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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
3.
Basic Structure of Vertex Algebras
3.1.
The State Space
3.1.1.
Vector Space Structure
3.1.2.
Grading Properties
3.1.3.
Filtration Concepts
3.1.4.
Basis Considerations
3.1.5.
Dimension Theory
3.2.
State-Field Correspondence
3.2.1.
The Vertex Operator Map
3.2.2.
Field Notation Y(a,z)
3.2.3.
Mode Expansions
3.2.4.
Locality Properties
3.2.5.
Field Algebra Structure
3.3.
Fundamental Elements
3.3.1.
The Vacuum Vector
3.3.1.1.
Definition and Uniqueness
3.3.1.2.
Vacuum Properties
3.3.1.3.
Role in Constructions
3.3.2.
The Translation Operator
3.3.2.1.
Definition and Action
3.3.2.2.
Derivation Properties
3.3.2.3.
Commutation Relations
3.3.2.4.
Infinitesimal Translations
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2. Mathematical Preliminaries
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4. Axioms and Fundamental Properties