Strong CP as an Infrared Holonomy: The $\theta$ Vacuum and Dressing in Yang-Mills Theory

We reformulate the strong $CP$ problem from an infrared viewpoint in which the vacuum angle $\theta$ is not treated as a local coupling but as a global Berry-type holonomy of the infrared-dressed state space over $\mathcal{A}\mathrm{/}\mathcal{G}$. Infrared dressing is described as adiabatic parallel transport of physical states in configuration space, generated by an infrared connection ${\mathcal{A}}_{\mathrm{I}\mathrm{R}}$. Using the Chern-Simons collective coordinate, we show that the Pontryagin index emerges as an integer infrared winding, such that the resulting holonomy phase is quantized by $Q\in \mathbb{Z}$ and reproduces the standard weight $e^{i\theta Q}$. A quantum rotor provides a controlled infrared example illustrating why broad classes of local correlators may remain insensitive to $\theta$, while global response functions, such as the vacuum energy curvature and the topological susceptibility, retain a nontrivial dependence. We contrast this picture with recent claims of $\theta$–independence based on the order of limits and show that it is consistent with both the rotor benchmark and the classic Witten-Veneziano perspective.