Single State Supermultiplet in 1+1 Dimensions

TL;DR

This paper investigates BPS soliton multiplet shortening in 1+1 dimensional N=1 models, revealing how boundary effects lead to single-state multiplets with abnormal statistics and proposing an index related to the Dirac operator on moduli space.

Contribution

It introduces an index for counting short multiplets in hybrid models with nonflat target space metrics, extending previous Landau-Ginzburg results and analyzing conditions for multiplet shortening.

Findings

Short multiplets lack fermion parity due to boundary effects.

The index relates to the Dirac operator on the reduced moduli space.

In many cases, the index vanishes, indicating no shortening.

Abstract

We consider multiplet shortening for BPS solitons in N=1 two-dimensional models. Examples of the single-state multiplets were established previously in N=1 Landau-Ginzburg models. The shortening comes at a price of loosing 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 is 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.