Memoirs of the American Mathematical Society 1997; 85 pp; softcover Volume: 126 ISBN10: 0821804847 ISBN13: 9780821804841 List Price: US$42 Individual Members: US$25.20 Institutional Members: US$33.60 Order Code: MEMO/126/600
 This work presents a rigorous account of quantum gauge field theory for bundles (both trivial and nontrivial) 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 wellbehaved areapreserving 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 infinitedimensional 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 paralleltransport 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  Introduction
 Terminology and basic facts
 The structure of bundles and connections over compact surfaces
 Quantum gauge theory on the disk
 A conditional probability measure
 The YangMills measure
 Invariants of systems of curves
 Loop expectation values I
 Some tools for the Abelian case
 Loop expectation values II
 Appendix
 Figures 1, 2, 3
 References
