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Geometry and Quantization of Moduli Spaces 1st ed. 2016 [Mīkstie vāki]

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  • Formāts: Paperback / softback, 220 pages, height x width: 240x168 mm, weight: 3959 g, 2 Illustrations, color; 60 Illustrations, black and white; X, 220 p. 62 illus., 2 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 06-Jan-2017
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3319335774
  • ISBN-13: 9783319335773
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Geometry and Quantization of Moduli Spaces 1st ed. 2016Palielināt
  • Formāts: Paperback / softback, 220 pages, height x width: 240x168 mm, weight: 3959 g, 2 Illustrations, color; 60 Illustrations, black and white; X, 220 p. 62 illus., 2 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 06-Jan-2017
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3319335774
  • ISBN-13: 9783319335773
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783319335773.html
Keywords:
This volume is based on four advanced courses held at the Centre de Recerca Matemątica (CRM), Barcelona. It presents both background information and recent developments on selected topics that are experiencing extraordinary growth within the broad research area of geometry and quantization of moduli spaces. The lectures focus on the geometry of moduli spaces which are mostly associated to compact Riemann surfaces, and are presented from both classical and quantum perspectives.

Loop groups, clusters, dimers and integrable systems.- Lectures on Klein surfaces and their fundamental group.- Five lectures on topological field theory.- Higgs bundles and local systems on Riemann surfaces.
Foreword v
1 Loop Groups, Clusters, Dimers and Integrable Systems
1(66)
Vladimir V. Fock
Audrey Marshakov
1.1 Introduction
1(6)
1.1.1 Integrable Systems on Poisson--Lie Groups
2(1)
1.1.2 Goncharov--Kenyon Integrable Systems
3(2)
1.1.3 Relations Between the Two Approaches
5(2)
1.2 Integrable Systems and r-Matrices
7(2)
1.3 Cluster Parametrisation of Double Bruhat Cells: Simple Groups
9(5)
1.3.1 Cartan--Weyl Generators of a Simple Group
9(1)
1.3.2 Construction of the Cluster Seeds
10(1)
1.3.3 Generators and Thurston Diagrams for the Group PGL(N)
11(1)
1.3.4 Example: Poisson submanifolds of PGL(3)
12(2)
1.4 Cluster Parameterisation of Double Bruhat Cells: Loop Groups
14(6)
1.4.1 The Coextended Affine Weyl Group, Wiring, and Thurston Diagrams
14(2)
1.4.2 Realisations of the Coextended Loop Group
16(2)
1.4.3 Integrable Systems on Double Bruhat Cells
18(2)
1.5 Dimers
20(11)
1.5.1 Recollection About Dimers
20(2)
1.5.2 Matrices and Dimers on a Disk
22(1)
1.5.3 Dimer Partition Functions with Signs
23(1)
1.5.4 PGL(N) and Dimers on a Cylinder
24(1)
1.5.5 Dimers and the Discrete Dirac Operator
25(2)
1.5.6 Spectral Submanifold and Face Partition Function
27(3)
1.5.7 Dual Surface, Double Partition Function, and Poisson Bracket
30(1)
1.6 Dimers and Integrable Systems for the Loop Groups
31(2)
1.7 Mutations and Discrete Flows
33(4)
1.7.1 Equivalence of Bipartite Graphs
33(1)
1.7.2 Discrete Flow τ
34(1)
1.7.3 General Discrete Flows
35(2)
1.8 Examples
37(15)
1.8.1 Triangle
37(2)
1.8.2 The Simplest Relativistic Toda Chain
39(2)
1.8.3 Degeneration to non-Affine Toda System
41(2)
1.8.4 Relativistic Toda Chain of Rank Two
43(4)
1.8.5 Parallelograms of Arbitrary Size and the Pentagram Map
47(5)
1.9 Appendix A: Cluster Varieties of Type Χ
52(1)
1.10 Appendix B: Relations among the Generators of a Simply Laced Lie Group
53(1)
1.11 Appendix C: Exchange Graphs and Decompositions of u ε W × W
53(2)
1.12 Appendix D: Thurston Diagrams
55(5)
1.13 Appendix E: Proofs of the Properties of the Minors Generating Functions
60(1)
1.14 Appendix F: Schwartz Coordinates and the Pentagram Map
61(6)
Bibliography
65(2)
2 Lectures on Klein Surfaces and Their Fundamental Group
67(42)
