From Self-Dual to Physical $\mathbb{C}{P}^{N-1}$: Anomalies, Boundary Stokes Phenomenon, and Global Structure of $\theta$-vacua

Abstract

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 $ϵ\ne g$, this model possesses a chiral imbalance, which may be viewed as a real topological deformation (imaginary-$\theta$). 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 $(\theta +2\pi n)\sim O(N)$ ($n$ 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.

Comments: 64 pages, 10 figures