Theta Functions And Knots

by Razvan Gelca

Overview

This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Răzvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology.Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern-Simons theory will find here an introduction using the simplest case, that of abelian Chern-Simons theory. Moreover, the construction of abelian Chern-Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done.Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study.

Product Details

ISBN-13:	9789814520577
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/24/2014
Pages:	468
Product dimensions:	5.90(w) x 9.00(h) x 1.10(d)

Table of 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 particles in a one-dimensional space 21

2.1.2 The Schrödinger representation 24

2.1.3 Weyl quantization 28

2.2 The quantization of finitely many free one-dimensional particles via holomorphic functions 30

2.2.1 The Segal-Bargmann quantization model 30

2.2.2 The Schrödinger representation and the Weyl quantization in the holomorphic setting 37

2.2.3 Holomorphic quantization in the momentum representation 40

2.3 Geometric quantization 41

2.3.1 Polarizations 42

2.3.2 The construction of the Hilbert space using geometric quantization 48

2.3.3 The Schrödinger representation from geometric considerations 52

2.3.4 Passing from real to Kahler polarizations 57

2.4 The Schrödinger representation as an induced representation 57

2.5 The Fourier transform and the representation of the symplectic group Sp(2n, R) 61

2.5.1 The Fourier transform defined by a pair of Lagrangian subspaces 61

2.5.2 The Maslov index 64

2.5.3 The resolution of the projective ambiguity of the representation of Sp(2n, R) 70

3 Surfaces and curves 81

3.1 The topology of surfaces 82

3.1.1 The classification of surfaces 82

3.1.2 The fundamental group 83

3.1.3 The homology and cohomology groups 85

3.1.4 The homology groups of a surface and the intersection form 90

3.2 Curves on surfaces 94

3.2.1 Isotopy versus homotopy 94

3.2.2 Multicurves on a torus 99

3.2.3 The first homology group of a surface as a group of curves 102

3.2.4 Links in the cylinder over a surface 108

3.3 The mapping class group of a surface 109

3.3.1 The definition of the mapping class group 109

3.3.2 Particular cases of mapping class groups 112

3.3.3 Elements of Morse and Cerf theory 114

3.3.4 The mapping class group of a closed surface is generated by Dehn twists 122

4 The theta functions associated to a Riemann surface 135

4.1 The Jacobian variety 135

4.1.1 De Fham cohomology 136

4.1.2 Hodge theory on a Riemann surface 137

4.1.3 The construction of the Jacobian variety 147

4.2 The quantization of the Jacobian variety of a Riemann surface in a real polarization 153

4.2.1 Classical mechanics on the Jacobian variety 153

4.2.2 The Hilbert space of the quantization of the Jacobian variety in a real polarization 156

4.2.3 The Schrödinger representation of the finite Heisenberg group 162

4.3 Theta functions via quantum mechanics 168

4.3.1 Theta functions from the geometric quantization of the Jacobian variety in a Kähler polarization 168

4.3.2 The action of the finite Heisenberg group on theta functions 173

4.3.3 The Segal-Bargmann transform on the Jacobian variety 180

4.3.4 The algebra of linear operators on the space of theta functions and the quantum torus 182

4.3.5 The action of the mapping class group on theta functions 184

4.4 Theta functions on the Jacobian variety of the torus 188

4.4.1 The theta functions and the action of the Heisenberg group 188

4.4.2 The action of the S map 189

4.4.3 The action of the T map 192

5 From theta functions to knots 195

5.1 Theta functions in the representation theoretical setting 195

5.1.1 Induced representations for finite groups 195

5.1.2 The Schrödinger representation of the finite Heisenberg group as an induced representation 199

5.1.3 The action of the mapping class group on theta functions in the representation theoretical setting 203

5.2 A heuristical explanation 210

5.2.1 From theta functions to curves 211

5.2.2 The idea of a skein module 214

5.3 The skein modules of the finking number 215

5.3.1 The definition of skein modules 215

5.3.2 The group algebra of the Heisenberg group as a skein algebra 221

5.3.3 The skein module of a handlebody 227

5.4 A topological model for theta functions 229

5.4.1 Reduced linking number skein modules 229

5.4.2 The Schrödinger representation in the topological perspective 234

5.4.3 The action of the mapping class group on theta functions in the topological perspective 241

6 Some results about 3- and 4-dimensional manifolds 251

6.1 3-dimensional manifolds obtained from Heegaard decompositions and surgery 251

