# Gauge Theory on Compact Surfaces

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*Ambar Sengupta*

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.

#### Table of Contents

# Table of Contents

## Gauge Theory on Compact Surfaces

- Table of Contents vii8 free
- 1. Introduction 110 free
- 2. Terminology and Basic Facts 514 free
- 3. The Structure of Bundles and Connections over Compact Surfaces 817
- 4. Quantum Gauge Theory on the Disk 2029
- 5. A Conditional Probability Mcasiuc 2938
- 6. The Yang-Mills Measure 4049
- 7. Invariants of Systems of Curves 4453
- 8. Loop Expectation Values I 4857
- 9. Some Tools for the Abelian Case 5968
- 10. Loop Expectation Values II 6372
- Appendix 6877
- Figures 1,2,3 8190
- References 8493