Theoretical Physics at the End of the Twentieth Century

by Saint-Aubin, Yvan; Vinet, Luc

• ISBN13: 9780387953113
• ISBN10: 0387953116
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: Springer Verlag

Summary

This volume provides a snapshot of topics engaging theoretical physicists at the end of the twentieth century and the beginning of the twenty-first. Based on seven of the courses given at the 1999 CRM Summer School in Banff, the chapters provide timely, diverse and exciting views of their fields. The contributions include:- Supersymmetric Yang-Mills theory and integrable systems (E. D'Hoker and D.H. Phong) - Branes, black holes and anti-De Sitter space (M.J. Duff)- Turbulence (K. Gawedzki)- Bose--Einstein condensation and coherent matter waves (A. Griffin) - Deformed Virasoro and elliptic algebras (S. Odake) - Mesoscopic physics (B. Simons, A. Atland) - QCD in extreme conditions (F. Wilczek) Young physicists will find in these chapters pedagogical introductions to subjects currently active in theoretical physics, and more seasoned physicists will find a chance to share the excitement of fields outside their immediate research interests.

Table of Contents

Series Preface	v
Preface	vii
Contributors	xvii
Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems	1	(126)
Eric D'Hoker
D.H. Phong
Introduction	1	(7)
Supersymmetry and the Standard Model	2	(2)
Supersymmetry and Unification of Forces	4	(3)
Supersymmetric Yang-Mills Dynamics	7	(1)
Supersymmetric Yang-Mills in 4 Dimensions	8	(20)
Supersymmetry Algebra	8	(1)
Massless Particle Representations	9	(1)
Massive Particle Representations	9	(2)
Field Contents of Supersymmetric Field Theories	11	(1)
N = 1 Supersymmetric Lagrangians	12	(1)
N = 1 Superfield Methods	13	(2)
Irreducible Superfields of N = 1	15	(3)
General N = 1 Susy Lagrangians via Superfields	18	(2)
Renormalizable N = 2,4 Susy Lagrangians	20	(1)
N = 2 Superfield Methods: Unconstrained Superspace	21	(3)
N = 2 Superfield Methods: Harmonic/Analytic Superspaces	24	(4)
Seiberg-Witten Theory	28	(19)
Wilson Effective Couplings and Actions	28	(2)
Holomorphicity and Nonrenormalization	30	(3)
Low Energy Dynamics of N = 2 Super-Yang-Mills	33	(1)
Particle and Field Contents	34	(2)
Form of the N = 2 Low Energy Effective Lagrangian	36	(3)
Physical Properties of the Prepotential	39	(1)
Electric-Magnetic Duality	40	(2)
Monodromy via Elliptic Curves for SU (2) Gauge Group	42	(2)
Physical Interpretation of Singularities	44	(1)
Hypergeometric Function Representation	45	(2)
More General Gauge Groups, Hypermultiplets	47	(10)
Model of Riemann Surfaces	48	(1)
Identifying Seiberg-Witten and Riemann Surface Data	49	(1)
SU(N) Gauge Algebras, Fundamental Hypermultiplets	50	(6)
Classical Gauge Algebras, Fundamental Hypermultiplets	56	(1)
Mechanical Integrable Systems	57	(13)
Lax Pairs with Spectral Parameter--Spectral Curves	58	(1)
The Toda Systems	59	(3)
The Calogero-Moser Systems for SU(N)	62	(2)
Relation between Calogero-Moser and Toda for SU(N)	64	(2)
Relations with KdV and KP Systems	66	(1)
Calogero-Moser Systems for General Lie Algebras	67	(1)
Scaling of Calogero-Moser to Toda for General Lie Algebras	68	(2)
Calogero-Moser Lax Pairs for General Lie Algebras	70	(17)
Lax Pairs with Spectral Parameter for Classical Lie Algebras	71	(2)
The General Ansatz	73	(3)
Lax Pairs for Untwisted Calogero-Moser Systems	76	(6)
Lax Pairs for Twisted Calogero-Moser Systems	82	(3)
Scaling Limits of Lax Pairs	85	(2)
Super-Yang-Mills and Calogero-Moser Systems	87	(10)
Correspondence of Seiberg-Witten and Integrable Systems	87	(2)
Calogero-Moser and Seiberg-Witten Theory for SU(N)	89	(1)
Four Fundamental Theorems	90	(5)
