Paris Marie Curie and visiting Newton Institute

Quantum Yang-Mills theory is an important part of the Standard model built
by physicists to describe elementary particles and their interactions. One
approach to this theory consists in constructing a probability measure on an
infinite-dimensional space of connections on a principal bundle over
space-time. However, in the physically realistic 4-dimensional situation,
the construction of this measure is still an open mathematical problem. The
subject of this talk will be the physically less realistic 2-dimensional
situation, in which the construction of the measure is possible, and fairly
well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the
distribution of a stochastic process with values in a compact Lie group (for

example the unitary group U(N)) indexed by the set of continuous closed
curves with finite length on a compact surface (for example a disk, a sphere
or a torus) on which one can measure areas. It can be seen as a Brownian
motion (or a Brownian bridge) on the chosen compact Lie group indexed by
closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills
measure is constructed, and describe it without assuming any prior
familiarity with the subject. I will then present a set of results obtained
in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel,
Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N
tends to infinity of the Yang-Mills measure constructed with the unitary
group U(N).