Theta functions and knots
This book centers around joint work of its author with Alejandro Uribe providing a low-dimensional topological interpretation for Weil's representation of the Heisenberg group action on the space of theta functions and related topics [\textit{R. Gelca} and \textit{A. Uribe}, Fundam. Math. 228, No. 2, 97--137 (2015; Zbl 1327.81240)], [``From classical theta functions to topological quantum field theory'', in ``The influence of Solomon Lefshetz in Geometry and Topology: 50 years of Mathematics at Cinvestav'', Contemp. Math., AMS, 2014, \url{arXiv:1006.3252}].NEWLINENEWLINEThe bulk of the book consists of a sequence of interlocking introductions to the subjects which its main results bring together: The theory of theta functions, quantum mechanics, low dimensional topology, and quantum topology. The reviewer finds the exposition to be brisk and insightful. These parts of the book may serve as supporting material for graduate courses on any of these topics.NEWLINENEWLINEThe main idea of the book is presented in Chapter 5.2, where we learn that theta functions can be represented as oriented framed multicurves in a handlebody. Various aspects of the theory of theta functions can then be expressed in this language. The space of theta functions on a Jacobian variety is seen to be represented as a `reduced linking number' skein module of the handlebody. The group algebra of the Heisenberg group becomes a `reduced linking number' skein algebra of the surface bounding the handlebody, which acts on the skein module. The discrete Fourier transform of a theta function defined by an element \(\gamma\) of the mapping class group (that is a special case of the Fourier--Mukai transform that is capturing the action of an element of the modular group on theta functions) is represented by pushing the multicurve representing the theta function to the boundary in all possible ways, Dehn twisting by \(\gamma\), taking the average, and pushing the result back into the interior.NEWLINENEWLINEThe next step, in Chapter 7, is to reinterpret the discrete Fourier transform in terms of Dehn surgery. This allows us to identify a \(3\)--manifold with an element in a skein \(\Omega\) which turns out to be isomorphic to the field of complex numbers. The exact Egorov identity gives invariance of this complex number under handleslides, and we have recovered the MOO invariant [\textit{H. Murakami} et al., Osaka J. Math. 29, No. 3, 545--572 (1992; Zbl 0776.57009)]. Physically, only quantum mechanics was used, without quantum field theory.NEWLINENEWLINEChapter 8 relates theta functions further to quantum groups and to the Reshetikhin--Turaev approach. Chapter 9 discusses the relationship of the associated topological quantum field theory, which physically arises from Weyl quantization, with Witten's abelian Chern--Simons Theory. An alternative approach to this relationship, via the heat equation, was discovered by \textit{J. E. Andersen} [Commun. Math. Phys. 255, No. 3, 727--745 (2005; Zbl 1079.53136)].
scientific article; zbMATH DE number 6330125