Fractional instantons in 2d $\mathbb{C}P^{N-1}$ model and 4d Yang-Mills theory with 't Hooft twists

We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the $\mathbb{C}P^{N-1}$ nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the $\mathbb{C}P^{N-1}$ 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 $k+p/N\in \mathbb{Z}+p/N$. The moduli space is globally determined as the $\mathbb{C}P^{Nk+p-1}$-fiber bundle on a $2$-torus, which is a K\"ahler 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 $\mathbb{C}P^{N-1}$ 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 $\mathbb{C}P^{N-1}$ 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 $\mathbb{C}P^{N-1}$ model as complex manifolds.