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

TL;DR

This paper explores the connection between the master field in large N matrix models and quantum groups, showing that the master field algebra is isomorphic to functions on the quantum group SU_q(2) at q=0, linking non-commutative probability with quantum group theory.

Contribution

It establishes a novel isomorphism between the master field algebra and the algebra of functions on SU_q(2) at q=0, revealing a deep connection between large N matrix models and quantum groups.

Findings

Master field algebra is isomorphic to functions on SU_q(2) at q=0.

The master field is a central element in the quantum group Hopf algebra.

Quantum Haar measure yields the Wigner semicircle distribution for the master field.

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 $SU_q(2)$ for $q=0$. The master field becomes a central element of the quantum group Hopf algebra. The quantum Haar measure on the $SU_q(2)$ for any $q$ gives the Wigner semicircle distribution for the master field. Coherent states on $SU_q(2)$ become coherent states in the master field theory.