Supersymmetries and Infinite-Dimensional Algebras (eBook)

Front Cover 1
Supersymmetries and Infinite-Dimensional Algebras 4
Copyright Page 5
Table of Contents 10
Preface 6
Contents of Volume I 16
Contents of Volume II 20
Part D: Lie Superalgebras, Lie Supergroups and their Applications 24
Chapter 20. Introduction to Superalgebras and Supermatrices 26
1 The notion of grading 26
2 Associative superalgebras 28
3 Grassmann algebras 30
4 Supermatrices 36
Chapter 21. General Properties of Lie Superalgebras 44
1 Lie superalgebras introduced 44
2 Definitions and immediate consequences 45
3 Subalgebras, direct sums and homomorphisms of Lie superalgebras 54
4 Graded representations of Lie superalgebras 58
5 The adjoint representation and the Killing form of a Lie superalgebra 64
Chapter 22. Superspace and Lie Supergroups 68
1 Grassmann variables as coordinates 68
2 Analysis on superspace 69
3 Linear Lie supergroups 91
Chapter 23. The Poincaré Superalgebras and Supergroups 102
1 Introduction 102
2 The N = 1, D = 4 Poincaré superalgebra and supergroup 103
3 Extended Poincaré superalgebras and Poincaré supergroups for D = 4 130
4 The Poincaré superalgebras and supergroups for Minkowski space-times of general dimension D 141
5 Irreducible representations of the unextended D = 4 Poincaré superalgebra 149
6 Irreducible representations of the extended D = 4 Poincaré superalgebras 161
7 Irreducible representations of the Poincaré superalgebras for general space-time dimensions 172
Chapter 24. Poincaré Supersymmetric Fields 184
1 Supersymmetric field theory 184
2 Supersymmetric multiplets 185
3 Superfields 204
4 Supersymmetric gauge theories 215
5 Spontaneous symmetry breaking 237
Chapter 25. Simple Lie Superalgebras 242
1 An outline of the presentation 242
2 The definition of a simple Lie superalgebra and some immediate consequences 243
3 Classical simple Lie superalgebras 246
4 Graded representations of basic classical simple complex Lie superalgebras 268
5 The classical simple real Lie superalgebras 290
6 The conformal, de Sitter and anti-de Sitter superalgebras 298
Part E: Infinite-Dimensional Lie Algebras and Superalgebras and their Applications 302
Chapter 26. The Structure of Kac-Moody Algebras 304
1 Introduction to infinite-dimensional Lie algebras 304
2 Construction of Kac-Moody algebras 305
3 Properties of general Kac-Moody algebras 312
4 Types of complex Kac-Moody algebras 322
5 Affine Kac-Moody algebras 330
6 Kac-Moody superalgebras 358
Chapter 27. Representations of Kac-Moody Algebras 362
1 Highest weight representations of general Kac-Moody algebras 362
2 Highest weight representations of affine Kac-Moody algebras 365
3 Character formulae 374
4 The vertex construction of the basic representation of a simply laced untwisted affine Kac-Moody algebra 376
5 Representations of untwisted affine Kac-Moody algebras in terms of fermion creation and annihilation operators 385
Chapter 28. The Virasoro Algebra and Superalgebras 392
1 The conformal algebras 392
2 Representations of the Virasoro algebra 396
3 Some constructions of highest weight representations of the Virasoro algebra 399
4 Virasoro superalgebras 405
Chapter 29. Algebraic Aspects of the Theory of Strings and Superstrings 412
1 Introduction 412
2 The bosonic string 412
3 The spinning string of Ramond, Neveu and Schwarz 434
4 The superstring of Green and Schwarz 440
5 The heterotic string 460
6 Further developments 469
Appendices 472
Appendix K: Proofs of Certain Theorems on Supermatrices and Lie Superalgebras 474
1 Proofs of Theorems I and IV of Chapter 20, Section 4 474
2 Proof of Theorem I of Chapter 21, Section 4 480
3 Proof of Theorem III of Chapter 21, Section 5 482
4 Proofs of Theorems II, III, IV and V of Chapter 25, Section 2 483
5 Proofs of Theorems VI, VII, VIII, IX, XVI, XX, XXII and XXIII of Chapter 25, Sections 3(a), 3(b) and 3(c) 486
6 Proofs of Theorems III and IV of Chapter 25, Section 4(a) 493
Appendix L: Clifford Algebras 498
1 The Clifford algebras of D-dimensional space–times 498
2 Irreducible representations for the case in which D is even 500
3 Irreducible representations for the case in which D is odd 512
4 Connections between representations of the D-dimensional Minkowski Clifford algebra and those of the (D — 2)-dimensional Euclidean Clifford algebra 519
5 A matrix identity for the D = 4 Minkowski Clifford algebra 524
Appendix M: Properties of the Classical Simple Complex Lie Superalgebras 526
1 The basic type 1 classical simple complex Lie superalgebras A(r/s), r > s = 0
2 The basic type I classical simple complex Lie superalgebras A(r/r),r = 1 532
3 The basic type II classical simple complex Lie superalgebras B(r/s), r = 0,s = 1 536
4 The basic type I classical simple complex Lie superalgebras C(s), s = 2 544
5 The basic type II classical simple complex Lie superalgebras D(r/s), r = 2, s = 1 548
6 The basic type II classical simple complex Lie superalgebras D(2/1 a), with a a complex parameter taking all values other than 0, — 1 and 8
7 The basic type II classical simple complex Lie superalgebra F(4) 560
8 The basic type II classical simple complex Lie superalgebra G(3) 565
9 The strange type I classical simple complex Lie superalgebras P(r), r = 2 568
10 The strange type II classical simple complex Lie superalgebras Q(r), r = 2 570
Appendix N: Properties of the Complex Affine Kac-Moody Algebras 574
1 The complex untwisted affine Kac-Moody algebra A1(1) 574
2 The complex untwisted affine Kac-Moody algebras Al(1), l = 2 575
3 The complex untwisted affine Kac-Moody algebras Bl(1), l = 3 576
4 The complex untwisted affine Kac-Moody algebras C(1), l = 2 577
5 The complex untwisted affine Kac-Moody algebras D(1)l' l = 4 579
6 The complex untwisted affine Kac-Moody algebra E(1)6 581
7 The complex untwisted affine Kac-Moody algebra E(1)7 582
8 The complex untwisted affine Kac-Moody 
583 
9 The complex untwisted affine Kac-Moody algebra F4(1) 584
10 The complex untwisted affine Kac-Moody algebra G(1)2 585
11 The complex twisted affine Kac-Moody algebra .(2)2 586
12 The complex twisted affine Kac-Moody algebras .(2)2l' l = 2 587
13 The complex twisted affine Kac-Moody algebras A(2)2l-1' l = 3 589
14 The complex twisted affine Kac-Moody algebras D(2)l+1' l = 2 591
15 The complex twisted affine Kac-Moody algebra E6(2) 593
16 The complex twisted affine Kac-Moody algebra D4(3) 594
Appendix O: Proofs of Certain Theorems on Kac-Moody and Virasoro Algebras 598
1 Proofs of Theorems I, III and IV of Chapter 27, Section 2 598
2 Proofs of Theorems I and II of Chapter 27, Section 4 602
3 Proof of Theorem I of Chapter 27, Section 5 612
4 Proofs of Theorems I and 11 of Chapter 28, Section 3 614
References 622
Subject Index 640