6.1.1 The Heegaard decompositions of a 3-dimensional manifold 251

6.1.2 3-dimensional manifolds obtained from surgery 253

6.2 The interplay between 3-dimensional and 4-dimensional topology 261

6.2.1 3-dimensional manifolds are boundaries of 4-dimensional handlebodies 261

6.2.2 The signature of a 4-dimensional manifold 268

6.3 Changing the surgery link 274

6.3.1 Handle slides 274

6.3.2 Kirby's theorem 279

6.4 Surgery for 3-dimensional manifolds with boundary 288

6.4.1 A relative version of Kirby's theorem 288

6.4.2 Cobordisms via surgery 295

6.5 Wall's formula for the nonadditivity of the signature of 4-dimensional manifolds 303

6.5.1 Lagrangian subspaces in the boundary of a 3-dimensional manifold 303

6.5.2 Wall's theorem 305

6.6 The structure of the linking number skein module of a 3-diniensional manifold 312

7 The discrete Fourier transform and topological quantum field theory 321

7.1 The discrete Fourier transform and handle slides 321

7.1.1 The discrete Fourier transform as a skein 321

7.1.2 The exact Egorov identity and handle slides 327

7.2 The Mmakarni-Ohtsuki-Okada invariant of a closed 3-dimensional manifold 330

7.2.1 The construction of the invariant 330

7.2.2 The computation of the invariant 332

7.3 The reduced linking number skein module of a 3-dimensional manifold 340

7.3.1 The Sikora isomorphism 340

7.3.2 The computation of the reduced linking number skein module of a 3-dimensional manifold 343

7.4 The 4-dimensional manifolds associated to discrete Fourier transforms 348

7.4.1 Fourier transforms from general surgery diagrams 348

7.4.2 A topological solution to the projectivity problem of the representation of the mapping class group on theta functions 350

7.5 Theta functions and topological quantum field theory 361

7.5.1 Empty skeins and the emergence of topological quantum field theory 361

7.5.2 Atiyah's axioms for a topological quantum field theory 363

7.5.3 The functor from the category of extended surfaces to the category of finite-dimensional vector spaces 365

7.5.4 The topological quantum field theory underlying the theory of theta functions 369

8 Theta functions in the quantum group perspective 383

8.1 Quantum groups 384

8.1.1 The origins of quantum groups 384

8.1.2 Quantum groups as Hopf algebras 387

8.1.3 The Yang-Baxter equation and the universal R-matrix 393

8.1.4 Link invariants and ribbon Hopf algebras 401

8.2 The quantum group associated to classical theta functions 415

8.2.1 The quantum group and its representation theory 415

8.2.2 The quantum group of theta functions is a quasi-triangular Hopf algebra 417

8.2.3 The quantum group of theta functions is a ribbon Hopf algebra 420

8.3 Modeling theta functions using the quantum group 425

8.3.1 The relationship between the linking number and the quantum group 426

8.3.2 Theta functions as colored oriented framed links in a handlebody 429

8.3.3 The Schrodinger representation and the action of the mapping class group via quantum group representations 430

9 An epilogue - Abelian Chern-Simons theory 437

9.1 The Jacobian variety as a moduli space of connections 437

9.2 Weyl quantization versus quantum group quantization of the Jacobian variety 441

Bibliography 445

Index 451