**Memoirs of the American Mathematical Society**

2003;
122 pp;
Softcover

MSC: Primary 58; 81; 60;

Print ISBN: 978-0-8218-3429-9

Product Code: MEMO/166/790

List Price: $62.00

Individual Member Price: $37.20

**Electronic ISBN: 978-1-4704-0388-1
Product Code: MEMO/166/790.E**

List Price: $62.00

Individual Member Price: $37.20

# Yang-Mills Measure on Compact Surfaces

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*Thierry Lévy*

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.

#### Table of Contents

# Table of Contents

## Yang-Mills Measure on Compact Surfaces

- Contents v6 free
- Introduction vii8 free
- Chapter 1. Discrete Yang-Mills measure 116 free
- 1.1. Notation 116
- 1.2. Discretization of a surface 318
- 1.3. Discrete holonomy and gauge transformations 520
- 1.4. Discrete Yang-Mills measure 722
- 1.5. Conditional Yang-Mills measure 823
- 1.6. Invariance under subdivision 1328
- 1.7. Invariance under area-preserving diffeomorphisms 1631
- 1.8. Two examples 1732
- 1.9. Discrete Abelian theory 2035

- Chapter 2. Continuous Yang-Mills measure 3550
- 2.1. Projective systems of probability spaces 3550
- 2.2. A Riemannian metric 3752
- 2.3. Piecewise geodesic Yang-Mills measure 3954
- 2.4. Lassos and small piecewise geodesic loops 4156
- 2.5. The space of paths 4560
- 2.6. Definition of the random holonomy 5166
- 2.7. Consistency with the discrete theory 5772
- 2.8. Binding up the conditional versions 6176
- 2.9. Surfaces with boundary 6277
- 2.10. The random holonomy process 6580

- Chapter 3. Abelian gauge theory 7590
- Chapter 4. Small scale structure in the semi-simple case 85100
- Chapter 5. Surgery of the Yang-Mills measure 99114
- Bibliography 121136

#### Readership

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