N ov 2 00 2 Supersymmetry on the Surface

1 Abstract By using symplectic Majorana spinors as Grassmann coordinates in a superspace associated with the supersymmetric extension of the isometry group on the spherical surface S 2 , it proves possible to formulate supersymmetric models on S 2 using superspace techniques. The formulation of gauge theories on a spherical surface [1] has provided insights into their properties. The kinetic term for all fields on this surface involves the operator L µν ≡ −x µ ∂ ν + x ν ∂ µ , which is the generator of the isometry group on the spherical surface. In formulating a supersymmetric extension of the isometry group on any surface of constant curvature, one must introduce a Fermionic operator Q which is the " square root " of the full angular momentum operator J µν , whose commutation relation is the same as that of L µν. Since J µν does not commute with Q (in contrast to the translation operator P µ in flat space [2]), closure of the algebra often requires introduction of further Bosonic operators that act as internal symmetry generators (which do not commute with Q) [3-5]. Supersymmetric models on several surfaces of constant curvature have been formulated using L µν directly in the kinetic term for component fields. In particular, supersymmetric models on S 2 and AdS 2 have been devised [6]. (This approach is distinct from supersymmet-ric models on a surface of constant curvature formulated by specializing the gravitational background field in a supergravity model; this approach has been used in [7].) The compo