Relating 't Hooft Anomalies of 4d Pure Yang-Mills and 2d ${\mathbb{C}\mathbb{P}}^{N-1}$ Model

It has recently been shown that a center-twisted compactification of the four-dimensional pure $SU(N)$ Yang-Mills theory on a three-torus gives rise to the two-dimensional ${\mathbb{C}\mathbb{P}}^{N-1}$-model on a circle with a flavor-twisted boundary condition. We verify the consistency of this statement non-perturbatively at theta angle $\theta =\pi$, in terms of the mixed 't Hooft anomalies for flavor symmetries and the time-reversal symmetry. This provides further support for the approach to the confinement of four-dimensional Yang-Mills theory from the two-dimensional ${\mathbb{C}\mathbb{P}}^{N-1}$-model.