(EBook PDF) Geometry and Physics Volume I A Festschrift in honour of Nigel Hitchin 1st edition by Jørgen Ellegaard Andersen, Andrew Dancer, Oscar GarcíaPrada 0192522361 9780192522368 full chapters

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ISBN-10 :  0192522361 

ISBN-13 ‏: ‎ 9780192522368

Author:   Jørgen Ellegaard Andersen, Andrew Dancer, Oscar GarcíaPrada 

Nigel Hitchin is one of the world’s foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin’s 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics.

Geometry and Physics: Volume I A Festschrift in honour of Nigel Hitchin 1st Table of contents:

Part I: Volume I
1: Boundary Value Problems for the Lorentzian Dirac Operator
1. Introduction
2. The Dirac operator on Lorentzian manifolds
2.1. Globally Hyperbolic Manifolds
2.2. Spinors
2.3. The Dirac Operator
2.4. The Cauchy Problem
3. Fredholm Pairs
4. Boundary Value Problems for the Dirac Operator
4.1. (Anti-) Atiyah–Patodi–Singer Boundary Conditions
4.2. Generalized Atiyah–Patodi–Singer Boundary Conditions
4.3. Boundary Conditions in Graph Form
4.4. Local Boundary Conditions
References
2: Torsion of Elliptic Curves and Unlikely Intersections
Introduction
1. Generalities
2. Examples and Evidence
3. A Geometric Approach to Effective Finiteness
4. Fields Generated by Elliptic Division
5. Intersections
6. General Weierstrass Families
References
3: Algebras of Quantum Monodromy Data and Character Varieties
1. Introduction
2. Algebras for SL2(C) Monodromy Data for Surfaces with Bordered Cusps
2.1. Darboux Coordinates in Dimension 2
2.2. Basic Relations
2.3. Composite Relations
3. Classical and Quantum R-Matrix Structures of SLk(C) Monodromy Data
3.1. R-Matrix Relations in the SLk(C) Case
3.2. Decorated Character Variety
3.3. Powers of Matrices
3.4. Semi-Classical limit
3.5. Casimirs of the Poisson Algebra of Monodromy Data
3.6. Reduction to the SL2 Decorated Character Variety
4. Examples of Algebras of Monodromy Data
4.1. Case of Only One Monodromy Datum
4.2. Monodromy Algebras for �0,s+1,1
4.2.1. Braid-group action in�0,s+1,1
4.2.2. IHX-relations
4.3. Monodromy Algebras for �0,2,2
4.4. Monodromy Algebra for �0,1,3
5. The Extended Riemann–Hilbert Correspondence
6. Acknowledgements
References
4: The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics
1. The Deformed Hermitian–Yang–Mills Equation and Mirror Symmetry
1.1. The D-Brane Effective Action and the Deformed Hermitian–Yang–Mills equation
1.2. The Semi-Flat Limit of SYZ Mirror Symmetry
2. Analytic Aspects of the dHYM Equation
3. Algebraic Aspects of the dHYM Equation
Acknowledgements
References
5: Quaternionic Geometry in Dimension 8
1. Introduction
Acknowledgements
2. TheWolf Spaces
2.1. Quaternionic Projective Plane
2.3. The Exceptional Wolf Space
3. Cohomogeneity One SU(3)-Actions
3.1. Quaternionic Projective Plane
3.2. Complex Grassmannian
3.3. The Exceptional Wolf Space
4. Nilpotent Perturbations
4.1. U(1)-Invariant Perturbations
5. New Closed Sp(2)Sp(1)-Structures
5.1. Perturbing with a Killing Vector Field
5.2. Perturbing the Exceptional Wolf Space
6. Relations to Other Special Geometries
References
6: Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy
1. The Volume Functional in Seven Dimensions
2. Reduction to Dimension 4
3. Reduction to Dimension 3
4. Further Remarks
4.1. Singularities
4.2. Connection with the Apostolov–Salamon Construction
4.3. A General Class of Equations and LeBrun’s Construction
References
7: A Hitchin Connection for a Large Class of Families of Kähler Structures
1. Introduction
2. Geometric Quantization
2.1. Prequantization
2.2. Kähler Quantization
3. Families of Kähler Structures
3.1. Smooth Families of Kähler Structures
3.2. The Canonical Line Bundle of a Family
3.3. Holomorphic Families of Kähler Structures
4. The Hitchin Connection
4.1. The Weakly Restricted Case
4.2. Hitchin Connection for Smooth Families of Complex Structures
5. The No-Go Theorem and Projective Flatness
6. Pull-Backs of the Hitchin Connection
7. Further Examples
References
8: The Taub–NUT Ambitoric Structure
1. Introduction
2. The Taub–NUT Hyperkähler Structure
2.1. The Standard Flat Hyperkähler Structure as the Completion of a Gibbons–Hawking Construction
2.2. The Taub–NUT Hyperkähler Structure as a Deformation of the Standard Flat Hyperkähler Struct
3. The Ricci-Flat Taub–NUT Kähler Metric
3.1. The Ricci-Flat Taub–NUT Kähler Metric as a Complete Kähler Metric on C2
3.2. The Ricci-Flat Taub–NUT Kähler Structure as a Toric Kähler Structure
4. The Bochner-Flat Taub–NUT Kähler Metric
4.1. The Bochner-Flat Taub–NUT Structure as a Bryant Bochner–Kähler Metric
4.2. The Bochner-Flat Taub–NUT Kähler Structure as a Toric Kähler Structure
5. The Taub–NUT Ambitoric Pair
6. The Gibbons–Hawking Ansatz
6.1. The General Gibbons–Hawking Construction
6.2. The Standard Deformation of a Gibbons–Hawking Metric
Acknowledgements