Partial Decoupling of Hypermultiplet, Product Groups	95	(2)
Calogero-Moser and Seiberg-Witten for General G	97	(6)
The General Case	97	(2)
Spectral Curves for Low Rank	99	(2)
Perturbative Prepotential for SO(2n)	101	(2)
References	103	(10)
Notations and Conventions	113	(2)
Spinors	113	(1)
Dirac Matrices in a Weyl Basis	114	(1)
Dirac Matrices in a Majorana Basis	114	(1)
Two-Component Spinors	114	(1)
Lie Algebra Theory	115	(4)
Elliptic Functions	119	(8)
Basic Definitions and Properties	119	(3)
Half and Third Period Functions	122	(1)
The Function Φ	123	(1)
The Functions Λ, Φ1, Φ2	124	(3)
Lectures on Branes, Black Holes, and Anti-de Sitter Space	127	(118)
M.J. Duff
Introduction	127	(13)
Supergravity, Supermembranes and M-Theory	127	(2)
The Kaluza-Klein Idea	129	(4)
The Field Content	133	(4)
The AdS/CFT Correspondence	137	(1)
Plan of the Lectures	138	(2)
Problems 1	140	(1)
Eleven-Dimensional Supergravity	140	(12)
Bosonic Field Equations	140	(1)
AdS4 x S7	140	(4)
Consistent Truncation to the Massless Modes	144	(2)
The Supermembrane Solution	146	(3)
AdS7 x S4	149	(1)
The Superfivebrane Solution	150	(2)
Problems 2	152	(1)
Type IIB Supergravity	152	(3)
Bosonic Field Equations	152	(1)
AdS5 x S5	153	(1)
The Self-Dual Superthreebrane Solution	154	(1)
Problems 3	155	(1)
The M2-Brane, D3-Brane and M5-Brane	155	(5)
The M2-Brane	155	(2)
The M5-Brane	157	(2)
The D3-Brane	159	(1)
Problems 4	159	(1)
ADS/CFT: The Membrane at the End of the Universe	160	(14)
Singletons Live on the Boundary	160	(2)
The Membrane as a Singleton: The Membrane/Supergravity Bootstrap	162	(3)
Doubletons and Tripletons Revisited	165	(1)
The Membrane at the End of the Universe	166	(3)
Near Horizon Geometry and p-Brane Aristocracy	169	(2)
Supermembranes with Fewer Supersymmetries. Skew-Whiffing	171	(2)
The Maldacena Conjecture	173	(1)
Problems 5	174	(1)
Anti-de Sitter Black Holes	174	(22)
Introduction	174	(3)
S5 Reduction of Type IIB Supergravity	177	(3)
D = 5 AdS Black Holes	180	(1)
Rotating D3-Brane	180	(3)
S7 Reduction of D = 11 Supergravity	183	(3)
D = 4 AdS Black Holes	186	(1)
Rotating M2-Brane	187	(1)
S4 reduction of D = 11 Supergravity	188	(2)
D = 7 AdS Black Holes	190	(1)
Rotating M5-Brane	191	(1)
Charge as Angular Momentum	192	(1)
Magnetic Black Holes	193	(1)
Kaluza-Klein States as Black Holes	194	(1)
Recent Developments	195	(1)
Problems 6	196	(1)
Solutions to Problems	196	(16)
Solutions 1	196	(2)
Solutions 2	198	(3)
Solutions 3	201	(3)
Solutions 4	204	(2)
Solutions 5	206	(1)
Solutions 6	207	(5)
References	212	(16)
The Lagrangian, Symmetries and Transformation Rules of D = 11 Supergravity	228	(2)
The Field Equations, Symmetries and Transformation Rules of Type IIB Supergravity	230	(3)
The Lagrangian, Symmetries and Transformation Rules of the M2-Brane	233	(1)
The Field Equations, Symmetries and Transformation Rules of the M5-Brane	234	(4)
The Lagrangian, Symmetries and Transformation Rules of the D3-Brane	238	(2)
D = 4, N = 2 Gauged Supergravity	240	(5)
Easy Turbulence	245	(32)
Krzysztof Gawedzki
Lecture 1 The Navier-Stokes Equations	245	(7)
Lecture 2 The Kolmogorov and Kraichnan-Batchelor Theories of Turbulence	252	(7)
Lecture 3 The Richardson Dispersion Law	259	(8)
Weakly Compressible Regime	265	(1)
Strongly Compressible Regime	265	(2)
Lecture 4 Cascades and Intermittency	267	(10)
BEC and the New World of Coherent Matter Waves	277	(30)
Allan Griffin
An Overview of Past and Recent Work	278	(7)
Some History before 1980	278	(3)
More Recent Developments (1980-1995)	281	(4)
Dynamics of the Pure Condensate	285	(9)
Static Condensate	287	(2)
Dynamics of the Condensate (Collective Modes)	289	(2)
Quantum Hydrodynamic Formulation	291	(2)
Interference of Coherent Matter Waves	293	(1)
