Theta functions and knots
type: Item
instance of: scholarly article
zbMATH Open document ID: 1311.57001
DOI: 10.1142/8872
author: Răzvan Gelca
publication date: 2014-08-14T00:00:00Z
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 6330125
zbMATH Keywords: theta functions
zbMATH Keywords: topological quantum field theory
zbMATH Keywords: Murakami-Ohtsuki-Okada invariant
zbMATH Keywords: Chern-Simons theory
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Schroedinger representation
zbMATH Keywords: Segal-Bargmann quantization
zbMATH Keywords: Weyl quantization
zbMATH Keywords: geometric quantization
zbMATH Keywords: exact Egorov identity
zbMATH Keywords: skein module
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1142/8872
OpenAlex ID: W596256088
title: Theta functions and knots
review text: 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.NEWLINENEWLIN…
reviewed by: Daniel Moskovich
description: scientific article; zbMATH DE number 6330125