March Supersymmetries and Infinite-Dimensional Algebras

1. Auflage 2013
ISBN: 978-1-4832-8837-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 3, 628 Seiten, Web PDF

Reihe: Techniques of Physics

Recent devopments, particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate.

Autoren/Hrsg.

March, N. H.

Inhaltsverzeichnis

1;Front Cover;1
2;Supersymmetries and Infinite-Dimensional Algebras;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;6
6;Contents of Volume I;16
7;Contents of Volume II;20
8;Part D: Lie Superalgebras, Lie Supergroups and their Applications;24
8.1;Chapter 20. Introduction to Superalgebras and Supermatrices;26
8.1.1;1 The notion of grading;26
8.1.2;2 Associative superalgebras;28
8.1.3;3 Grassmann algebras;30
8.1.4;4 Supermatrices;36
8.2;Chapter 21. General Properties of Lie Superalgebras;44
8.2.1;1 Lie superalgebras introduced;44
8.2.2;2 Definitions and immediate consequences;45
8.2.3;3 Subalgebras, direct sums and homomorphisms of Lie superalgebras;54
8.2.4;4 Graded representations of Lie superalgebras;58
8.2.5;5 The adjoint representation and the Killing form of a Lie superalgebra;64
8.3;Chapter 22. Superspace and Lie Supergroups;68
8.3.1;1 Grassmann variables as coordinates;68
8.3.2;2 Analysis on superspace;69
8.3.3;3 Linear Lie supergroups;91
8.4;Chapter 23. The Poincaré Superalgebras and Supergroups;102
8.4.1;1 Introduction;102
8.4.2;2 The N = 1, D = 4 Poincaré superalgebra and supergroup;103
8.4.3;3 Extended Poincaré superalgebras and Poincaré supergroups for D = 4;130
8.4.4;4 The Poincaré superalgebras and supergroups for Minkowski space-times of general dimension D;141
8.4.5;5 Irreducible representations of the unextended D = 4 Poincaré superalgebra;149
8.4.6;6 Irreducible representations of the extended D = 4 Poincaré superalgebras;161
8.4.7;7 Irreducible representations of the Poincaré superalgebras for general space-time dimensions;172
8.5;Chapter 24. Poincaré Supersymmetric Fields;184
8.5.1;1 Supersymmetric field theory;184
8.5.2;2 Supersymmetric multiplets;185
8.5.3;3 Superfields;204
8.5.4;4 Supersymmetric gauge theories;215
8.5.5;5 Spontaneous symmetry breaking;237
8.6;Chapter 25. Simple Lie Superalgebras;242
8.6.1;1 An outline of the presentation;242
8.6.2;2 The definition of a simple Lie superalgebra and some immediate consequences;243
8.6.3;3 Classical simple Lie superalgebras;246
8.6.4;4 Graded representations of basic classical simple complex Lie superalgebras;268
8.6.5;5 The classical simple real Lie superalgebras;290
8.6.6;6 The conformal, de Sitter and anti-de Sitter superalgebras;298
9;Part E: Infinite-Dimensional Lie Algebras and Superalgebras and their Applications;302
9.1;Chapter 26. The Structure of Kac-Moody Algebras;304
9.1.1;1 Introduction to infinite-dimensional Lie algebras;304
9.1.2;2 Construction of Kac-Moody algebras;305
9.1.3;3 Properties of general Kac-Moody algebras;312
9.1.4;4 Types of complex Kac-Moody algebras;322
9.1.5;5 Affine Kac-Moody algebras;330
9.1.6;6 Kac-Moody superalgebras;358
9.2;Chapter 27. Representations of Kac-Moody Algebras;362
9.2.1;1 Highest weight representations of general Kac-Moody algebras;362
9.2.2;2 Highest weight representations of affine Kac-Moody algebras;365
9.2.3;3 Character formulae;374
9.2.4;4 The vertex construction of the basic representation of a simply laced untwisted affine Kac-Moody algebra;376
9.2.5;5 Representations of untwisted affine Kac-Moody algebras in terms of fermion creation and annihilation operators;385
9.3;Chapter 28. The Virasoro Algebra and Superalgebras;392
9.3.1;1 The conformal algebras;392
9.3.2;2 Representations of the Virasoro algebra;396
9.3.3;3 Some constructions of highest weight representations of the Virasoro algebra;399
9.3.4;4 Virasoro superalgebras;405
