Emergent Phase Space Description of Unitary Matrix Model

TL;DR

This paper introduces a geometric droplet-based phase space description of large N phases in unitary matrix models, connecting droplet topology with phase transitions and Young diagram representations.

Contribution

It establishes a novel phase space framework linking droplet geometries to large N phases and effective quantum mechanics in unitary matrix models.

Findings

Droplet geometries encode phase information of UMM.

Large N phase transitions correspond to droplet dynamics.

Explicit analysis of Chern-Simons theory phases.

Abstract

We show that large $N$ phases of a $0$ dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ($(0 + 1)$ dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large $N$. Therefore, large $N$ phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to $U(N)$ representations and hence different large $N$ states of the theory in momentum space. We explicitly consider two examples : single plaquette model with $\text{Tr} U^2$ terms and Chern-Simons theory on $S^3$. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.