THETA FUNCTIONS KNOTS AND 8872hc_9789814520577_tp.indd 1 25/2/14 9:00 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk THETA FUNCTIONS KNOTS AND Ra˘ zvan Gelca Texas Tech University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8872hc_9789814520577_tp.indd 2 25/2/14 9:00 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. In-house Editor: Elizabeth Lie Printed in Singapore March13,2014 17:56 WorldScientificBook-9inx6in wsbro To Alejandro Uribe and Charles Frohman, from whom I learned to love mathematical physics and topology. v May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk March13,2014 17:56 WorldScientificBook-9inx6in wsbro 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 March13,2014 17:56 WorldScientificBook-9inx6in wsbro 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]. March13,2014 17:56 WorldScientificBook-9inx6in wsbro 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