Arithmetic and Geometry Around Quantization (eBook)

Preface 6
Contents 8
Mirror Duality via G2 and Spin( 7) Manifolds 10
1 Introduction 10
2 Associative and complex distributions in G2 manifolds 11
3 Mirror duality in G2 manifolds 14
4 Mirror duality in Spin(7) manifolds 23
References 29
2-Gerbes and 2-Tate Spaces 31
1 Introduction 31
2 Group actions on gerbes and central extensions 32
3 Group actions on 2-gerbes and central extensions of 2- groups 34
4 2-Tate spaces and 2-groups 38
References 42
The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds 44
1 Introduction and results 44
2 Themetric space Z(Sp) 53
3 The metric of Z( Q) and the Hofer metric 63
References 70
Towards Quantum Cohomology of Real Varieties 72
1 Introduction 72
Part I: Moduli spaces of pointed complex and real curves 76
2 Moduli space of pointed complex curves 77
3 Moduli space of pointed real curves 81
Part II: Quantum cohomology of real varieties 92
4 Gromov–Witten classes 92
5 Gromov–Witten–Welschinger classes 96
Part III: Yet again, mirror symmetry! 102
References 104
Weyl Modules and Opers without Monodromy 107
1 Introduction 107
2 Some results on opers 110
3 Proof of the main theorem 112
4 Exactness 116
5 Computation of characters 119
6 Proof of Theorem 2 122
References 126
Differentiable Operads, the Kuranishi Correspondence, and Foundations of Topological Field Theories Based on Pseudo- Holomorphic Curves 128
1 Introduction 128
2 Smooth correspondence and chain level intersection theory 130
3 Bott Morse theory: a baby example 134
4 A8 spaces. 137
5 A8 algebra. 144
6 A8 correspondence. 146
7 A8 homomorphism. 149
8 A8 homotopy. 157
9 Filtered A8 algebra and Filtered A8 correspondence. 161
10 Kuranishi correspondence. 167
11 Floer theory of Lagrangian submanifolds. 176
12 Transversality. 179
13 Orientation. 192
14 Variations and generalizations. 197
References 202
Notes on the Self-Reducibility of the Weil Representation and Higher- Dimensional Quantum Chaos 206
1 Introduction 206
2 Preliminaries 216
3 Self-reducibility of the Weil representation 220
4 Bounds on Higher-Dimensional Exponential Sums 225
5 The Hannay–Berry model 228
6 The Hecke quantum unique ergodicity theorem 231
References 236
Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields 238
1 Introduction 238
2 Quantization of symplectic vector spaces over finite fields 244
3 Geometric intertwining morphisms 251
4 Proof of the geometric kernel sheaf theorem 253
References 255
Noncommutative Geometry in the Framework of Differential Graded Categories 257
1 Introduction 257
2 Noncommutative geometry in a DG framework 258
3 On noncommutative motives 266
References 276
Multiplicative Renormalization and Hopf Algebras 280
1 Introduction 280
2 Hopf algebra of Green's functions 282
Appendix: Hopf algebras 294
References 295