From Self-Dual to Physical $\mathbb{C}P^{N-1}$: Anomalies, Boundary Stokes Phenomenon, and Global Structure of $θ$-vacua
We introduce a two-coupling generalization of $\mathbb{C}P^{N-1}$ model that continuously interpolates between the self-dual ($ε=0$) and the physical ($ε=g$) theories as a useful nonperturbative tool. At $ε\neq g$, this model possesses a chiral imbalance, which may be viewed as a real topological deformation (imaginary-$θ$). We demonstrate that exact quantum equivalence between first- and second-order formulations strictly requires a topological counterterm sourced by a bosonic chiral anomaly. Solving this deformed theory at large $N$ yields two primary results. First, we analytically determine the nonperturbative vacuum structure of the self-dual theory, a self-dual vacuum with a dynamically generated field-strength condensate. Second, we resolve a fundamental paradox where saddles with $(θ+ 2πn) \sim O(N)$ ($n$ is branch number) spuriously yield lower energy densities than the physical ground state. Because the effective action possesses an essential singularity at $F=0$, we show that the Lefschetz thimble analysis must be generalized to include boundary thimbles. A boundary Stokes phenomenon renders the problematic saddles topologically inactive, fully restoring the validity of the large-$N$ expansion for strongly coupled theories.