Alphabeta MathCFT Liouville And Noncompact CFT
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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
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
Complex Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Complex Subspace Realification
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Dual Spaces Adjoint Operators and Annihilators
- Gelfand Theory and Commutative C Star Algebras
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Quantum Algebra and Representation Theory
Quantum Field Theory Mathematics
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- 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 Fundamental Theorems of Calculus
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Gaussian free field on the Riemann sphere in the Liouville convention
Lemma
The Liouville constant-mode integral and total-charge bound
Definition
Spherical Gaussian multiplicative chaos interaction data
Lemma
L2 limits of spherical Gaussian multiplicative chaos test functionals
Definition
Probabilistic Liouville CFT on the sphere
Lemma
Absolute convergence of the Liouville Upsilon defining integral
Definition
Spherical Virasoro conformal blocks from Shapovalov matrices
Definition
Radial Hilbert-space data and Liouville Hamiltonian
Definition
DOZZ structure constant in the probabilistic Liouville convention
Lemma
Circle Fourier GMC positive and inverse moments
Lemma
Local disk GMC and uniform anchored inverse moments
Theorem
Radial Liouville Poisson spectral resolution
Lemma
Liouville charged Ward identities and radial overlaps
Theorem
Probabilistic Liouville three-point identity and its primary boundary value
Theorem
Probabilistic/bootstrap equality for Liouville four-point functions in a proved domain
Sources
- Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Conformal Bootstrap in Liouville Theory, arXiv:2005.11530v1
- Rémi Rhodes and Vincent Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, arXiv:1602.07323v1
- Guillarmou, Kupiainen, Rhodes and Vargas, Conformal Bootstrap in Liouville Theory
- Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Integrability of Liouville theory: proof of the DOZZ formula
- Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Conformal Bootstrap in Liouville Theory
- Antti Kupiainen, Rémi Rhodes, and Vincent Vargas, Integrability of Liouville theory: proof of the DOZZ formula, arXiv:1707.08785