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Index Theory with Applications to Mathematics and Physics

David D. Bleecker and Bernhelm Booß-Bavnbek
International Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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This titanic book (almost 800 pages) is something of a marvel. None other than Yuri Manin pithily appraises the book as follows:

This treatise, written with ambition, wit and (mathematical) eloquence, strives to combine the qualities of a guide-book, historical chronicles, and a hiking manual for enthusiastic travelers and budding future explorers of this vast territory. Any reader possessing will and enthusiasm can profit from studying (parts of) this book and enjoy finding his or her own path through this land.

Indeed, these phrases, found in the back pages of the book in question, characterize Index Theory with Applications to Mathematics and Physics to a tee. The book can be read with great profit by strong graduate students in differential geometry, global analysis, operator theory, and so on, and provides a wonderful resource for seasoned scholars in these fields given its encyclopedic scope. Additionally, as Manin mentions, there is a strong historical dimension to the book, which makes for a heightened experience and a deeper understanding.

Bleecker and Booß-Bavnbeck advertise that as far as Parts I–III (up to p. 362) of their opus are concerned, the prerequisites to the reader include no more than familiarity with smooth functions and complex separable Hilbert spaces, coupled with “a will to acquire specialized topics in functional analysis, algebraic topology, elliptic operator theory, global analysis, Riemannian geometry, complex variables, and some other subjects” (hence the above claim that a graduate student client should be “strong” even if he’s not all that far along yet).

The remainder of the book, Part IV, at over 300 pages, requires more maturity:

choosing one or two chapters of this [p]art… of the book would make a suitable text for a graduate course in selected topics of global analysis… Part [IV] is written for graduate students, PhD students, and other experienced learners, interested in low-dimensional topology and gauge-theoretic particle physics [:] We try to explain the very place of index theory in geometry and for revisiting quantum field theory.

Regarding Part IV, then, the chapters to choose from are titled (Chapter 14) “Physical motivation and overview” (a lot of physics, including string theory, quantum gravity, GUTs, QM, the standard model, and Feynman diagrams); (Chapter 15) “Geometric preliminaries” (Riemannian geometry, Hodge theory, characteristic classes); (Chapter 16) “Gauge theoretic interactions” (meet Yang-Mills); (Chapter 17) “The local index theorem for twisted Dirac operators” (Clifford algebras, spinors, enter the heat kernel, sundry variations of the index theorem, a number of very deep connections (Hirzebruch’s Signature Formula, Gauß-Bonnet-Chern, Yang-Mills generalized, and Hirzebrurch-Riemann-Roch for Kähler manifolds)); and (Chapter 18) “Seiberg-Witten theory” (the opening section is titled, “Background and survey: intersection form and homotopy type of compact oriented simply-connected four manifolds” and it’s off to races after that). It is indeed the case that these chapters contain ample material for the more advanced course (or seminar) Beecker and Booß-Bavnbeck recommend in this connection.

The preliminary and preparatory part of the book, i.e. its first 362 pages, serves to provide a very thorough, elegant, and very ramified discussion of the index theorem and its addenda, paying proper attention to both the K-theoretic approach to the theorem (and its fascinating and famous history: Atiyah, Singer, Bott, Hirzebruch, and so on) and the approach via the heat kernel (McKean, Singer, Patodi). It is the latter (and historically later) approach to the index theorem that perhaps most explicitly evokes its centrality in mathematics in general, given the ubiquity of the heat (or Gaußian) kernel, occurring everywhere from probability to number theory, and from physics (of course) to PDE.

It is irresistible to note that some particular gems are to be found in Parts I–III, as follows: Chapters 1and 3 deal with the all-important Fredholm operators, Chapter 9 is a culmination of preceding chapters in the form of a discussion of compact operators over closed manifolds, Chapter 10 presents K-theory. We get to the index theorem itself in Chapter 12 with the focus falling on closed manifolds, and, capping it all of, Chapter 13 is an appealing survey titled “Classical applications.” These applications make for the appearance of such things as Chern classes, the Euler characteristic and signature, abelian integrals in the context of Riemann surfaces, Riemann-Roch, the Lefschetz fixed point formula, and even some analysis on symmetric spaces. Quite amazing.

Thus, Index Theory with Applications to Mathematics and Physics certainly appears to live up to the very enthusiastic comments appended at the very back of the book, appraisals of the text by a number of scholars (such as Yuri Manin, above), expressing uniform praise for this work. Here’s what Edward Witten has to say about it: “Readers from a wide range of backgrounds will find much to learn here.” And so it is. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.