Lectures on Chern-Weil theory and Witten deformations (Q2768761)

The last twenty years have seen widespread interest in the use of analytic techniques to calculate and refine invariants arising in algebraic and differential topology. Insights from mathematical physics have driven much of this activity. The book under review is an introduction to this field. This book is noteworthy for its combination of brevity, clarity, accessibility, and depth. This combination rests in part on the elegance of the writing and in part on the author's insight in using techniques due to J.-M. Bismut and collaborators [see, e.g., \textit{J.-M. Bismut} and \textit{G. Lebeau}, Publ. Math., Inst. Hautes Étud. Sci. 74, 1-297 (1991; Zbl 0784.32010) and \textit{J.-M. Bismut} and \textit{W. Zhang}, Geom. Funct. Anal. 4, No. 2, 136-212 (1994; Zbl 0830.58030), to develop ideas of \textit{E. Witten}, J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)].NEWLINENEWLINENEWLINECentral themes of this book include transgression, localization, and deformation of the de Rham complex. Among the topics covered are Chern-Weil theory (including characteristic classes in odd degree), the Berline-Vergne localization formula, the Bott residue formula, the Duistermaat-Heckman formula, the Berezin integral, the Mathai-Quillen Thom form, the Gauss-Bonnet-Chern theorem, an analytic proof of the Poincaré-Hopf index formula, an analytic proof of the Morse inequalities, the Thom-Smale complex, Witten's instanton complex, and Atiyah's theorem on the Kervaire semi-characteristic. Although brief, this book includes many sufficiently detailed proofs. Each chapter lists references to the relevant original papers.NEWLINENEWLINENEWLINEThe reader of this book should be familiar with the material usually covered in a first course on differential topology and geometry. In the hands of a teacher able to provide motivating comments, examples, and problems, this book would provide the foundation for an excellent course. Someone unfamiliar with the motivation for the book's content would find self-study difficult, although in principle the references could provide the motivation. A reader aware of the theorems discussed in the book but looking to understand their proofs through the eyes of an expert will find this book fascinating. This book is a valuable contribution to the literature.