displaystyle Q_{-}^{1}} given by
Q ± 1 = e i ν ± Q ± + e − i ν ± Q ¯ ± {\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 ν + {\displaystyle \nu _{+}} and ν − {\displaystyle \nu _{-}} . By definition, both supercharges are real, i.e. ( Q ± 1 ) † = Q ± 1 {\displaystyle (Q_{\pm }^{1})^{\dagger }=Q_{\pm }^{1}} . They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by
{ Q ± 1 , Q ± 1 } = 2 ( H ± P ) , { Q + 1 , Q − 1 } = Z 1 , {\displaystyle {\begin{aligned}&\{Q_{\pm }^{1},Q_{\pm }^{1}\}=2(H\pm P),\\&\{Q_{+}^{1},Q_{-}^{1}\}=Z^{1},\end{aligned}}}
where Z 1 {\displaystyle Z^{1}} is a real central charge.
The N=(0,1) and N=(1,0) subalgebras These algebras can be obtained from the N = ( 1 , 1 ) {\displaystyle {\mathcal {N}}=(1,1)} subalgebra by removing Q − 1 {\displaystyle Q_{-}^{1}} resp. Q + 1 {\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