دانلود کتاب Kontsevich’s Deformation Quantization and Quantum Field Theory (Lecture Notes in Mathematics)

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Preface
Acknowledgments
Contents
1 Introduction
2 Foundations of Differential Geometry
2.1 Smooth Manifolds
2.1.1 Charts and Atlases
2.1.2 Pullback and Push-forward
2.1.3 Tangent Space
2.2 Vector Fields and Differential 1-Forms
2.2.1 Tangent Bundle
2.2.2 Vector Bundles
2.2.3 Vector Fields
2.2.4 Flow of a Vector Field
2.2.5 Cotangent Bundle
2.2.6 Differential 1-Forms
2.3 Tensor Fields
2.3.1 Tensor Bundle
2.3.2 Multivector Fields and Differential s-Forms
2.4 Integration on Manifolds and Stokes\' Theorem
2.4.1 Integration of Densities
2.4.2 Integration of Differential Forms
2.4.3 Stokes\' Theorem
2.5 de Rham\'s Theorem
2.5.1 (Co)chain Complexes
2.5.2 Singular Homology
2.5.3 de Rham Cohomology and de Rham\'s Theorem
2.6 Hodge Theory for Real Manifolds
2.6.1 Riemannian Manifolds
2.6.2 Hodge Dual
2.6.3 Hodge Decomposition
2.7 Lie Groups and Lie Algebras
2.7.1 Lie Groups
2.7.2 Lie Algebras
2.7.3 The Exponential Map
2.7.3.1 Matrix Lie Groups
2.7.4 Smooth Actions
2.7.5 Adjoint and Coadjoint Representations
2.7.6 Principal Bundles
2.7.7 Lie Algebra Cohomology
2.7.7.1 The Chevalley–Eilenberg Complex
2.8 Connections and Curvature on Vector Bundles
2.8.1 The Affine Case
2.8.2 Generalization to Vector Bundles
2.8.3 Interpretation as Differential Forms
2.9 Distributions and Frobenius\' Theorem
2.9.1 Plane Distributions
2.9.2 Frobenius\' Theorem
2.10 Connections and Curvature on Principal Bundles
2.10.1 Vertical and Horizontal Subbundles
2.10.2 Ehresmann Connection and Curvature
2.11 Basics of Category Theory
2.11.1 Definition of a Category
2.11.2 Functors
2.11.3 Monoidal Categories
3 Symplectic Geometry
3.1 Symplectic Manifolds
3.1.1 Symplectic Form
3.1.2 Symplectic Vector Spaces
3.1.3 Symplectic Manifolds
3.1.4 Symplectomorphisms
3.2 The Cotangent Bundle as a Symplectic Manifold
3.2.1 Tautological and Canonical Forms
3.2.2 Symplectic Volume
3.3 Lagrangian Submanifolds
3.3.1 Lagrangian Submanifolds of Cotangent Bundles
3.3.2 Conormal Bundle
3.3.3 Graphs and Symplectomorphisms
3.4 Local Theory
3.4.1 Isotopies and Vector Fields
3.4.2 Tubular Neighborhood Theorem
3.4.3 Homotopy Formula
3.5 Moser\'s Theorem
3.5.1 Equivalences for Symplectic Structures
3.5.2 Moser\'s Trick
3.6 Weinstein\'s Tubular Neighborhood Theorem
3.6.1 Weinstein\'s Lagrangian Neighborhood Theorem
3.6.2 Weinstein\'s Tubular Neighborhood Theorem
3.6.3 Some Applications
3.7 Classical Mechanics
3.7.1 Lagrangian Mechanics and Variational Principle
3.7.2 Hamiltonian and Symplectic Vector Fields
3.7.3 Hamiltonian Mechanics
3.7.4 Relation of Lie Brackets and Further Structure
3.7.5 Integrable Systems
3.8 Moment Maps
3.8.1 Symplectic and Hamiltonian Actions
3.8.2 Hamiltonian Actions II and Moment Maps
3.9 Symplectic Reduction
3.9.1 Quotient Manifold by Group Action
3.9.2 The Marsden–Weinstein Theorem
3.9.3 Noether\'s Theorem
3.10 Kähler Manifolds and Complex Geometry
3.10.1 Complex Structures
3.10.2 Kähler Manifolds
3.10.2.1 Mumford\'s Criterion
3.10.2.2 Kähler Forms
3.11 Hodge Theory for Complex Manifolds
3.11.1 Hodge Dual and Hodge Laplacian
3.11.2 Hodge Decomposition and Hodge Diamond
3.11.2.1 Application: Cohomology of Complex Torus
4 Poisson Geometry
4.1 Poisson Manifolds
4.1.1 Poisson Structures and the Schouten–NijenhuisBracket
4.1.2 Examples of Poisson Structures
4.2 Symplectic Leaves and Local Structure of Poisson Manifolds
4.2.1 Local and Regular Poisson Structures
4.2.2 Local Splitting and Symplectic Foliation
4.3 Poisson Morphisms and Completeness
4.4 Poisson Cohomology
4.4.1 Definition and Existence
4.4.2 Interpretation
4.4.2.1 Zeroth Cohomology Group
4.4.2.2 First Cohomology Group
4.4.2.3 Second Cohomology Group
4.4.2.4 Third Cohomology Group
4.5 Symplectic Groupoids and Integration of Poisson Manifolds
4.5.1 Lie Algebroids and Lie Groupoids
4.5.2 Symplectic Groupoids and Integrability Conditions
4.6 Dirac Manifolds
4.6.1 Courant Algebroids
4.6.2 Dirac Structures
4.6.3 Dirac Structures for Constrained Manifolds
4.7 Morita Equivalence for Poisson Manifolds
4.7.1 Morita Equivalence of Symplectic Groupoids
