Ceyhan / Manin / Marcolli Arithmetic and Geometry Around Quantization

2010
ISBN: 978-0-8176-4831-2
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 292 Seiten

Reihe: Progress in Mathematics

This volume comprises both research and survey articles originating from the conference on Arithmetic and Geometry around Quantization held in Istanbul in 2006. A wide range of topics related to quantization are covered, thus aiming to give a glimpse of a broad subject in very different perspectives.

Autoren/Hrsg.

Ceyhan, Özgür

Marcolli, Matilde

Manin, Yu. I.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Mirror Duality via G2 and Spin( 7) Manifolds;10
3.1;1 Introduction;10
3.2;2 Associative and complex distributions in G2 manifolds;11
3.3;3 Mirror duality in G2 manifolds;14
3.4;4 Mirror duality in Spin(7) manifolds;23
3.5;References;29
4;2-Gerbes and 2-Tate Spaces;31
4.1;1 Introduction;31
4.2;2 Group actions on gerbes and central extensions;32
4.3;3 Group actions on 2-gerbes and central extensions of 2- groups;34
4.4;4 2-Tate spaces and 2-groups;38
4.5;References;42
5;The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds;44
5.1;1 Introduction and results;44
5.2;2 Themetric space Z(Sp);53
5.3;3 The metric of Z( Q) and the Hofer metric;63
5.4;References;70
6;Towards Quantum Cohomology of Real Varieties;72
6.1;1 Introduction;72
6.2;Part I: Moduli spaces of pointed complex and real curves;76
6.2.1;2 Moduli space of pointed complex curves;77
6.2.2;3 Moduli space of pointed real curves;81
6.3;Part II: Quantum cohomology of real varieties;92
6.3.1;4 Gromov–Witten classes;92
6.3.2;5 Gromov–Witten–Welschinger classes;96
6.4;Part III: Yet again, mirror symmetry!;102
6.5;References;104
7;Weyl Modules and Opers without Monodromy;107
7.1;1 Introduction;107
7.2;2 Some results on opers;110
7.3;3 Proof of the main theorem;112
7.4;4 Exactness;116
7.5;5 Computation of characters;119
7.6;6 Proof of Theorem 2;122
7.7;References;126
8;Differentiable Operads, the Kuranishi Correspondence, and Foundations of Topological Field Theories Based on Pseudo- Holomorphic Curves;128
8.1;1 Introduction;128
8.2;2 Smooth correspondence and chain level intersection theory;130
8.3;3 Bott Morse theory: a baby example;134
8.4;4 A8 spaces.;137
8.5;5 A8 algebra.;144
8.6;6 A8 correspondence.;146
8.7;7 A8 homomorphism.;149
8.8;8 A8 homotopy.;157
8.9;9 Filtered A8 algebra and Filtered A8 correspondence.;161
8.10;10 Kuranishi correspondence.;167
8.11;11 Floer theory of Lagrangian submanifolds.;176
8.12;12 Transversality.;179
8.13;13 Orientation.;192
8.14;14 Variations and generalizations.;197
8.15;References;202
9;Notes on the Self-Reducibility of the Weil Representation and Higher- Dimensional Quantum Chaos;206
9.1;1 Introduction;206
9.2;2 Preliminaries;216
9.3;3 Self-reducibility of the Weil representation;220
9.4;4 Bounds on Higher-Dimensional Exponential Sums;225
9.5;5 The Hannay–Berry model;228
9.6;6 The Hecke quantum unique ergodicity theorem;231
9.7;References;236
10;Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields;238
10.1;1 Introduction;238
10.2;2 Quantization of symplectic vector spaces over finite fields;244
10.3;3 Geometric intertwining morphisms;251
10.4;4 Proof of the geometric kernel sheaf theorem;253
10.5;References;255
11;Noncommutative Geometry in the Framework of Differential Graded Categories;257
11.1;1 Introduction;257
11.2;2 Noncommutative geometry in a DG framework;258
11.3;3 On noncommutative motives;266
11.4;References;276
12;Multiplicative Renormalization and Hopf Algebras;280
12.1;1 Introduction;280
12.2;2 Hopf algebra of Green's functions;282
12.3;Appendix: Hopf algebras;294
12.4;References;295