Florent Schaffhauser
2.1 Klein Surfaces and Real Algebraic Curves
68(12)
2.1.1 Algebraic Curves and Two-Dimensional Manifolds
68(3)
2.1.2 Topological Types of Real Curves
71(4)
2.1.3 Dianalytic Structures on Surfaces
75(5)
2.2 The Fundamental Group of a Real Algebraic Curve
80(21)
2.2.1 A Short Reminder on the Fundamental Group of a Riemann Surface
80(6)
2.2.2 The Fundamental Group of a Klein Surface
86(12)
2.2.3 Galois Theory of Real Covering Spaces
98(3)
2.3 Representations of the Fundamental Group of a Klein Surface
101(8)
2.3.1 Linear Representations of Fundamental Groups of Real Curves
101(2)
2.3.2 Real Structure of the Usual Representation Variety
103(4)
Bibliography
107(2)
3 Five Lectures on Topological Field Theory
109(56)
Constantin Teleman
3.1 Introduction and Examples
109(9)
3.1.1 Definition
109(1)
3.1.2 Example: Finite Group Gauge Theory
110(1)
3.1.3 Baby Classification, D = 1
111(1)
3.1.4 D = 2 and Frobenius Algebras
112(1)
3.1.5 Finite Group Gauge Theory in 2D
113(1)
3.1.6 Finite Higher-Groupoid Theories
113(1)
3.1.7 Yang-Mills Theory in 2D
114(1)
3.1.8 Variant of TQFT: Cohomological Field Theories
115(3)
3.2 Two-Dimensional Gauge Theory
118(9)
3.2.1 Interpretation in K-Theory
119(1)
3.2.2 Integration from the Index
120(1)
3.2.3 Index Formulas on φ(Σ; G)
121(1)
3.2.4 The Verlinde Ring
122(1)
3.2.5 Twisted K-Theory, a Crash Course
122(2)
3.2.6 Twisted KG(G)
124(2)
3.2.7 Generalizations: Higgs Bundles as an Example
126(1)
3.3 Extended TQFT and Higher Categories
127(10)
3.3.1 Higher Categories
128(1)
3.3.2 Strict Versus Weak Categories
129(1)
3.3.3 Finite Group Gauge Theory in 2D
129(1)
3.3.4 The Correspondence 2-Category
130(2)
3.3.5 Inadequacy of Strict Categories
132(1)
3.3.6 The Quadratic Map π2 → π3
133(1)
3.3.7 A Braided Tensor Category from X
134(1)
3.3.8 Ribbon Structure
135(1)
3.3.9 Finite Homotopy Types
136(1)
3.4 The Cobordism Hypothesis in Dimensions 1 and 2
137(14)
3.4.1 Duals and 1D framed TQFT's
139(2)
3.4.2 Cobordism Hypothesis in 1D
141(1)
3.4.3 O(1) Action on Dualizable Objects
141(1)
3.4.4 Cobordism Hypothesis in 2D
142(1)
3.4.5 The Serre Twist
143(1)
3.4.6 Oriented and r-Spin Theories
144(1)
3.4.7 Adjunction in Pictures: Oriented Handles
144(1)
3.4.8 Framed Handles
145(2)
3.4.9 Adjunction: Algebraic Conditions
147(2)
3.4.10 Oriented TQFT's from Frobenius Algebras
149(1)
3.4.11 Finite Gauge Theory Revisited
149(1)
3.4.12 The Serre Functor on a Scheme
150(1)
3.4.13 Vector Spaces Associated to the Circle
150(1)
3.5 Cobordism Hypothesis in Higher Dimension
151(14)
3.5.1 Reduction to co-Dimension 2
152(1)
3.5.2 3D Example
153(4)
3.5.3 Tensor and Module Categories
157(1)
3.5.4 Tensor Product of Categories
157(1)
3.5.5 2-Dualizability
158(1)
3.5.6 Drinfeld Center and Hochschild Homology
158(2)
3.5.7 Fusion Categories
160(1)
3.5.8 The Serre Automorphism Tv
161(2)
Bibliography
163(2)
4 Higgs Bundles and Local Systems on Riemann Surfaces
165
Richard A. Wentworth
4.1 Preface
165(3)
4.2 The Dolbeault Moduli Space
168(20)
4.2.1 Higgs Bundles
168(5)
4.2.2 The Moduli Space
173(5)
4.2.3 The Hitchin--Kobayashi Correspondence
178(10)
4.3 The Betti Moduli Space
188(12)
4.3.1 Representation Varieties
188(1)
4.3.2 Local Systems and Holomorphic Connections
188(4)
4.3.3 The Corlette--Donaldson Theorem
192(5)
4.3.4 Hyperkahler Reduction
197(3)
4.4 Differential Equations
200
4.4.1 Uniformization
200(2)
4.4.2 Higher Order Equations
202(3)
4.4.3 Opers
205(8)
4.4.4 The Eichler--Shimura Isomorphism
213(2)
Bibliography
215
Vladimir V. Fock  is Professor at the Institut de Recherche Mathématique Avancée in Strasbourg. Andrey Marshakov is Professor at the National Research University Higher School of Economics in Moscow.Florent Schaffhauser is Associate Professor at Universidad de Los Andes in Bogotį.





Constantin Teleman is Professor at the University of California in Berkeley.

Richard A. Wentworth is Professor at the University of Maryland.