Supersymmetry in the space-time R×S³

Abstract

Global N=1 supersymmetry is implemented on the surface of a four-dimensional sphere, in order to obtain a rotationally invariant and infrared-regulated version of such theories. We find that only R-invariant theories have supersymmetric actions in this space, and that the R-operator explicitly appears in the supersymmetry algebra (although the flat space superalgebra is recovered in the limit of large radius). Going to R×S³ therefore provides a rationale for only considering R-invariant theories, as is usually done for phenomenological reasons in flat space. The vacuum structure is analyzed at the tree level in various theories, and in most cases, it turns out to be simpler than in flat space. Supersymmetry breaking is considered, and it is found necessary to redefine Witten's index. This is due to the unusual superalgebra in R×S³, and its consequence, that the number of bosonic and fermionic states are not equal at any energy level. Finally, superspace is defined, and the scalar and vector multiplets are described in terms of chiral superfields. The supersymmetric transformations and lagrangian can be reexpressed in this language.