The master field on the plane

Thierry Lévy (Université Pierre et Marie Curie)
Wednesday, May 16, 2012 - 4:15pm
TU Berlin, MA041 Strasse des 17. Juni 136, 10623 Berlin

The Yang-Mills field on the plane is a collection of random matrices indexed by the set of loops based at the origin on the plane. These matrices belong to a fixed compact group, for example the unitary group U(N), and the Yang-Mills field can be thought of as a random unitary representation of the group of reduced rectifiable loops, the group operation being concatenation. I will describe the large N limit of this random reprensentation, in particular the fact that it converges almost surely towards a deterministic limit. This limit, which is one instance of what physicists call the master field, takes the form of a plain deterministic real-valued function on the set of loops. This function can be computed by a recursive algorithm based on the Makeenko-Migdal equations, which are a graphical expression in terms of the geometry of loops of the algebraic structure of freeness.