Yang-Mills measure on compact surfaces
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 2003: Volume 166, Number 790
ISBNs: 978-0-8218-3429-9 (print); 978-1-4704-0388-1 (online)
MathSciNet review: 2006374
MSC: Primary 58D20; Secondary 60F20, 60G60, 60H40, 81T13, 81T27
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.
Table of Contents
- 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