ON A ONE-PARAMETER FAMILY OF EXOTIC SUPERSPACES IN TWO DIMENSIONS

Using Jordan algebraic techniques we define and study a family of exotic superspaces in two dimensions with two bosonic and two fermionic coordinates. They are defined by the one-parameter family of Jordan superalgebras JD(2/2)_α. For two special values of α the JD(2/2)_α can be realized in terms of a single fermionic or a single bosonic oscillator, respectively. For other values of α it can be interpreted as defining an exotic oscillator algebra. The derivation, reduced structure and Möbius superalgebras of JD(2/2)_α are identified with the rotation, Lorentz and finite-dimensional conformal superalgebras of the corresponding superspaces. The conformal superalgebras turn out to be the superalgebras D(2,1;α) with the even subgroup SO(2,2)×SU(2). We give an explicit differential operator realization of the actions of D(2,1;α) on these superspaces.