Theta functions and knots:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Hackensack, NJ [u.a.]
World Scientific
2014
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 445 - 449 |
| Physical Description: | XIV, 454 S. graph. Darst. |
| ISBN: | 9789814520577 |
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| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c 2014 | |
| 300 | |a XIV, 454 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| _version_ | 1804152455689142272 |
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| adam_text | Titel: Theta functions and knots
Autor: Gelca, Răzvan
Jahr: 2014
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 Schrödinger representation .......... 24
2.1.3 Weyl quantization.................. 28
2.2 The quantization of finitely many free one-dimensional par-
ticles 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 rep-
resentation ...................... 40
x Theta Functions and Knots
2.3 Geometrie quantization ................... 41
2.3.1 Polarizations..................... 42
2.3.2 The construetion of the Hubert space using geo-
metric 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 sym-
plectic group Sp(2n,R) ................... 61
2.5.1 The Fourier transform defined by a pair of La-
grangian subspaces ................. 61
2.5.2 The Maslov index.................. 64
2.5.3 The resolution of the projeetive 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 inter-
section 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 gen-
erated by Dehn twists................ 122
4. The theta functions associated to a Riemann surface 135
4.1 The Jacobian variety..................... 135
Contents xi
4.1.1 De Rham 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 Jaco-
bian 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 Kahler 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 Heisen-
berg group...................... 188
4.4.2 The action of the S map.............. 189
4.4.3 The action of the T map.............. 192
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 linking number ......... 215
5.3.1 The definition of skein modules .......... 215
xii Theta Functions and Knots
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 decom-
positions 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-
dimensional manifold..................... 312
7. The discrete Fourier transform and topological quantum
field theory 321
7.1 The discrete Fourier transform and handle slides..... 321
Contents xiii
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 Murakami-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
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
xiv Theta Functions and Knots
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 Schrödinger representation and the action of
the mapping class group via quantum group rep-
resentations ..................... 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
|
| any_adam_object | 1 |
| author | Gelca, Răzvan 1967- |
| author_GND | (DE-588)122178068 |
| author_facet | Gelca, Răzvan 1967- |
| author_role | aut |
| author_sort | Gelca, Răzvan 1967- |
| author_variant | r g rg |
| building | Verbundindex |
| bvnumber | BV042033527 |
| classification_rvk | SK 300 |
| ctrlnum | (OCoLC)890066627 (DE-599)GBV778247716 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042033527 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:10:59Z |
| institution | BVB |
| isbn | 9789814520577 |
| language | English |
| lccn | 2014005501 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027474956 |
| oclc_num | 890066627 |
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| owner | DE-19 DE-BY-UBM DE-29T DE-703 DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-29T DE-703 DE-11 |
| physical | XIV, 454 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Gelca, Răzvan 1967- Verfasser (DE-588)122178068 aut Theta functions and knots Răzvan Gelca Hackensack, NJ [u.a.] World Scientific 2014 XIV, 454 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 445 - 449 Knotentheorie (DE-588)4164318-5 gnd rswk-swf Thetafunktion (DE-588)4185175-4 gnd rswk-swf Thetafunktion (DE-588)4185175-4 s Knotentheorie (DE-588)4164318-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027474956&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gelca, Răzvan 1967- Theta functions and knots Knotentheorie (DE-588)4164318-5 gnd Thetafunktion (DE-588)4185175-4 gnd |
| subject_GND | (DE-588)4164318-5 (DE-588)4185175-4 |
| title | Theta functions and knots |
| title_auth | Theta functions and knots |
| title_exact_search | Theta functions and knots |
| title_full | Theta functions and knots Răzvan Gelca |
| title_fullStr | Theta functions and knots Răzvan Gelca |
| title_full_unstemmed | Theta functions and knots Răzvan Gelca |
| title_short | Theta functions and knots |
| title_sort | theta functions and knots |
| topic | Knotentheorie (DE-588)4164318-5 gnd Thetafunktion (DE-588)4185175-4 gnd |
| topic_facet | Knotentheorie Thetafunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027474956&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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