Fractional instantons in 2d CPN-1 model and 4d Yang-Mills theory with 't Hooft twists

We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the CPN-1 nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the CPN-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 CPNk+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 CPN-1 limit, while the other makes the modular invariance manifest. We also discuss the implications of our finding for the $4$d $SU(N)$ Yang-Mills theory on the $4$-torus with 't~Hooft twists. By tuning the aspect ratio of the 4-torus, fractional instantons in the CPN-1 model with a non-Fubini-Study metric are obtained through the dimensional reduction of $4$d Yang-Mills theory, whose moduli space coincides with the one obtained for the standard CPN-1 model as complex manifolds.