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.
N=(2,2) algebra[edit]
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[edit]
The N=(0,2) and N=(2,0) subalgebras[edit]
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[edit]
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[edit]
These algebras can be obtained from the
subalgebra by removing
resp.
from the generators.
See also[edit]
References[edit]
- 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