Theta Functions and Knots

469 Pages·2014·6.02 MB·English

by  Razvan Gelca

THETA FUNCTIONS
KNOTS
AND
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THETA FUNCTIONS
KNOTS
AND
Ra˘ zvan Gelca
Texas Tech University, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
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Library of Congress Cataloging-in-Publication Data
Gelca, Răzvan, 1967– author.
Theta functions and knots / by Răzvan Gelca (Texas Tech University, USA).
pages cm
Includes bibliographical references and index.
ISBN 978-9814520577 (hard cover : alk. paper)
1. Functions, Theta. 2. Knot theory. I. Title.
QA345.G35 2014
515'.984--dc23
2014005501
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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In-house Editor: Elizabeth Lie
Printed in Singapore
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To Alejandro Uribe and Charles Frohman, from whom I
learned to love mathematical physics and topology.
v
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Preface
Theta Functions and Knots combines two ways of mathematical reasoning,
one analytical, based on computations and formulas, one geometric, based
on spacial and combinatorial intuition. As the title discloses, the book
is a discussion about how several areas of mathematics interact, rather
than just an intensive study of one particular area. Quantum mechanics,
complex analysis, low dimensional topology, and representation theory are
brought together by theta functions. There are other fields of mathemat-
ics such as algebraic geometry, number theory, differential equations, also
related to theta functions, that the book does not discuss because so many
other sources do. Instead it focuses on some ideas of the author and his
collaborators that allow representation theory to connect theta functions
with low dimensional topology.
The book can be read in two different perspectives: as a text on theta
functions that emphasizes their quantum mechanical, representation theo-
retical, and topological aspects, or as an introduction to Edward Witten’s
Chern-Simonstheorythroughitssimplestcase,sometimescalledthe“triv-
ial” case, abelian Chern-Simons theory. It is the unifying force of Chern-
Simons theory that is responsible for the many facets of mathematics that
come together in this book. The physical nature of Chern-Simons the-
ory motivated the author to introduce Riemann’s theta functions through
quantum mechanics, in the spirit of Yuri Manin, although a purely alge-
braic geometric approach is also possible, and is actually embedded in the
quantum mechanical approach.
Theta Functions and Knots addresses some developments in the theory
of theta functions that occurred during the two decades preceeding the
writing of the book, thus filling a gap in the literature. The book is an
introductiontoboththetheoryofthetafunctionsandtherelatedconcepts
vii
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viii Theta Functions and Knots
from other fields of mathematics. It is accessible to all those curious about
the interplay between theta functions and knots, as well as to those who
want to become initiated in Chern-Simons theory.
The main concepts and results are covered in depth, to offer a self-
contained presentation and to make the reader aware of all difficulties and
subtleties. This means that the parts of complex analysis, low dimensional
topology, quantum mechanics, and representation theory that lie outside
the background of a mathematician without a particular interest in these
areas are discussed in detail. Historical and bibliographical information is
given whenever possible.
ThenarrativeisfocusedonthethetafunctionsassociatedtoaRiemann
surface, on the action of the finite Heisenberg group on theta functions
discoveredbyAndr´eWeil,andontheactionofthemodulargroupontheta
functions, which is the product of nineteenth century mathematics dating
backtotheworksofJacobi. Theyarediscussedfromvariouspointsofview.
The central role is played by the representation theory of the Heisenberg
group from which the entire abelian Chern-Simons theory is recovered.
The author would like to thank Alastair Hamilton, Jozef Przytycki,
ZhenghanWang,ErnestoLupercio,MaraNeusel,RogerBarnard,andJohn
McClearyfortheirsuggestionsandtheirhelpinthecompletionofthisbook.
The author would also like to thank the many contributors to Wikipedia
for offering guidance through the labyrinth of contemporary mathematics.
An active errata will be kept on author’s web page. Any corrections,
additions, or comments can be sent to [email protected].
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Contents
Preface vii
1. Prologue 1
1.1 The history of theta functions . . . . . . . . . . . . . . . . 1
1.1.1 Elliptic integrals and theta functions . . . . . . . 1
1.1.2 The work of Riemann . . . . . . . . . . . . . . . . 5
1.2 The linking number . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 The definition of the linking number. . . . . . . . 7
1.2.2 The Jones polynomial . . . . . . . . . . . . . . . . 12
1.2.3 Computing the linking number from skein relations 14
1.3 Witten’s Chern-Simons theory . . . . . . . . . . . . . . . 16
2. A quantum mechanical prototype 21
2.1 The quantization of a system of finitely many free one-
dimensional particles . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 The classical mechanics of finitely many free par-
ticles in a one-dimensional space . . . . . . . . . . 21
2.1.2 The Schr¨odinger representation . . . . . . . . . . 24
2.1.3 Weyl quantization . . . . . . . . . . . . . . . . . . 28
2.2 Thequantizationoffinitelymanyfreeone-dimensionalpar-
ticles via holomorphic functions . . . . . . . . . . . . . . . 30
2.2.1 The Segal-Bargmann quantization model . . . . . 30
2.2.2 The Schr¨odinger representation and the Weyl
quantization in the holomorphic setting . . . . . . 37
2.2.3 Holomorphic quantization in the momentum rep-
resentation . . . . . . . . . . . . . . . . . . . . . . 40
ix