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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
6.
Fundamental Examples
6.1.
Commutative Vertex Algebras
6.1.1.
Definition and Structure
6.1.2.
Polynomial Algebras
6.1.3.
Trivial OPE Structure
6.1.4.
Simple Examples
6.2.
Heisenberg Vertex Algebra
6.2.1.
The Heisenberg Lie Algebra
6.2.1.1.
Canonical Commutation Relations
6.2.1.2.
Central Extension
6.2.1.3.
Generators and Structure
6.2.2.
Fock Space Construction
6.2.2.1.
Vacuum Vector
6.2.2.2.
Creation Operators
6.2.2.3.
Annihilation Operators
6.2.2.4.
Fock Space Basis
6.2.3.
Vertex Operator Realization
6.2.3.1.
Explicit Vertex Operators
6.2.3.2.
Mode Expansions
6.2.3.3.
OPE Calculations
6.2.3.4.
Normal Ordering Relations
6.3.
Lattice Vertex Algebras
6.3.1.
Even Lattice Theory
6.3.1.1.
Lattice Definitions
6.3.1.2.
Bilinear Forms
6.3.1.3.
Root Systems
6.3.1.4.
Lattice Automorphisms
6.3.2.
Vertex Operator Construction
6.3.2.1.
Lattice Vertex Operators
6.3.2.2.
Cocycle Factors
6.3.2.3.
Twisted Group Algebra
6.3.2.4.
State-Field Correspondence
6.3.3.
Examples of Lattice Algebras
6.3.3.1.
Root Lattice Constructions
6.3.3.2.
Weight Lattice Cases
6.3.3.3.
Exceptional Lattices
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5. Core Operations and Concepts
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7. Affine Vertex Algebras