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 -graded Lie superalgebra. The most common ways to do this are discussed below.
Let the Lie algebra of IO(1,1) be generated by the following generators:
- is the generator of the time translation,
- is the generator of the space translation,
- is the generator of Lorentz boosts.
For the commutators between these generators, see Poincaré algebra.
The supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) , which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators and transform as left-handed Weyl spinors, while and transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus[1]: 283
where all remaining commutators vanish, and and are complex central charges. The supercharges are related via . , , and are Hermitian.
Subalgebras of the N=(2,2) algebra
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The N=(0,2) and N=(2,0) subalgebras
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The subalgebra is obtained from the algebra by removing the generators and . Thus its anti-commutation relations are given by[1]: 289
plus the commutation relations above that do not involve or . Both generators are left-handed Weyl spinors.
Similarly, the subalgebra is obtained by removing and and fulfills
Both supercharge generators are right-handed.
The N=(1,1) subalgebra
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The subalgebra is generated by two generators and given by
for two real numbers and .
By definition, both supercharges are real, i.e. . They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by[1]: 287
where is a real central charge.
The N=(0,1) and N=(1,0) subalgebras
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These algebras can be obtained from the subalgebra by removing resp. from the generators.
- 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