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

A bstract 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.