Alphabeta MathPointed Curves Chiral Algebras And Coinvariants
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Tensor Products of Modules
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
Category Theory
- Abelian Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
Complex Analysis
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Mittag-Leffler and Runge's Theorem
- Riemann Surfaces, Branched Maps, and Differentials
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
Group Theory
Homological Algebra
Linear Algebra
Partial Differential Equations
Quantum Algebra and Representation Theory
Quantum Field Theory Mathematics
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Classification of Compact Connected Surfaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Formal-coordinate change group Aut O
Definition
Integrally graded conformal module
Definition
Smooth projective complex curve with formal-coordinate bundle
Lemma
Virasoro modes give an Aut O action with field covariance
Lemma
Formal residue is invariant under coordinate change
Lemma
Full local chiral Jacobi from vertex Jacobi in the right D convention
Lemma
A smooth curve has a formal coordinate at every point
Lemma
A formal-coordinate fibre is an Aut O torsor
Definition
Pointed curve with formal coordinates
Lemma
A formal chiral operation glues over a pointed curve
Definition
Completed local mode Lie algebra of a conformal vertex algebra
Definition
Coordinate-independent chiral algebra attached to a VOA
Lemma
The completed local mode algebra acts on conformal modules
Definition
Global outgoing mode algebra and residue action
Lemma
The outgoing mode algebra is coordinate independent and acts
Definition
Coinvariants and conformal blocks on a pointed curve
Lemma
The coinvariant dual is the invariant-functional space
Sources
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §10.1
- Edward Frenkel, Vertex Algebras and Algebraic Curves, Séminaire Bourbaki 875 (2000), §§4.1–4.3
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §§9.1–9.4 and 10.1–10.5
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §§10.1–10.5
- Beilinson and Drinfeld, Chiral Algebras, preliminary Chapter 3, §3.1
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §§10.1–10.9
- Beilinson and Drinfeld, Chiral Algebras, preliminary Chapter 3
- Frenkel, Vertex Algebras and Algebraic Curves, Bourbaki survey, §§4.1–4.3 and 5.1
- Beilinson and Drinfeld, Chiral Algebras, preliminary Chapter 3, §§3.1.1–3.1.2 and 3.3.1–3.3.3
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §§10.6–10.9
- Beilinson and Drinfeld, Chiral Algebras, preliminary Chapter 4, §§4.4.4–4.4.5
- Frenkel, Vertex Algebras and Algebraic Curves, Bourbaki survey, §5.3
- Edward Frenkel, Vertex Algebras and Algebraic Curves, Séminaire Bourbaki 875 (2000), §§4.1–4.3 and §7
- Bin Gui, Lectures on Vertex Operator Algebras and Conformal Blocks, arXiv:2305.03822v1, §§11.10–11.18
- Frenkel, Vertex Algebras and Algebraic Curves, Bourbaki survey, §§5.1–5.3
- Beilinson and Drinfeld, Chiral Algebras, preliminary Chapter 4, §4.4.4