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We report the results of the lattice simulation of the ${\mathbb C} P^{N-1}$sigma model on $S_{s}^{1}$(large) $\times$ $S_{\tau}^{1}$(small). We take asufficiently large ratio of the circumferences to approximate the model on${\mathbb R} \times S^1$. For periodic boundary condition imposed in the$S_{\tau}^{1}$ direction, we show that the expectation value of the Polyakovloop undergoes a deconfinement crossover as the compactified circumference isdecreased, where the peak of the associated susceptibility gets sharper forlarger $N$. For ${\mathbb Z}_{N}$ twisted boundary condition, we find that,even at relatively high $\beta$ (small circumference), the regular $N$-sidedpolygon-shaped distributions of Polyakov loop leads to small expectation valuesof Polyakov loop, which implies unbroken ${\mathbb Z}_{N}$ symmetry ifsufficient statistics and large volumes are adopted. We also argue theexistence of fractional instantons and bions by investigating the dependence ofthe Polyakov loop on $S_{s}^{1}$ direction, which causes transition between${\mathbb Z}_{N}$ vacua.

Lattice study on the twisted ${\mathbb C}P^{N-1}$ models on ${\mathbb R}\times S^{1}$