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Résumé |
The two-dimensional Yang-Mills measure is one of the few examples of a functional integral issued from gauge theories which can be constructed in a mathematically rigorous manner. A. Migdal had rightly identified in 1975 that this construction should involve in an essential way the heat semigroup on the structure group. This naturally raised the interest of probabilists, who are particularly fond of this semigroup and the associated diffusion, the Brownian motion. In this talk, I will discuss two distinct threads which both lead from the two-dimensional Yang-Mills measure to combinatorial objects, namely random ramified coverings on surfaces. The first thread is a beautiful duality between the unitary groups and the symmetric groups, known as the Schur-Weyl duality, which in principle allows one to translate any computation on a unitary group into a computation on a symmetric group. In 1994, D. Gross and W. Taylor interpreted this duality as a duality between the two-dimensional Yang-Mills theory and a string theory, and they exhibited wonderful formulae for the expectation of Wilson loops, the simplest of which can be proved mathematically. The second thread is a reflection on the role of the semi-group property in the construction of the Yang-Mills measure, and an extension of this construction to diffusions on the structure group which are more general than the Brownian motion. This extension builds a correspondence between a class of random fields which share the essential properties of the Yang-Mills measure, in particular a two-dimensional version of the classical Markov property -- a class of diffusions on compact groups known as Lévy processes (Paul Lévy's 125th birthday was celebrated a few weeks ago)-- and a class of almost topological quantum field theories, which act as a kind of algebraic skeleton of the corresponding random fields. |
