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Mathematics
Vertex Algebras
1. Foundations and Motivation
2. Mathematical Preliminaries
3. Basic Structure of Vertex Algebras
4. Axioms and Fundamental Properties
5. Core Operations and Concepts
6. Fundamental Examples
7. Affine Vertex Algebras
8. Virasoro and Conformal Structure
9. Module Theory and Representations
10. Advanced Constructions
11. Intertwining Operators and Fusion
12. Zhu's Theory
13. Vertex Operator Superalgebras
14. Rationality and Finiteness
15. Moonshine and Connections
16. Computational Aspects
17. Applications and Related Areas
5.
Core Operations and Concepts
5.1.
Operator Product Expansion
5.1.1.
OPE Definition
5.1.2.
Singular Parts
5.1.3.
Regular Parts
5.1.4.
Pole Structure
5.1.5.
Residue Calculations
5.1.6.
OPE Coefficients
5.2.
Products of Operators
5.2.1.
n-th Products
5.2.2.
Product Formulas
5.2.3.
Calculation Methods
5.2.4.
Associativity Properties
5.3.
Normal Ordering
5.3.1.
Definition and Motivation
5.3.2.
Normal Ordered Products
5.3.3.
Wick's Theorem
5.3.4.
Contraction Rules
5.3.5.
Applications to Free Fields
5.4.
Commutator Relations
5.4.1.
Mode Commutators
5.4.2.
Field Commutators
5.4.3.
Dong's Lemma
5.4.4.
Skew-Symmetry
5.4.5.
Borcherds Identity
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4. Axioms and Fundamental Properties
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6. Fundamental Examples