Alphabeta MathModular Categories, Verlinde Formula, and TQFT
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 3 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- 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
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Chern and Pontryagin Classes by Splitting and Complexification
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- 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
- Spectra and Stable Homotopy Groups
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Thom Spectra and Unoriented Bordism Detection
- Topological Vector Bundles and Grassmannian Classification
Braid Groups
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Exactness and the Member Calculus
- Limits and Colimits
- Monoidal Categories and Monoidal Functors
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Strictification and Mac Lanes Coherence Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
- Chains, Antichains, Sperner and Dilworth
- Finite Counting, Factorials and Binomial Coefficients
- Formal Power Series
- Graphs, Walks and Connectivity
- Lattice Paths and Catalan Numbers
- Linear Recurrences and Rational Generating Functions
- Plane Graphs, Euler's Formula and the Five Colour Theorem
- Trees, Forests and Spanning Trees
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
- Periods, Jacobians, and Abel--Jacobi Theory
- 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
Coxeter Groups
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- The Gauss Bonnet Theorem for Riemannian Surfaces
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Characteristic Numbers and Cobordism Obstructions
- Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations
- Exotic Smooth Structures and Milnor Spheres
- Formal Immersions and the Smale Hirsch Theorem
- Gradient Like Vector Fields and Morse Trajectories
- Handle Cancellation Slides and Elementary Moves
- Handle Decompositions Duality and Rearrangement
- Intersection Pairings Self Intersection and Euler Classes
- Isotopy Extension and Embedding Theory Beyond Whitney
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Oriented and Mod Two Intersection Numbers
- Reeb Stability and Global Foliation Constructions
- Smooth Cobordism Relations Groups and Rings
- Smooth Surgery Traces and Handle Trading
- Stable Unstable Manifolds and Morse Smale Transversality
- Sublevel Deformation and the Handle Attachment Theorem
- The Hirzebruch Signature Theorem
- The Whitney Trick and Surgery Below the Middle Dimension
- Thom Spaces Normal Data and Collapse Maps
- Vanishing Cycles, Novikov and Taut Foliations
Functional Analysis
Group Theory
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- 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
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
Number Theory
Partial Differential Equations
Quantum Algebra and Representation Theory
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
- Constant Rank, Submersions, Immersions and Regular Level Sets
- 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
- Darboux, L'Hôpital, and Taylor's Theorem
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- 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
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- 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
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- 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 Ascoli–Arzelà Theorem
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Modular ribbon fusion category and Hopf-link data
Definition
Pointed ribbon categories from abelian 3-cocycles
Lemma
Finite sine-transform orthogonality at level ell
Lemma
Distinct fusion-ring characters are orthogonal under duality
Lemma
A linear Lagrangian relation sends a Lagrangian subspace to a Lagrangian subspace
Lemma
The four-Lagrangian Maslov cocycle identity
Lemma
The boundary kernel of a compact oriented 3-manifold is Lagrangian
Lemma
The middle-boundary Maslov correction composes for Lagrangian relations
Definition
Weighted Lagrangian ribbon bordisms
Lemma
Simple ribbon fusion dimensions are nonzero
Theorem
A pointed metric-group category is modular exactly when its radical is trivial
Definition
RT state spaces from a modular category
Lemma
Balancing and the Hopf-link character identity
Theorem
Every closed oriented smooth three-manifold bounds a compact oriented smooth four-manifold
Lemma
Closed oriented smooth one- and two-manifolds bound
Lemma
The fifth stable offset in integral Eilenberg–Mac Lane homology vanishes
Theorem
Signature classifies integral oriented smooth four-bordism
Lemma
Projective-plane stabilization gives a five-bordism with the prescribed three-boundary
Lemma
Relative one-parameter Morse paths with finitely many elementary events
Lemma
Finite comparison of parametrized elementary relative handle bands
Lemma
Relative descending paths with finitely many same-index passages
Lemma
Circumcision by a framed core and zero-framed meridian preserves the boundary parametrization
Lemma
An isolated transverse same-index trajectory passage gives the framed band sum
Lemma
Framed circles in a four-dimensional two-handle trace move to its fixed base
Lemma
Interior one-handle trades and the fully framed circle-surgery pair
Lemma
Relative Morse handle normalization with the full vertical height fixed
Lemma
Parametrized link-surgery existence for compact oriented three-manifolds
Lemma
A boundary-marked five-bordism gives a common middle two-handle trace
Lemma
Continuous one-handle surgery gives relative Kirby moves
Lemma
Changing the base arc in a one-handle replacement gives marked Kirby moves
