Topological Quantum Field Theory and Four Manifolds: (25 Mathematical Physics Studies)

The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

Table of Contents:
Topological Aspects of Four-Manifolds.- The Theory of Donaldson Invariants.- The Theory of Seiberg-Witten Invariants.- Supersymmetry in Four Dimensions.- Topological Quantum Field Theories in Four Dimensions.- The Mathai-Quillen Formalism.- The Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory.- The u-plane Integral.- Some Applications of the u-plane Integral.- Further Developments in Donaldson—Witten Theory.

Review :
From the reviews of the first edition: "The present book … starts with a survey of important topological topics, then reviews the theories of Donaldson and Seiberg-Witten, and describes various aspects of supersymmetery … . Graduate students, post-docs and junior faculty interested in the interaction of physics and mathematics … 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." (Gert Roepstorff, Zentralblatt MATH, Vol. 1087, 2006) "The book is written to be accessible either for physicists who want to know about the topological consequences of supersymmetric quantum field theory, or for mathematicians curious about where the links between Donaldson and Seiberg-Witten theory come from. For both groups it should be a good point of entry to the literature. … the authors manage to give an end-to-end treatment of the relation between Donaldson and Seiberg-Witten invariants, including a detailed computation of the formula connecting the two for manifolds of Seiberg-Witten simple type." (Andrew Neitzke, Mathematical Reviews, Issue 2006 f)