4.7.2 Morita Equivalence of Poisson Manifolds
4.7.3 Symplectic Realization of Morita Equivalent Poisson Manifolds
5 Deformation Quantization
5.1 Star Products
5.1.1 Formal Power Series
5.1.1.1 Formal Derivatives and Formal Integrals
5.1.2 Formal Deformations
5.1.3 The Moyal Product
5.1.4 The Canonical Star Product on g*
5.1.5 Equivalent Star Products
5.1.6 Fedosov\'s Globalization Approach
5.1.6.1 Weyl Bundles and the Operators δ and δ*
5.1.6.2 Fedosov\'s Connection
5.1.6.3 Fedosov\'s Main Theorems and the Global Star Product
5.1.7 Symmetries of Star Products
5.1.8 -Hamiltonians and Quantum Moment Maps
5.2 Formality
5.2.1 Some Formal Setup
5.2.2 Differential Graded Lie Algebras
5.2.3 L∞-Algebras
5.2.4 The DGLA of Multivector Fields V
5.2.4.1 The Case of Poisson Bivector Fields
5.2.5 The DGLA of Multidifferential Operators D
5.2.6 The Hochschild–Kostant–Rosenberg Map
5.2.7 The Dual Point of View
5.2.8 Formality of D and Classification of Star Products on Rd
5.3 Kontsevich\'s Star Product
5.3.1 Data for the Construction
5.3.1.1 Admissible Graphs
5.3.1.2 The Multidifferential Operators B
5.3.1.3 Weights of Graphs
5.3.1.4 Configuration Spaces
5.3.2 Proof of Kontsevich\'s Formula
5.3.2.1 U1 Coincides with U1(0)
5.3.2.2 Checking the Degrees
5.3.2.3 Reformulation of the L∞-Condition in Terms of Graphs
5.3.2.4 The Key Is Stokes\' Theorem
5.3.2.5 Classification of Boundary Strata
5.3.2.6 A Trick Using Logarithms
5.3.2.7 Last Step: Vanishing Terms for Type S2 Strata
5.3.3 Logarithmic Formality
5.4 Globalization of Kontsevich\'s Star Product
5.4.1 The Product, Connection and Curvature Maps
5.4.2 Construction of Solutions for a Fedosov-Type Equation
5.5 Operadic Approach to Formality and Deligne\'s Conjecture
5.5.1 Operads and Algebras
5.5.2 Topological Operads
5.5.3 The Little Disks Operad
5.5.4 Deligne\'s Conjecture
5.5.5 Formality of Chain Operads and Relation to Deformation Quantization
6 Quantum Field Theoretic Approach to Deformation Quantization
6.1 The Atiyah–Segal Definition of a Functorial Quantum Field Theory
6.2 Feynman Path Integrals and Perturbative Quantum Field Theory
6.2.1 Functional Integrals and Expectation Values
6.2.2 Gaussian Integrals
6.2.2.1 Green\'s Functions
6.2.2.2 Infinite-Dimensional Case
6.2.3 Integration of Grassmann Variables
6.3 The Moyal Product as a Path Integral Quantization
6.3.1 The Propagator
6.3.2 Expectation Values
6.3.3 Digression on the Divergence of Vector Fields
6.3.4 Independence of Evaluation Points
6.3.5 Associativity
6.3.6 The Evolution Operator as an Application
6.3.7 Perturbative Evaluation of Integrals
6.3.8 Feynman Diagrams and Perturbative Expansion
6.3.9 Infinite-Dimensional Case
6.3.10 Generalizing the Expansion
6.3.10.1 Quantum Mechanics
6.4 Symmetries and Gauge Formalisms
6.4.1 The Main Construction: Faddeev–PopovGhost Method
6.4.2 The BRST Formalism
6.4.2.1 The BRST Operator and the Proof of Theorem 6.4.4
6.4.3 Infinite-Dimensional Case
6.4.3.1 The Trivial Poisson Sigma Model on the Plane
6.4.3.2 Expectation values
6.4.3.3 The Trivial Poisson Sigma Model on the Upper Half-Plane
6.4.3.4 Some Generalizations
6.5 The Poisson Sigma Model
6.5.1 Formulation of the Model
6.5.2 Observables
6.6 Phase Space Geometry and Symplectic Groupoids
6.6.1 Hamiltonian Formulation of the Poisson Sigma Model
6.6.2 The Phase Space and Its Symplectic GroupoidStructure
6.7 Deformation Quantization for Affine Poisson Structures
6.7.1 Gauge-Fixing and Feynman Diagrams
6.7.2 Independence of the Evaluation Point
6.7.3 Associativity
6.8 The Cattaneo–Felder Construction
6.8.1 Supermanifolds and Graded Manifolds
6.8.1.1 Sheaves and Presheaves
6.8.1.2 Supermanifolds
6.8.1.3 Berezinian Integration
6.8.1.4 Change of Variables
6.8.1.5 Divergence of Vector Fields on Supermanifolds
6.8.1.6 Odd-symplectic Supermanifolds
6.8.2 The Batalin–Vilkovisky Formalism
6.8.2.1 The BV Laplacian
6.8.2.2 BV Integrals and the BV Theorem
6.8.2.3 BRST via BV
6.8.2.4 Faddeev–Popov via BV
6.8.3 Formulation of the Main Theorem
6.8.4 Proof Sketch of the Main Theorem
6.8.5 Other Similar Constructions
6.9 A More General Approach: The AKSZ Construction
6.9.1 Differential Graded Symplectic Hamiltonian Manifolds
6.9.2 AKSZ Sigma Models
Bibliography
Index