We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the CP^N-1 nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the CP^N-1 model by an N-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial 't~Hooft fluxes p/N for the U (1) gauge field, and the topological charge becomes fractionalized as k+p/N Z+p/N. The moduli space is globally determined as the CP^Nk+p-1-fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension Nk + p as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the CP^N-1 limit, while the other makes the modular invariance manifest. We also discuss the implications of our finding for the 4d SU (N) Yang-Mills theory on the 4-torus with 't~Hooft twists. By tuning the aspect ratio of the 4-torus, fractional instantons in the CP^N-1 model with a non-Fubini-Study metric are obtained through the dimensional reduction of 4d Yang-Mills theory, whose moduli space coincides with the one obtained for the standard CP^N-1 model as complex manifolds.
Hayashi et al. (Thu,) studied this question.