Alphabeta MathHall Integration, Stability, DT/BPS Invariants, and Wall Crossing: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- 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, 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 Field of Fractions and Localisation
Algebraic Geometry
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- 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
Commutative Algebra
Homological Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- 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
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- 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 Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- 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
2 · Logical flowchart
3 · Items
Example
One vertex with no arrows gives the quantum dilogarithm
Example
The A₂ quiver gives the quantum-dilogarithm pentagon
Counterexample
Omitting strictly semistable objects breaks Hall factorization at the A₂ wall
Counterexample
Commutativizing the target erases the A₂ wall-crossing identity
Sources
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §2.2
- Olivier Schiffmann, Lectures on Hall Algebras, Example 3.14
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §4.6
- Olivier Schiffmann, Lectures on Hall Algebras, Example 3.15
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§4.4–4.6
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §§2 and 4