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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
7.
Affine Vertex Algebras
7.1.
Affine Kac-Moody Algebras
7.1.1.
Loop Algebra Construction
7.1.2.
Central Extension
7.1.3.
Cartan Subalgebra
7.1.4.
Root System Structure
7.1.5.
Weyl Group Action
7.2.
Vacuum Representations
7.2.1.
Highest Weight Theory
7.2.2.
Verma Modules
7.2.3.
Irreducible Quotients
7.2.4.
Character Formulas
7.3.
Vertex Algebra Structure
7.3.1.
State-Field Correspondence
7.3.2.
Current Fields
7.3.3.
OPE Relations
7.3.4.
Mode Algebras
7.4.
Sugawara Construction
7.4.1.
Energy-Momentum Tensor
7.4.2.
Sugawara Operators
7.4.3.
Virasoro Relations
7.4.4.
Central Charge Formula
7.4.5.
Coset Constructions
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6. Fundamental Examples
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8. Virasoro and Conformal Structure