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 TrU2 terms and Chern-Simons theory on S3. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.
Highlights
A unitary matrix model (UMM) is a statistical ensemble of unitary matrices, defined by the partition function
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
We find that for a given UMM, it is possible to construct an effective unitary matrix quantum mechanics (UMQM) such that its phase space distributions match with droplets of UMM on different time slices at large N
Summary
2.1 Unitary matrix model and eigenvalue distributions at large N A UMM is a statistical ensemble of unitary matrices, defined by the partition function. The partition function in this basis is given by. The partition function can be thought of as describing a system of N particles interacting by a repulsive potential (Coulomb repulsion) − ln | sin(θi − θj)/2| and moving in a common. If we neglected the Coulomb repulsion, all eigenvalues would have sat at the minima of potential V (θ). In large N limit one defines a set of continuous variables i θ(x) = θi, where x = ∈ [0, 1] N and in terms of these continuous variables the effective action becomes,. Which captures information about distributions of eigenvalues between −π and π in large N limit. Different saddle points correspond to different eigenvalue distributions and different phases of the theory
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