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Theta Functions and Knots

Răzvan Gelca
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Given that, back in the early 1980s, I did my doctoral thesis on Weil’s 1966 formulation of what is sometimes called the inverse Hecke correspondence, I have had a deep affinity for all things Hecke for on the order of thirty years. “Hecke” is Erich Hecke, one of Hilbert’s premier students and a grandmaster of analytic number theory. One of the most Hecke things of all is the study of theta series and theta functions, with Hecke’s powerful use of Fourier analytic tools heavily featured throughout.

Over the last few decades, low-dimensional topology has experienced an explosion, or rather a sequence of explosions on several fronts, three of which are the theory of knots, the influx of results and methods from physics (with the redoubtable Ed Witten taking center stage), and the revolutions in differential geometry leading up to the climactic proof of “Poincaré-3” by Grigori Perelman. The book under review is in large part concerned with the first two movements, with the second of these concerned specifically with Chern-Simons theory as championed by Witten. Says the author:

The narrative is focused on theta functions associated to a Riemann surface, on the action of the finite Heisenberg group on theta functions discovered by André Weil, and on the action of the modular group on theta functions … [a] product of nineteenth century mathematics dating back to … Jacobi. The central role is played by the representation theory of the Heisenberg group from which the entire abelian Chern-Simons theory is recovered.

The latter statement is very, very deep indeed, seeing that the (unitary) representation theory of the Heisenberg group was first explicated in 1964 in Weil’s seminal Acta Mathematica article, “Sur certains groups d’opérateurs unitaires,” introducing the (projective) Weil representation of the associated symplectic group, and Weil himself noted that this representation had its inception in the context of a physics article by David Shale dealing with the behavior of bosons. To learn that this network of representation-theoretic results is in fact also related to Chern-Simons theory, a 3-dimensional topological quantum field theory (as Wikipedia tells us) which has long had connections to knot theory (cf. Witten’s 1989 paper, “Quantum Field Theory and the Jones polynomial”), is nothing short of stunning — what a wonderful interplay of deep mathematics and physics! In point of fact, with these themes in play it is hard to decide whether the present book is more in the way of number theory (or representation theory, I guess, of the indicated type), or, well, what to call it?, perhaps the common playground of low-dimensional topology and quantum field theory. Happily we don’t have to make this call: it is simply about very beautiful mathematics.

I am certainly a big fan, then, and propose to study this book very carefully, given that it brings up a number of themes relevant to my own researches. More importantly, it looks like a really good book, presenting its many themes in a very accessible and clear fashion, replete with plenty of pictures (how could it be otherwise with knots at center stage?) and lots of wonderful theorems and proofs from representation theory as well as differential geometry and the kind of functional analysis (e.g. à la Hermann Weyl, talking about another of Hilbert’s star pupils) needed to do quantum physics. It’s pretty beefy at over 400 pages, but that clearly can’t be helped: Gelca takes his readers through large areas of algebraic geometry (of the older but never outdated sort, having to do with thetas and Riemann surfaces: very beautiful stuff), representation theory, quantum mechanics and (more generally) quantum field theory, and of course knots and skeins and all the topology going with it. But it’s certainly all worth it.

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

  • Prologue
  • A Quantum Mechanical Prototype
  • Surfaces and Curves
  • The Theta Functions Associated to a Riemann Surface
  • From Theta Functions to Knots
  • Some Results About 3- and 4-Dimensional Manifolds
  • The Discrete Fourier Transform and Topological Quantum Field Theory
  • Theta Functions in the Quantum Group Perspective
  • An Epilogue — Abelian Chern–Simons Theory