Twisted CP(N-1) instanton projectors and the N-level quantum density matrix

Twisted classical solutions to the $\mathbb{C}P^{N-1}$ model play a key role in the analysis of such models on the spatially compactified cylinder $\mathbb{S}_L^1 \times {\mathbb{R}^1}$ and have recently been shown to be important for the resurgent structure of this quantum field theory. Instantons and non-self-dual solutions both fractionalize, and domain walls formed by such topological solutions can be associated with $N$-vacua having maximally repulsive energy eigenvalues. The purpose of this paper is to reinforce this view through the investigation of a number of parallels between the $\mathbb{C}P^{N-1}$ model and $N$-level quantum mechanical density matrices. Specifically, we demonstrate the existence of a time-evolution equation for the $\mathbb{C}P^{N-1}$ instanton projector analogous to the Liouville-von Neumann equation in the quantum mechanical formalism. The group theoretical analysis of density matrices and the $\mathbb{C}P^{N-1}$ model are also closely related. Finally, we explore the emergence of geometrical (Berry) phases in both systems and their interrelationship.