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 -graded Lie superalgebra. The most common ways to do this are discussed below.

N=(2,2) algebra

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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.

See also

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References

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  • 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. ^ a b c Mirror symmetry. Hori, Kentaro. Providence, RI: American Mathematical Society. 2003. ISBN 9780821829554. OCLC 52374327.{{cite book}}: CS1 maint: others (link)