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Yang-Mills Measure on Compact Surfaces
Thierry Lévy, Université de Strasbourg, France
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Memoirs of the American Mathematical Society
2003; 122 pp; softcover
Volume: 166
ISBN-10: 0-8218-3429-0
ISBN-13: 978-0-8218-3429-9
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/166/790

In this memoir we present a new construction and new properties of the Yang-Mills measure in two dimensions.

This measure was first introduced for the needs of quantum field theory and can be described informally as a probability measure on the space of connections modulo gauge transformations on a principal bundle. We consider the case of a bundle over a compact orientable surface.

Our construction is based on the discrete Yang-Mills theory of which we give a full acount. We are able to take its continuum limit and to define a pathwise multiplicative process of random holonomy indexed by the class of piecewise embedded loops.

We study in detail the links between this process and a white noise and prove a result of asymptotic independence in the case of a semi-simple structure group. We also investigate global Markovian properties of the measure related to the surgery of surfaces.