Supersymmetry algebras in 1 + 1 dimensions explained

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

algebra

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

H = P _{0} is the generator of the time translation,P = P _{1} is the generator of the space translation,M = M _{01} is the generator of Lorentz boosts. For the commutators between these generators, see Poincaré algebra.

The

l{N}=(2,2) supersymmetry algebra over this space is a supersymmetric extension of this Lie algebra with the four additional generators (supercharges) Q _{+,} Q _{-,} \overline{Q} _{+,} \overline{Q} _{-} , which are odd elements of the Lie superalgebra. Under Lorentz transformations the generators Q _{+} and \overline{Q} _{+} transform as left-handed Weyl spinors, while Q _{-} and \overline{Q} _{-} transform as right-handed Weyl spinors. The algebra is given by the Poincaré algebra plus^[1] \begin{align} &\begin{align} &Q ^{2} _{+} = Q ^{2} _{-} = \overline{Q} ^{2} _{+} = \overline{Q} ^{2} _{-} =0 , \\ &\{ Q _{\pm} , \overline{Q} _{\pm} \} = H \pm P, \\ \end{align} \\ &\begin{align} &\{\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{align} \end{align}

where all remaining commutators vanish, and

Z and \tilde{Z} are complex central charges. The supercharges are related via Q ^{\dagger} _{\pm} = \overline{Q} _{\pm} . H , P , and M are Hermitian.

Subalgebras of the algebra

The and subalgebras

The

l{N} = ( 0 , 2 ) subalgebra is obtained from the l{N} = ( 2 , 2 ) algebra by removing the generators Q _{-} and \overline{Q} _{-} . Thus its anti-commutation relations are given by\begin{align} &Q ^{2} _{+} = \overline{Q} ^{2} _{+} = 0 , \\ &\{ Q _{+} , \overline{Q} _{+} \} = H + P \\ \end{align}

plus the commutation relations above that do not involve

Q _{-} or \overline{Q} _{-} . Both generators are left-handed Weyl spinors.

Similarly, the

l{N} = ( 2 , 0 ) subalgebra is obtained by removing Q _{+} and \overline{Q} _{+} and fulfills\begin{align} &Q ^{2} _{-} = \overline{Q} ^{2} _{-} = 0 , \\ &\{ Q _{-} , \overline{Q} _{-} \} = H - P. \\ \end{align}

Both supercharge generators are right-handed.

The subalgebra

The

l{N} = ( 1 , 1 ) subalgebra is generated by two generators Q ^{1} _{+} and Q ^{1} _{-} given by\begin{align} Q ^{1} _{\pm} = e ^{i \nu _{\pm}} Q _{\pm} + e ^{- i \nu _{\pm}} \overline{Q} _{\pm} \end{align} for two real numbers \nu _{+} and \nu _{-} .

By definition, both supercharges are real, i.e.

(Q ^{1 )} _{\pm} ^{\dagger} = Q ^{1} _{\pm} . They transform as Majorana-Weyl spinors under Lorentz transformations. Their anti-commutation relations are given by\begin{align} &\{ Q ^{1} _{\pm} , Q ^{1} _{\pm} \} = 2 (H \pm P), \\ &\{ Q ^{1} _{+} , Q ^{1} _{-} \} = Z ^{1, \end{align}}

where

Z ^{1} is a real central charge.

The and subalgebras

These algebras can be obtained from the

l{N} = ( 1 , 1 ) subalgebra by removing Q ^{1} _{-} resp. 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. Book: Mirror symmetry. 2003. American Mathematical Society. Hori, Kentaro.. 9780821829554. Providence, RI. 52374327.