The Master Field for Large $N$ Matrix Models and Quantum Groups

Abstract

In recent works by Singer, Douglas and Gopakumar and Gross an application of results of Voiculescu from non-commutative probability theory to constructions of the master field for large $N$ matrix field theories have been suggested. In this note we consider interrelations between the master field and quantum groups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum group $S{U}_q(2)$ for $q=0$. The master field becomes a central element of the quantum group Hopf algebra. The quantum Haar measure on the $S{U}_q(2)$ for any $q$ gives the Wigner semicircle distribution for the master field. Coherent states on $S{U}_q(2)$ become coherent states in the master field theory.

10 pages, LaTex

References in corpus (4)

• Mastering the Master Field
• Free Variables and the Two Matrix Model
• Non--Commutative (Quantum) Probability, Master Fields and Stochastic Bosonization
• The Master Field for Rainbow Diagrams and Free Non-Commutative Random Variables

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• Free coherent states and $p$-adic numbers
• Equivalence of $q$-bosons using the exponential phase operator
• The Master Field For 2D QCD On The Sphere
• Noncommutative probability, matrix models, and quantum orbifold geometry
• The space of free coherent states is isomorphic to the space of distributions on p-adic numbers
• Ward-Takahashi Identities in Large N Field Theories
• Dynamical systems where time is a quantum group and quantum ergodicity
• The Master Field for the Half-Planar Approximation for Large $N$ Matrix Models and Boltzmann Field Theory