Alphabeta MathSupersymmetric Moduli, Seiberg–Witten Data and Duality Questions
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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
- Finite Averaging and Character-Theory Prerequisites
- 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, 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
- The Group Algebra and Representations of Finite Groups
Advanced QFT and Supersymmetry
Algebraic Topology
Classical Electromagnetism
Classical Electromagnetism — Mathematics
Combinatorics
Complex Analysis
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- 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
Fourier Analysis
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Banach-Space Differential Calculus and Banach Manifolds
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- Complex Norms and Realification
- Distributions Test Functions and Differentiation
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- Finite Dimensional Normed Spaces and Riesz Lemma
- Fourier Transform Convolution and Approximate Identities
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Schwartz Space and the Plancherel Theorem
- Spectral Measures and Borel Functional Calculus
- Tempered Distributions and the Fourier Transform
- The Analytic Hahn Banach Theorem
- Unbounded Self Adjoint Operators and Stones Theorem
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
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- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Non-relativistic Quantum Mechanics — Mathematics
Partial Differential Equations
- Euclidean Surface Measure, Divergence, and Green Identities
- Fundamental Solutions Newtonian Potentials and Green Functions
- Harmonic Functions and Mean Values in Rn
- Interior and Boundary Sobolev Elliptic Regularity
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- Maximum Principles Harnack and Liouville in Rn
- Poisson Problems and Interior Harmonic Estimates
- Schauder and Lᵖ Elliptic Estimates
- Smooth Approximation and Sobolev Extension
- Sobolev Poincare and Morrey Inequalities
- Sobolev Traces and Zero Boundary Values
- Weak Derivatives and Sobolev Spaces
- Weak Elliptic Maximum Principles and Holder Regularity
Quantum Field Theory
Quantum Field Theory Mathematics
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- 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
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- 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
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Regular Surfaces and Surface Integrals
- 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 Divergence Theorem and Classical Stokes
- The Exponential Function
- 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
- Trigonometric and Oscillatory Examples in One Variable
- Volumes of Elementary Solids and Solids of Revolution
Set Theory
Special Relativity
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Ascoli–Arzelà Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Postulate
Independent pure N=2 SU(2) ultraviolet vector-multiplet datum
Definition
Classical and quantum supersymmetric vacuum-moduli data
Definition
Pure N=2 SU(2) Coulomb-branch and period data
Definition
Local dual U(1) monopole-Higgs effective data
Definition
Finite-energy dual-Higgs flux-tube sector
Postulate
Source-proposed pure SU(2) N=2 Seiberg–Witten low-energy data
Physics theorem
N=2 SU(2) central-charge mass bound from Seiberg–Witten periods
Physics theorem
Flux quantization and positive tension for an admitted dual vortex
Physics theorem
Conditional monopole Higgs vacuum after a small N=1 perturbation
Physics theorem
Critical monopole truncation: BPS bound, minimizing tubes and nonlinear obstruction
Definition
Nonlinear local dual-Higgs magnetostatic flux cylinders
Physics theorem
Nonlinear local electric-flux confinement from planar energy infima
Sources
- N. Seiberg and E. Witten, Electric-Magnetic Duality, Monopole Condensation, and Confinement in N=2 Supersymmetric Yang-Mills Theory
- David Tong, Supersymmetric Field Theory
- Adel Bilal, Duality in N=2 SUSY SU(2) Yang-Mills Theory
- David Tong, TASI Lectures on Solitons: Instantons, Monopoles, Vortices and Kinks
- David Tong, TASI Lectures on Solitons, Lecture 3: Vortices
- David Tong, TASI Lectures on Solitons, Lecture3 Vortices