The single-state supermultiplet in (1 + 1) dimensions

We consider multiplet shortening for Bogomol'nyi-Prasad-Sommerfield solitons in N = 1 two-dimensional models. Examples of single-state multiplets were established previously in N = 1 Landau-Ginzburg models. The shortening comes at the price of losing the fermion parity (-1)^F due to boundary effects. This implies the disappearance of the boson-fermion classification resulting in abnormal statistics. We discuss an appropriate index that counts such short multiplets. A broad class of hybrid models which extend the Landau-Ginzburg models to include a nonflat metric on the target space are considered. Our index turns out to be related to the index of the Dirac operator on the soliton reduced moduli space (the moduli space is reduced by factoring out the translational modulus). The index vanishes in most cases, implying the absence of shortening. In particular, it vanishes when there are only two critical points on the compact target space and the reduced moduli space has nonvanishing dimension. We also generalize the anomaly in the central charge to take into account the target space metric.