Coupled Dynamics of the Condensate and Noncondensate	294	(9)
References	303	(4)
Beyond CFT: Deformed Virasoro and Elliptic Algebras	307	(144)
Satoru Odake
Introduction	307	(3)
Conformal Field Theory and Virasoro Algebra	310	(24)
Conformal Field Theory	310	(1)
Virasoro Algebra	310	(8)
Free Field Realization	318	(16)
Deformed Virasoro Algebra (A1(1) Type)	334	(21)
Definition and Consistency	334	(3)
Conformal Limit	337	(1)
Representation Theory	338	(2)
Free Field Realization	340	(13)
Higher DVA Currents	353	(2)
Solvable Lattice Models and Elliptic Algebras	355	(35)
Solvable Lattice Models and Yang-Baxter Equation	355	(4)
Corner Transfer Matrices and Vertex Operators	359	(14)
Introduction to Quasi-Hopf Algebra	373	(5)
Elliptic Quantum Groups	378	(12)
Free Field Approach to ABF Model	390	(22)
ABF Model	391	(2)
Vertex Operators	393	(6)
Local Height Probability	399	(4)
Form Factor	403	(4)
OPE and Trace	407	(2)
Screening Operators and Vertex Operators	409	(3)
DVA (A2(2) Type) and Dilute AL Models	412	(18)
DVA (A2(2))	412	(2)
Free Field Realization	414	(4)
Dilute AL Models	418	(2)
Free Field Approach	420	(8)
OPE and trace	428	(2)
Conclusion	430	(3)
References	433	(8)
Some Formulas	441	(10)
Some Functions	442	(1)
Delta Function	443	(1)
Some Summations	444	(1)
Some Integrals	445	(2)
Hausdorff Formula	447	(1)
Trace Technique	447	(4)
Mesoscopic Physics	451	(116)
B.D. Simons
A. Altland
Introduction to Mesoscopic Physics	451	(31)
Manifestations of Phase Coherence in Mesoscopics	454	(5)
Qualitative Considerations: Feynman Paths	459	(14)
Interaction Phenomena	473	(7)
Impurity Diagram Technique	480	(2)
Field Theory of Disordered Conductors	482	(15)
Derivation of the σ-Model Action	484	(8)
Applications of the σ-Model	492	(4)
Random Matrix Theory	496	(1)
Quantum Chaos	497	(17)
Spectral Statistics: A Brief History	500	(3)
Semi-Classics and the Trace Formula	503	(3)
Ballistic σ-Model	506	(1)
Perturbation Theory	506	(3)
Quantum Hall Effect	509	(5)
Coulomb Interaction Phenomena	514	(12)
Matsubara Field Integral	515	(3)
Plasma Theory of the Free Electron Gas	518	(3)
Plasma Theory of the Disordered Electron Gas	521	(2)
Gauge Fixing: Low-Energy Saddle-Point	523	(1)
Zero-Bias Anomaly	524	(2)
Novel Symmetry Classes	526	(32)
Realizations of Novel Symmetries	529	(2)
Dirty Superconductivity	531	(5)
Hybrid Superconductor-Normal Systems	536	(6)
Dirty d-Wave Superconductivity	542	(1)
Chiral Symmetry and Sublattice Models	543	(9)
Random Classical Dynamics	552	(6)
References	558	(9)
QCD In Extreme Conditions	567
Frank Wilczek
Introduction	567
Symmetry and the Phenomena of QCD	569
Apparent and Actual Symmetries	569
Asymptotic Freedom	572
Confinement	577
Chiral Symmetry Breaking	583
Chiral Anomalies and Instantons	587
High Temperature QCD: Asymptotic Properties	592
Significance of High Temperature QCD	592
Numerical Indications for Quasi-Free Behavior	593
Ideas About Quark-Gluon Plasma	594
Screening Versus Confinement	596
Models of Chiral Symmetry Breaking	598
More Refined Numerical Experiments	601
High-Temperature QCD: Phase Transitions	603
Yoga of Phase Transitions and Order Parameters	603
Application to Glue Theories	607
Application to Chiral Transitions	608
Close Up on Two Flavors	611
A Genuine Critical Point! (?)	615
High-Density QCD: Methods	618
Hopes, Doubts, and Fruition	618
Another Renormalization Group	620
Pairing Theory	621
Taming the Magnetic Singularity	623
High-Density QCD: Color-Flavor Locking and Quark-Hadron Continuity	624
Gauge Symmetry (Non)Breaking	625
Symmetry Accounting	626
Elementary Excitations	628
A Modified Photon	629
Quark-Hadron Continuity	630
Remembrance of Things Past	630
More Quarks	631
Fewer Quarks, and Reality	631