Emergent Phase Space Description of Unitary Matrix Model

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.

arXiv comment52 pages, 15 figures, v2 Introduction and discussions extended, References added

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Cited by in corpus (8)

• Chern-Simons Theory on Seifert Manifold and Matrix Model
• From Phase Space to Integrable Representations and Level-Rank Duality
• Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories
• Higher Spin Gravity in $Ad{S}_3$ and Folds on Fermi Surface
• New phase in Chern-Simons theory on lens space
• Toward the construction of the general multi-cut solutions in Chern-Simons Matrix Models
• Quantum Mechanics of Plancherel Growth
• A Unitary Matrix Model for $q$-deformed Plancherel Growth