Topological quantum field theory and four manifolds.
The origin of topological quantum field theory may be seen in E. Witten's reformulation of the Donaldson invariants for 4-manifolds in the 1980's which enormously shortened a complicated but fascinating theory and proved dramatically more elegant. Why concentrate on 4-manifolds? Because most of the previous techniques used in five and higher dimensions fail in four. Since this breakthrough, topological quantum field theory has led to many striking new results in mathematics (rather than in physics as many of us would have hoped). Today the main applications are seen within the context of closed, compact and orientable Riemannian 4-manifolds, their topology and classification up to diffeomorphisms. The present book therefore starts with a survey of important topological topics, then reviews the theories of Donaldson and Seiberg-Witten, and describes various aspects of supersymmetry in general and twisted \({\mathcal N}=2\) supersymmetric models in particular. As an interlude, a quite different approach, the Mathai-Quillen formalism, is introduced to show that there are alternative ways of constructing topological quantum fields relevant to the present goal. Another chapter deals with the supersymmetric Yang-Mills model and its exact solution by Seiberg and Witten. Also, the so-called \(u\)-plane integral, analyzed in detail by Moore and Witten, is discussed and its importance as a systematic tool emphasized. Graduate students, post-docs and junior faculty interested in the interaction of physics and mathematics, if not intimidated by the use of demanding mathematical tools, will greatly benefit from this coherent treatment of the subject and the thorough evaluation of its virtues which is, to my knowledge, the first of its kind.
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