Lemma
One-handle slides become Kirby slides after interior trading
Lemma
Parametric index grouping preserves a relative descending path
Lemma
Index-grouped descending paths give based framed handle movies
Lemma
Finite relative mixed-handle movies give marked Kirby moves
Lemma
Marked relative surgery presentations and complete moves for three-dimensional ribbon bordisms
Lemma
The modular Hopf-link matrix squares to D² times duality
Lemma
Truncated sl2 fusion data give the sine Hopf-link matrix
Lemma
Multicolored ribbon graphs with arbitrary coupons have a unique evaluation
Lemma
The complete Kirby-color band-slide identity with open colored graph coupons
Lemma
Modular Gauss sums, the Kirby color, and its slide identity
Theorem
The algebraic Verlinde formula
Theorem
The standard root-of-unity sl2 tilting ribbon quotient is modular
Lemma
Colored signed twist surgery, circumcision, and paired-handle cancellation
Lemma
RT capped operators: geometric Gram factors and exact signature gluing
Lemma
Wall nonadditivity, RT cap references, and the Walker weight sign
Lemma
Decorated ribbon graphs lift through a marked relative surgery calculus
Lemma
RT handlebody pairings and surgery gluing
Theorem
The weighted Reshetikhin–Turaev TQFT and modular-functor factorization
Lemma
Modular Hom state spaces have exact sewing inverses and the dimension formula
Corollary
RT torus operators and the state-space dimension formula
Sources
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.13–8.14
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§3.1–3.2
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.4 and 8.13
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§2.2 and 3.1
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.3
- Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §3.3
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Lemma 8.14.1
- Etingof, Nikshych, and Ostrik, On Fusion Categories, §2.1
- Teruji Thomas, The Maslov index as a quadratic space
- Kevin Walker, On Witten's 3-manifold invariants, §18
- Teruji Thomas, The Maslov index as a quadratic space, §§2, 5, 7
- Allen Hatcher, Algebraic Topology, Theorem 3.43 and §3.3
- J. Peter May, A Concise Course in Algebraic Topology, Chapter 21 §4
- Walker, On Witten’s 3-Manifold Invariants, §§1 and 18
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§4.1 and 5.7
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §7.21
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Example 8.13.5
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.1
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §5.1
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §8.15
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Propositions 8.13.8 and 8.13.10
- Benson Farb and Dan Margalit, A Primer on Mapping Class Groups, author draft Version 5.0, 4 January 2011
- Jiayong Li and Jordan Alan Watts, The orientation-preserving diffeomorphism group of S2 deforms to SO(3) smoothly, arXiv version 2
- John Milnor, Topology from the Differentiable Viewpoint
- Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces
- Allen Hatcher, Spectral Sequences in Algebraic Topology, Chapter 1
- Daniel S. Freed, Bordism: Old and New
- Justin Roberts, Kirby calculus in manifolds with boundary
- Jean Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, §I.3
- András Juhász, Defining and classifying TQFTs via surgery, §step 1.1.2–2.3
- András Juhász, Defining and classifying TQFTs via surgery, §§2.2–2.3
- Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, Chapter 2
- Justin Roberts, Kirby calculus in manifolds with boundary, Theorem 1 and Stage 1
- Thomas Kerler, Bridged Links and Tangle Presentations of Cobordism Categories, §2.2.2
- Robion Kirby, A Calculus for Framed Links in S3, Inventiones 45 (1978), 35–56
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Proposition 8.14.2
- Etingof, Nikshych, and Ostrik, On Fusion Categories, §2.2
- Turaev, Quantum Invariants of Knots and 3-Manifolds, I §§3.1–3.5 and 4.9
- Virelizier, Kirby elements and quantum invariants, §§1.5–1.6, Theorem 2.5 and Theorem 3.4
- Turaev, Quantum Invariants of Knots and 3-Manifolds, I §§3.5 and 4.9
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.14–8.15
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Corollaries 8.14.4–8.14.5
- Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §§1–5
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§3.3–3.4
- Turaev, Quantum Invariants of Knots and 3-Manifolds, I §3.5, II §3.3, IV §§2.2 and 2.7
- Bakalov–Kirillov, Lectures on Tensor Categories and Modular Functor, Lemma 3.1.5
- Turaev, Quantum Invariants of Knots and 3-Manifolds, IV §§1.4, 1.7–1.8 and 2.1–2.9
- Bakalov–Kirillov, Lectures on Tensor Categories and Modular Functor, §4.4
- Turaev, Quantum Invariants of Knots and 3-Manifolds, IV §§3.5 and 4.1–4.3
- Walker, On Witten’s 3-manifold invariants, §18
- Wall, Non-additivity of the signature, Inventiones Mathematicae 7 (1969), 269–274
- Lambropoulou–Rourke, Markov’s theorem in 3-manifolds, §5.2, Definition 5.6, Remarks 5.7 and Theorem 5.8
- Lambropoulou–Rourke, Algebraic Markov equivalence for links in 3-manifolds, pp. 2–3
- Turaev, Quantum Invariants of Knots and 3-Manifolds, II §§2.3 and 3.1
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§5.1–5.7
- V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, Chapters II–IV
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§5.1–5.8
- Kevin Walker, On Witten's 3-manifold invariants, §§7 and 18–19
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.14 and 8.16
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§5.8 and 6.1