From 4d Yang-Mills to 2d ℂℙ^{N − 1} model: IR problem and confinement at weak coupling

We study four-dimensional SU(N) Yang-Mills theory on ℝ×T3=ℝ×SA1×SB1×SC1, with a twisted boundary condition by a ℤ_N center symmetry imposed on S_B ¹ × S_C ¹. This setup has no IR zero modes and hence is free from IR divergences which could spoil trans-series expansion for physical observables. Moreover, we show that the center symmetry is preserved at weak coupling regime. This is shown by first reducing the theory on T²= S_A× S_B , to connect the model to the two-dimensional ℂℙ^{N− 1}-model. Then, we prove that the twisted boundary condition by the center symmetry for the Yang-Mills is reduced to the twisted boundary condition by the ℤ_N global symmetry of ℂℙ^{N− 1}. There are N classical vacua, and fractional instantons connecting those N vacua dynamically restore the center symmetry. We also point out the presence of singularities on the Borel plane which depend on the shape of the compactification manifold, and comment on its implications.