References
9: Mirror Symmetry with Branes by Equivariant Verlinde Formulas
1. Introduction
2. Background
3. Computation for i < g −1
4. The Symmetry for i, j < g −1
5. Computation over the Moduli Stack
6. Symmetry for i = g −1
7. Reflection of Mirror Symmetry
References
10: Deformation Theory of Lie Bialgebra Properads
1. Introduction
1.1. Deformation Theory of Lie Bialgebras and Graph Complexes
1.2. Main Theorems
1.2.1. Theorem
1.2.2. Theorem
1.3. Some Applications
1.4. Some Notation
1.5. Remark
Acknowledgements
2. Properads of Lie Bialgebras and Graph Complexes
2.1. Lie n-Bialgebras
2.2. Properads of (Involutive) Lie Bialgebras
2.3. Complete Variants
2.4. Directed Graph Complexes
2.4.1. Remark
2.5. Oriented Graph Complexes
3. Deformation Complexes of Properads and Directed Graph Complexes
3.1. Deformation Complexes of Properads
3.2. Complete Variants
3.3. A Map from the Graph Complex GCor_c+d+1 to Der(Holieb_c,d)
3.4. A Map from the Graph Complex GCorc+d+1[[¯h]] to Der(Holieb�c,d)
3.4. A Map from the Graph Complex GCorc+d+1[[¯h]] to Der(Holieb�c,d)
4. Computations of the Cohomology of Deformation Complexes
4.1. The Proof of Theorem 1.2.1
4.1.1. Proposition
4.1.2. Proposition
4.1.3. Remark
4.2. The Proof of Theorem 1.2.2
4.2.1. Proposition
4.2.2. Proposition
4.3. Some Applications
4.3.1. Proposition
4.3.2. On the Unique Non-Trivial Deformation of�Holieb_odd
References
11: Vertex Algebras and 4-Manifold Invariants
1. Introduction andMotivation
1.1. Unorthodox Invariants of Smooth 4-Manifolds
1.2. 4d TQFT
1.3. G2 Perspective
2. Flux Vacua of T[M4]2.1. Theory T[M4]2.1.1. Example:M4 = S2 ×S2
2.1.2. Examples with torsion in homology
2.2. Holomorphic Differentials and Fluxes
3. Seiberg–Witten Invariants and the Kondo Problem
3.1. Electric and Magnetic Impurities
3.2. Half-TwistedModel with Target Space C∗ R×S1
3.3. Anomalies: M2-Branes Ending on M5-Branes and Embedded Surfaces
3.3.1. Foams and knot cobordisms
3.4. Anomalies: Intersecting M5-Branes and Basic Classes
3.5. Gauge Theoretic Invariants of 4-Manifolds as 2d Correlators
3.5.1. Seiberg–Witten invariants from half-twisted correlators
3.5.2. Multi-monopole invariants
3.5.3. Multiple U(1) groups and multiple monopoles
4. Equivariant Multi-Monopole Invariants
4.1. A Cure for Non-Compactness
4.1.1. Moduli space
4.1.2. A vanishing theorem
4.1.3. The universal bundle and its Chern class
4.1.4. Equivariant formand the integral
4.2. Computation for M4 of Simple Type
4.3. Simply connected Kähler Surfaces
4.3.1. Dimensionality
4.3.2. Connectedness
4.4. Multi-Monopole Homology of 3-Manifolds
5. Non-Abelian Generalizations
5.1. Coulomb-Branch Index and New 4-Manifold Invariants
Acknowledgments
A. Curvature of the Canonical Connection on the Universal Bundle
A.1. SU(Nf ) Invariance and the Moment Map
B. Topological Twist of SQED
References
12: Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups
1. Introduction
2. Introduction to Hyperfunctions
3. Hyperfunctions Arising from Localization of Hamiltonian Group Actions
4. �G and its Hamiltonian Group Action
5. Fixed Points Sets of Rank 1 Subtori
6. An Explicit Example: The Loop Space of SU(2)
7. Isotropy Representation of T×S1
8. An Application of the Hyperfunction Fixed Point Localization Formula to�SU(2)
References
13: Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence
1. Introduction
1.1. Motivations
1.1.1. Quantum integrable systems
1.1.2. Geometric Langlands programme
1.1.3. Relation to conformal field theory
1.2. Main Results
2. Separation of Variables for the Classical Hitchin Integrable System
2.1. Integrability and Special Geometry
2.2. Integrability of the Hitchin System
2.3. Algebraic Integrability of Jacobian Fibrations
2.4. Separation of Variables
2.5. Punctures
3. Quantization of Hitchin’s Integrable System
3.1. Genus zero—The SL(2,C) Gaudin Model
3.2. Quantization of Hitchin’s Hamiltonians and the Geometric Langlands Correspondence
3.2.1. Opers
3.2.2. Geometric Langlands correspondence
4. Quantum Separation of Variables
4.1. Genus Zero
4.2. Higher Genus
5. Quantization Conditions
5.1. Natural Choices of Quantization Conditions
5.2. Quantization versus Classification of Real Projective Structures
5.2.1. Single-valuedness
5.3. Reformulation in Terms of Complex Fenchel–Nielsen Coordinates
5.3.1. Complex Fenchel–Nielsen coordinates
5.3.2. Quantization conditions in terms of complex Fenchel–Nielsen coordinates
6. Generating Functions of Varieties of Opers
6.1. Generating Function W
6.2. Quantization Conditions in Terms of the Generating Function W
7. Global Definition of the Yang’s Function
7.1. Global Issues in the Definition of the Functions W
7.2. Use of the Glueing Construction
8. Concluding Remarks
8.1. Semi-classical Limit
8.2. Real versus Complex Integrable Systems
8.3. Relation to Conformal Field Theory
8.4. Real Geometric Langlands

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