9.4;Chapter 29. Algebraic Aspects of the Theory of Strings and Superstrings;412
9.4.1;1 Introduction;412
9.4.2;2 The bosonic string;412
9.4.3;3 The spinning string of Ramond, Neveu and Schwarz;434
9.4.4;4 The superstring of Green and Schwarz;440
9.4.5;5 The heterotic string;460
9.4.6;6 Further developments;469
10;Appendices;472
10.1;Appendix K: Proofs of Certain Theorems on Supermatrices and Lie Superalgebras;474
10.1.1;1 Proofs of Theorems I and IV of Chapter 20, Section 4;474
10.1.2;2 Proof of Theorem I of Chapter 21, Section 4;480
10.1.3;3 Proof of Theorem III of Chapter 21, Section 5;482
10.1.4;4 Proofs of Theorems II, III, IV and V of Chapter 25, Section 2;483
10.1.5;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
10.1.6;6 Proofs of Theorems III and IV of Chapter 25, Section 4(a);493
10.2;Appendix L: Clifford Algebras;498
10.2.1;1 The Clifford algebras of D-dimensional space–times;498
10.2.2;2 Irreducible representations for the case in which D is even;500
10.2.3;3 Irreducible representations for the case in which D is odd;512
10.2.4;4 Connections between representations of the D-dimensional Minkowski Clifford algebra and those of the (D — 2)-dimensional Euclidean Clifford algebra;519
10.2.5;5 A matrix identity for the D = 4 Minkowski Clifford algebra;524
10.3;Appendix M: Properties of the Classical Simple Complex Lie Superalgebras;526
10.3.1;1 The basic type 1 classical simple complex Lie superalgebras A(r/s), r > s = 0;526
10.3.2;2 The basic type I classical simple complex Lie superalgebras A(r/r),r = 1;532
10.3.3;3 The basic type II classical simple complex Lie superalgebras B(r/s), r = 0,s = 1;536
10.3.4;4 The basic type I classical simple complex Lie superalgebras C(s), s = 2;544
10.3.5;5 The basic type II classical simple complex Lie superalgebras D(r/s), r = 2, s = 1;548
10.3.6;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;555
10.3.7;7 The basic type II classical simple complex Lie superalgebra F(4);560
10.3.8;8 The basic type II classical simple complex Lie superalgebra G(3);565
10.3.9;9 The strange type I classical simple complex Lie superalgebras P(r), r = 2;568
10.3.10;10 The strange type II classical simple complex Lie superalgebras Q(r), r = 2;570
10.4;Appendix N: Properties of the Complex Affine Kac-Moody Algebras;574
10.4.1;1 The complex untwisted affine Kac-Moody algebra A1(1);574
10.4.2;2 The complex untwisted affine Kac-Moody algebras Al(1), l = 2;575
10.4.3;3 The complex untwisted affine Kac-Moody algebras Bl(1), l = 3;576
10.4.4;4 The complex untwisted affine Kac-Moody algebras C(1), l = 2 ;577
10.4.5;5 The complex untwisted affine Kac-Moody algebras D(1)l' l = 4;579
10.4.6;6 The complex untwisted affine Kac-Moody algebra E(1)6;581
10.4.7;7 The complex untwisted affine Kac-Moody algebra E(1)7;582
10.4.8;8 The complex untwisted affine Kac-Moody

algebra E8(1);583
10.4.9;9 The complex untwisted affine Kac-Moody algebra F4(1);584
10.4.10;10 The complex untwisted affine Kac-Moody algebra G(1)2;585
10.4.11;11 The complex twisted affine Kac-Moody algebra .(2)2;586
10.4.12;12 The complex twisted affine Kac-Moody algebras .(2)2l' l = 2;587
10.4.13;13 The complex twisted affine Kac-Moody algebras A(2)2l-1' l = 3;589
10.4.14;14 The complex twisted affine Kac-Moody algebras D(2)l+1' l = 2;591
10.4.15;15 The complex twisted affine Kac-Moody algebra E6(2);593
10.4.16;16 The complex twisted affine Kac-Moody algebra D4(3);594
10.5;Appendix O: Proofs of Certain Theorems on Kac-Moody and Virasoro Algebras;598
10.5.1;1 Proofs of Theorems I, III and IV of Chapter 27, Section 2;598
10.5.2;2 Proofs of Theorems I and II of Chapter 27, Section 4;602
10.5.3;3 Proof of Theorem I of Chapter 27, Section 5;612
10.5.4;4 Proofs of Theorems I and 11 of Chapter 28, Section 3;614
11;References;622
12;Subject Index;640