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

Abstract

We consider multiplet shortening forBogomol'nyi-Prasad-Sommerfield solitons in \U0001d4a9 = 1 two-dimensional models. Examples of single-statemultiplets were established previously in \U0001d4a9 = 1Landau-Ginzburg models. The shortening comes at the price oflosing the fermion parity (-1)F due to boundary effects. Thisimplies the disappearance of the boson-fermion classificationresulting in abnormal statistics. We discuss an appropriateindex that counts such short multiplets.A broad class of hybrid models which extend the Landau-Ginzburgmodels to include a nonflat metric on the target space areconsidered. Our index turns out to be related to the index ofthe Dirac operator on the soliton reduced moduli space (themoduli space is reduced by factoring out the translationalmodulus). The index vanishes in most cases, implying the absenceof shortening. In particular, it vanishes when there are onlytwo critical points on the compact target space and the reducedmoduli space has nonvanishing dimension.We also generalize the anomaly in the central charge totake into account the target space metric.