Vertex Operator Algebras and the Monster (eBook)

Front Cover 1
Vertex Operator Algebras and the Monster 4
Copyright Page 5
Contents 8
Preface 12
Introduction 16
Notational Conventions 52
Chapter 1. Lie Algebras 56
1.1. Algebras 57
1.2. Modules 59
1.3. Algebra Constructions 61
1.4. Module Constructions 64
1.5. Induced Modules 66
1.6. Affine Lie Algebras 72
1.7. Heisenberg Algebras 76
1.8. Contravariant Forms 81
1.9. The Virasoro Algebra 86
1.10. Graded Dimension 97
Chapter 2. Formal Calculus 102
2.1. Formal Series 103
2.2. Derivations 109
2.3. Affine Lie Algebras via Formal Variables 113
Chapter 3. Realizations of l(2)ˆ by Twisted Vertex Operators 116
3.1. The Affine Lie Algebra SI(2)ˆ 117
3.2. The Twisted Vertex Operators Xz+½ (a, z) 122
3.3. Normal Ordering 128
3.4. Some Commutators 131
3.5. Irreducible Representations of SI(2)ˆ[O2] 136
Chapter 4. Realizations of SI(2)ˆ by Untwisted Vertex Operators 138
4.1. The Untwisted Vertex Operators Xz(a, z) 139
4.2. Normal Ordering 144
4.3. Some Commutators 147
4.4. Irreducible Representations of SI(2)ˆ and SI(2)ˆ[01] 151
4.5. Isomorphism of Two Constructions 152
Chapter 5. Central Extensions 156
5.1. 2-Cocycles 157
5.2. Commutator Maps 159
5.3. Extraspecial 2-Groups 162
5.4. Automorphisms of Central Extensions 166
5.5 . Representations of Central Extensions 172
Chapter 6. The Simple Lie Algebras An, Dn, En 176
6.1. Lattices 177
6.2. A Class of Lie Algebras 181
6.3. The Cases An, Dn, En 188
6.4. A Group of Automorphisms of g 192
Chapter 7. Vertex Operator Realizations of Ân, Dn, Ên 198
7.1. The Untwisted Vertex Operators X,(a, z 199
7.2. Construction of Ân, Dn, Ên 206
7.3. The Twisted Vertex Operators Xz+½(a, z) 212
7.4. Construction of Ân[0], Dn[0], Ê[0] 216
Chapter 8. General Theory of Untwisted Vertex Operators 226
8.1. Expansions of Zero 230
8.2. Exponentials of Derivations 233
8.3. Projective Changes of Variable and Higher Derivatives of Composite Functions 237
8.4. Commutators of Untwisted Vertex Operators 245
8.5. General Vertex Operators 253
8.6. Commutators of General Vertex Operators 259
8.7. The Virasoro Algebra Revisited 271
8.8. The Jacobi Identity 281
8.9. Cross-brackets and Commutative Affinization 291
8.10. Vertex Operator Algebras and the Rationality, Commutativity and Associativity Properties 299
Chapter 9. General Theory of Twisted Vertex Operators 310
9.1. Commutators of Twisted Vertex Operators 311
9.2. General Twisted Vertex Operators 318
9.3. Commutators of General Twisted Vertex Operators 330
9.4. The Virasoro Algebra: Twisted Construction Revisited 342
9.5. The Jacobi Identity and Cross-Brackets: Twisted Case 347
Chapter 10. The Moonshine Module 352
10.1. The Golay Code 353
10.2. The Leech Lattice 357
10.3. The Moonshine Module Vh and the Griess Algebra B 366
10.4. The Group C and Its Actions on Vh and on B 374
10.5. The Graded Character of the C-Module Vh 384
Chapter 11. Triality 396
11.1. The Setting 397
11.2. Construction of s1:VL0.VL0 405
11.3. Construction of s1:V1L0 . VL1 411
11.4 Construction of s1:V1L1.V1L1 419
11.5. Summary 426
Chapter 12. The Main Theorem 428
12.1 The Main Setting 429
12.2. The Extra Automorphism s 442
12.3. The Monster M and the Statement of the Main Theorem 455
12.4. The M-Invariant Q-Form VkQ 463
12.5. The M-Invariant Positive Definite Hermitian Form 465
Chapter 13. Completion of the Proof 472
13.1. Reduction to Two Lemmas 473
13.2. Groups Acting on WL : Proof of Lemma 13.1.2 482
13.3. Action of D6/4 on W£ 489
13.4. Groups Acting on W£ 496
13.5. Some Group Cohomology 506
13.6. A Splitting of a Sequence: Proof of Lemma 13.4.7 513
Appendix: Complex Realization of Vertex Operator Algebras 516
A.1. Linear Algebra of Infinite Direct Sums 517
A.2. The Vertex Operator Algebra VL 521
A.3. Relation to the Formal Variable Approach 532
Bibliography 538
List of Frequently Used Symbols 548
Index 554
Pure and Applied Mathematics 558