Alphabeta MathCharacters, Sewing and Modular Functor Data
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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
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- 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
- Topological Vector Bundles and Grassmannian Classification
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- 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 Beltrami Equation and Measurable Riemann Mapping
- 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
- Chern–Weil Theory and Characteristic Forms
- Compact Laplace-type operators and harmonic representatives
- Connections Levi Civita and Parallel Transport
- Integration of Forms and the General Stokes Theorem
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Group Theory
Homological Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
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 Inverse Function Theorem Completed
- 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
Ordinary VOA character
Lemma
Simultaneous fusion-matrix diagonalization implies the Verlinde formula
Remark
Logarithmic and quasi-lisse modularity have distinct scopes
Lemma
The Laurent determinant cocycle has Virasoro central weight two
Open problem
Nondegeneracy of the annular pairing beyond finite rigidity
Lemma
Modular covariance of the three genus-one one-point Ward identities
Definition
Chiral modular functor with sewing and anomaly
Definition
Equivalence of chiral modular functors
Sources
- Paul Ginsparg, Applied Conformal Field Theory
- Yi-Zhi Huang, Vertex operator algebras and the Verlinde conjecture
- Yi-Zhi Huang, Modular invariance of (logarithmic) intertwining operators
- Tomoyuki Arakawa, Jethro van Ekeren, and Hao Li, Modular invariance of characters of quasi-lisse vertex algebras
- Chiara Damiolini and Lukas Woike, Modular Functors from Conformal Blocks of Rational Vertex Operator Algebras
- Damiolini, Gibney and Tarasca, Conformal blocks from vertex algebras and their connections
- Lukas Müller and Lukas Woike, Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness
- Adrien Brochier and Lukas Woike, A Classification of Modular Functors via Factorization Homology
- Bojko Bakalov and Alexander Kirillov Jr., Lectures on Tensor Categories and Modular Functor, Chapter 5
- Bin Gui and Hao Zhang, Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-q-traces
- Joachim Kock, Notes on Psi Classes
- Brochier and Woike, A Classification of Modular Functors via Factorization Homology