# Gauge theory on compact surfaces

### About this Title

**Ambar Sengupta**

Publication: Memoirs of the American Mathematical Society

Publication Year
1997: Volume 126, Number 600

ISBNs: 978-0-8218-0484-1 (print); 978-1-4704-0185-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0600

MathSciNet review: 1346931

MSC (1991): Primary 58D20; Secondary 81S20, 81T13

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This work presents a rigorous account of quantum gauge field theory for bundles (both trivial and non-trivial) over compact surfaces. The Euclidean quantum field measure describing this theory is constructed and loop expectation values for a broad class of Wilson loop configurations are computed explicitly. Both the topology of the surface and the topology of the bundle are encoded in these loop expectation values. The effect of well-behaved area-preserving homeomorphisms of the surface is to take these loop expectation values into those for the pullback bundle. The quantum gauge field measure is constructed by conditioning an infinite-dimensional Gaussian measure to satisfy constraints imposed by the topologies of the surface and of the bundle. Holonomies, in this setting, are defined by interpreting the usual parallel-transport equation as a stochastic differential equation.

Readership

Graduate students, research mathematicians and physicists interested in quantum field theory, gauge theory or stochastic geometry.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. Terminology and basic facts
- 3. The structure of bundles and connections over compact surfaces
- 4. Quantum gauge theory on the disk
- 5. A conditional probability measure
- 6. The Yang-Mills measure
- 7. Invariants of systems of curves
- 8. Loop expectation values I
- 9. Some tools for the abelian case
- 10. Loop expectation values II
- Appendix