Alphabeta MathZero-FI Regular ADHM Data and Framed ASD Connections
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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
- Finite Fields and Cyclotomic Extensions
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
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- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
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- Normal Subgroups and Quotient Groups
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- 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
- 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
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- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
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- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
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- Local Coefficients, Twisted Homology, and Duality
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- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
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Category Theory
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Combinatorics
Commutative Algebra
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Complex Analysis
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Differential Geometry
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- Sard Theorem and Transversality
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- Smooth Vector Bundles and Sections
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- Tensor Fields Exterior Algebra and Differential Forms
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Differential Topology
Fourier Analysis
Functional Analysis
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Homological Algebra
- Chain Complexes and Homology
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Lie Theory
Linear Algebra
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Measure Theory
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Number Theory
Numerical Differential Equations and Stochastic Dynamics
Partial Differential Equations
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- Weak Derivatives and Sobolev Spaces
Quantum Field Theory Mathematics
- ADHM Linear Data, Positive-FI Quotients and Framed Monads
- Coupled Penrose vanishing on the round four-sphere
- Faithful flatness of finite-type algebraic-to-analytic local rings
- Finite-dimensional coherent cohomology on compact projective space
- Instanton Charge, Self-Dual Bound, and BPST Example
- Local coherent acyclicity on sufficiently small polydiscs
- Non-Abelian Principal-Connection Forms and Curvature
- Oka coherence and local coherent free resolutions
- Positive real linear monad normal forms
- Projective Dolbeault comparison and scalar structure-sheaf cohomology
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
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- Convexity
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- Foundations of the Real Numbers for Analysis
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- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
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- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Real Gamma and Beta Functions
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
- Volumes of Elementary Solids and Solids of Revolution
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Blowups, Exceptional Divisors, and Strict Transforms
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
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
- Well-Founded Relations, Rank, and the Cumulative Hierarchy
Topology
- Classification of Covering Spaces
- Compactness
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- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- 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
Donaldson's ADHM matrix equations in the source convention
Lemma
Local holomorphic frames for an integrable smooth Dolbeault operator
Lemma
Entrywise conjugation does not match the raw ADHM real moment map
Definition
Zero-FI regular ADHM and framed twistor data
Lemma
Determinant flatness and charge normalization for framed U(N) ASD data
Lemma
Positive scalar term eliminates the coupled Laplace kernel on round S4
Lemma
Smooth framed bundle of sections over real twistor lines
Lemma
Exact raw ADHM block, real monad and stereographic ASD projection
Theorem
Conditional framed Penrose–Ward correspondence on the round four-sphere
Lemma
Chern character and charge of the normalized twistor monad
Lemma
Conditional Penrose vanishing from positive scalar curvature
Lemma
Conditional twistor vanishing and normalized monad reconstruction
Lemma
Real monad normalization, regularity, framing and charge
Theorem
Conditional zero-FI regular ADHM and framed ASD correspondence
Lemma
Rank-one regular zero-FI ADHM and ordinary ASD boundary
Sources
- Simon K. Donaldson, The ADHM construction of Yang-Mills instantons
- Jean-Louis Koszul and Bernard Malgrange, Sur certaines structures fibrées complexes
- Hiraku Nakajima and Kota Yoshioka, Lectures on Instanton Counting
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Annales de l’Institut Fourier 6 (1956), 1–42
- Nikita A. Nekrasov, Seiberg–Witten Prepotential from Instanton Counting
- Simon K. Donaldson, The ADHM construction of Yang–Mills instantons
- J. H. Rawnsley, On the Atiyah-Hitchin-Drinfeld-Manin Vanishing Theorem for Cohomology Groups of Instanton Bundles