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Yang-Mills Measure on Compact Surfaces

Thierry Lévy Université de Strasbourg, Strasbourg, France
Available Formats:
Electronic ISBN: 978-1-4704-0388-1
Product Code: MEMO/166/790.E
List Price: $62.00 MAA Member Price:$55.80
AMS Member Price: $37.20 Click above image for expanded view Yang-Mills Measure on Compact Surfaces Thierry Lévy Université de Strasbourg, Strasbourg, France Available Formats:  Electronic ISBN: 978-1-4704-0388-1 Product Code: MEMO/166/790.E  List Price:$62.00 MAA Member Price: $55.80 AMS Member Price:$37.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1662003; 122 pp
MSC: Primary 58; 81; 60;

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.

Graduate students and research mathematicians interested in geometry, topology, and analysis.

• Chapters
• 1. Discrete Yang-Mills measure
• 2. Continuous Yang-Mills measure
• 3. Abelian gauge theory
• 4. Small scale structure in the semi-simple case
• 5. Surgery of the Yang-Mills measure
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Volume: 1662003; 122 pp
MSC: Primary 58; 81; 60;

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.