The Master Field for Large N Matrix Models and Quantum Groups

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 SUq(2) for q = 0. The master field becomes a central element of the quantum group Hopf algebra. The quantum Haar measure on the SUq(2) for any q gives the Wigner semicircle distribution for the master field. Coherent states on SUq(2) become coherent states in the master field theory. Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow, e-mail: [email protected] Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow, On leave absence from Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow