Quantum Fields and Strings: A Course for Mathematicians. Vol. 1

Preface......Page all_7369_to_00063.cpc0007.djvu
Brief Contents......Page all_7369_to_00063.cpc0009.djvu
Cross-Reference Codes......Page all_7369_to_00063.cpc0011.djvu
Contents......Page all_7369_to_00063.cpc0013.djvu
Introduction......Page all_7369_to_00063.cpc0023.djvu
Glossary......Page all_7369_to_00063.cpc0029.djvu
Notes on Supersymmetry (P. Deligne and J. Morgan)......Page all_7369_to_00063.cpc0063.djvu
1.1. The sign rule......Page all_7369_to_00126.cpc0004.djvu
1.2. Categorical approach......Page all_7369_to_00126.cpc0006.djvu
1.3. Examples of the categorical approach......Page all_7369_to_00126.cpc0007.djvu
1.4. Free modules......Page all_7369_to_00126.cpc0012.djvu
1.6. The trace......Page all_7369_to_00126.cpc0013.djvu
1.7. Even rules......Page all_7369_to_00126.cpc0015.djvu
1.8. Examples of the even rules principle......Page all_7369_to_00126.cpc0017.djvu
1.10. The Berezinian of an automorphism......Page all_7369_to_00126.cpc0018.djvu
1.11. The Berezinian of a free module......Page all_7369_to_00126.cpc0020.djvu
Appendix. Graded super vector spaces......Page all_7369_to_00126.cpc0021.djvu
2.1-2.7. Super manifolds as ringed spaces......Page all_7369_to_00126.cpc0024.djvu
2.8-2.9. The functor of points approach to super manifolds......Page all_7369_to_00126.cpc0027.djvu
2.10. Super Lie groups......Page all_7369_to_00126.cpc0028.djvu
2.11. Classical series of super Lie groups......Page all_7369_to_00126.cpc0029.djvu
3.2. Vector bundles......Page all_7369_to_00126.cpc0030.djvu
3.3. The tangent bundle, the cotangent bundle and the de Rham complex......Page all_7369_to_00126.cpc0031.djvu
3.5. Distributions......Page all_7369_to_00126.cpc0034.djvu
3.6. Connections on vector bundles......Page all_7369_to_00126.cpc0036.djvu
3.7. Actions of super Lie algebras; vector fields and flows; Lie derivative......Page all_7369_to_00126.cpc0037.djvu
3.8. Super Lie groups and Harish-Chandra pairs......Page all_7369_to_00126.cpc0038.djvu
3.10. Change of variables formula for densities......Page all_7369_to_00126.cpc0039.djvu
3.11. The Lie derivative of sections of Ber(Ωⁱ_M)......Page all_7369_to_00126.cpc0042.djvu
3.12. Integral forms......Page all_7369_to_00126.cpc0043.djvu
3.13. A second definition of integral forms......Page all_7369_to_00126.cpc0044.djvu
3.15. Integral forms as functions of infinitesimal submanifold elements......Page all_7369_to_00126.cpc0045.djvu
4.1-4.3. Real structures and *-operations......Page all_7369_to_00126.cpc0048.djvu
4.4. Super Hilbert spaces......Page all_7369_to_00126.cpc0049.djvu
4.5. SUSY quantum mechanics......Page all_7369_to_00126.cpc0050.djvu
4.6. Real and complex super manifolds......Page all_7369_to_00126.cpc0051.djvu
4.7. Complexification, in infinite dimensions......Page all_7369_to_00126.cpc0052.djvu
4.9. Integration on cs manifolds; examples......Page all_7369_to_00126.cpc0053.djvu
References......Page all_7369_to_00126.cpc0056.djvu
Notes on Spinors (P. Deligne)......Page all_7369_to_00126.cpc0058.djvu
1. Overview......Page all_7369_to_00126.cpc0060.djvu
2. Clifford Modules......Page all_7369_to_00189.cpc0003.djvu
3. Reality of Spinorial Representations and Signature Modulo 8......Page all_7369_to_00189.cpc0009.djvu
4. Pairings and Dimension Modulo 8, Over ℂ......Page all_7369_to_00189.cpc0015.djvu
5. Passage to Quadratic Spaces......Page all_7369_to_00189.cpc0023.djvu
6. The Minkowski Case......Page all_7369_to_00189.cpc0025.djvu
References......Page all_7369_to_00189.cpc0031.djvu
Classical Field Theory (P. Deligne and D. Freed)......Page all_7369_to_00189.cpc0033.djvu
1.1. The nonrelativistic particle......Page all_7369_to_00189.cpc0039.djvu
1.2. The relativistic particle......Page all_7369_to_00189.cpc0042.djvu
1.3. Noether\'s theorem......Page all_7369_to_00189.cpc0043.djvu
1.4. Synthesis......Page all_7369_to_00189.cpc0046.djvu
2.1. Dimensional analysis......Page all_7369_to_00189.cpc0049.djvu
2.2. Densities and twisted differential forms......Page all_7369_to_00189.cpc0050.djvu
2.3. Fields and lagrangians......Page all_7369_to_00189.cpc0051.djvu
2.4. First order lagrangians......Page all_7369_to_00189.cpc0058.djvu
2.5. Hamiltonian theory......Page all_7369_to_00189.cpc0059.djvu
2.6. Symmetries and Noether\'s theorem......Page all_7369_to_00189.cpc0061.djvu
2.7. More on symmetries......Page all_7369_to_00252.cpc0004.djvu
2.8. Computing Noether\'s current by gauging symmetries......Page all_7369_to_00252.cpc0008.djvu
2.9. The energy-momentum tensor......Page all_7369_to_00252.cpc0011.djvu
2.10. Finite energy configurations, classical vacua, and solitons......Page all_7369_to_00252.cpc0016.djvu
2.11. Dimensional reduction......Page all_7369_to_00252.cpc0020.djvu
Appendix: Takens\' acyclicity theorem......Page all_7369_to_00252.cpc0021.djvu
3.1. Coordinates on Minkowski spacetime......Page all_7369_to_00252.cpc0024.djvu
3.2. Real scalar fields......Page all_7369_to_00252.cpc0025.djvu
3.3. Complex scalar fields......Page all_7369_to_00252.cpc0027.djvu
3.4. Spinor fields......Page all_7369_to_00252.cpc0028.djvu
3.5. Abelian gauge fields......Page all_7369_to_00252.cpc0031.djvu
4.1. Classical electromagnetism......Page all_7369_to_00252.cpc0034.djvu
4.2. Principal bundles and connections......Page all_7369_to_00252.cpc0037.djvu
4.3. Pure Yang-Mills Theory......Page all_7369_to_00252.cpc0040.djvu
4.4. Electric and magnetic charge......Page all_7369_to_00252.cpc0042.djvu
5.1. Nonlinear σ-models......Page all_7369_to_00252.cpc0044.djvu
5.2. Gauge theory with bosonic matter......Page all_7369_to_00252.cpc0046.djvu
6.1. Gauge theory......Page all_7369_to_00252.cpc0048.djvu
6.2. Wess-Zumino-Witten terms......Page all_7369_to_00252.cpc0050.djvu
6.3. Smooth Deligne cohomology......Page all_7369_to_00252.cpc0051.djvu
7. Wick Rotation: From Minkowski Space to Euclidean Space......Page all_7369_to_00252.cpc0054.djvu
7.2. Potential terms......Page all_7369_to_00252.cpc0055.djvu
7.4. Kinetic terms for fermions......Page all_7369_to_00252.cpc0056.djvu
References......Page all_7369_to_00252.cpc0058.djvu
Supersolutions (P. Deligne and D. Freed)......Page all_7369_to_00252.cpc0060.djvu
1.1. Super Minkowski spaces and super Poincare groups......Page all_7369_to_00315.cpc.djvu
1.2. Superfields, component fields, and lagrangians......Page all_7369_to_00315.cpc0006.djvu
1.3. A simple example......Page all_7369_to_00315.cpc0011.djvu
2.1. M³², M⁴⁴, M⁶⁽⁸⁰⁾ and their complexifications......Page all_7369_to_00315.cpc0013.djvu
2.2. Dimensional reduction......Page all_7369_to_00315.cpc0015.djvu
2.3. Coordinates on M³²......Page all_7369_to_00315.cpc0016.djvu
2.4. Coordinates on M⁴⁴......Page all_7369_to_00315.cpc0019.djvu
2.5. Coordinates on M⁶⁽⁸⁰⁾......Page all_7369_to_00315.cpc0024.djvu
2.6. Low dimensions......Page all_7369_to_00315.cpc0027.djvu
3.1. Preliminary remarks on linear algebra......Page all_7369_to_00315.cpc0031.djvu
3.2. The free supersymmetric σ-model......Page all_7369_to_00315.cpc0033.djvu
3.3. Nonlinear supersymmetric σ-model......Page all_7369_to_00315.cpc0037.djvu
3.4. Supersymmetric potential terms......Page all_7369_to_00315.cpc0043.djvu
3.6. Dimensional reduction......Page all_7369_to_00315.cpc0046.djvu
4.1. Fields and supersymmetry transformations on M³²......Page all_7369_to_00315.cpc0049.djvu
4.3. The potential term on M³²......Page all_7369_to_00315.cpc0052.djvu
4.4. Analysis of the classical theory......Page all_7369_to_00315.cpc0054.djvu
4.5. Reduction to M²⁽¹¹⁾......Page all_7369_to_00315.cpc0059.djvu
5.1. Fields and supersymmetry transformations on M⁴⁴......Page all_7369_to_00315.cpc0061.djvu
5.2. The σ-model action on M⁴⁴......Page all_7369_to_00315.cpc0063.djvu
5.3. The superpotential term on M⁴⁴......Page all_7369_to_00378.cpc0003.djvu
5.4. Analysis of the classical theory......Page all_7369_to_00378.cpc0004.djvu
6.1. The minimal theory in components......Page all_7369_to_00378.cpc0006.djvu
6.2. Gauge theories with matter......Page all_7369_to_00378.cpc0010.djvu
6.3. Superspace construction......Page all_7369_to_00378.cpc0016.djvu
7.1. Constrained connections on M³²......Page all_7369_to_00378.cpc0020.djvu
7.2. The Yang-Mills action on M³²......Page all_7369_to_00378.cpc0024.djvu
7.3. Gauge theory with matter on M³²......Page all_7369_to_00378.cpc0025.djvu
8.1. Constrained connections on M4\'4......Page all_7369_to_00378.cpc0028.djvu
8.2. The Yang-Mills action on M⁴⁴......Page all_7369_to_00378.cpc0032.djvu
8.3. Gauge theory with matter on M⁴⁴......Page all_7369_to_00378.cpc0034.djvu
9.1. Dimensional reduction of bosonic Yang-Mills......Page all_7369_to_00378.cpc0038.djvu
9.2. Constrained connections on M²⁽²²⁾......Page all_7369_to_00378.cpc0040.djvu
9.3. The reduced Yang-Mills action......Page all_7369_to_00378.cpc0041.djvu
10.1. Constrained connections on M⁶⁽⁸⁰⁾......Page all_7369_to_00378.cpc0044.djvu
10.2. Reduction to M⁴⁸......Page all_7369_to_00378.cpc0047.djvu
10.3. More theories on M⁴⁴ with extended supersymmetry......Page all_7369_to_00378.cpc0051.djvu
11.1. Complements on M⁶⁽⁸⁰⁾......Page all_7369_to_00378.cpc0054.djvu
11.2. Constrained connections......Page all_7369_to_00378.cpc0055.djvu
11.3. An auxiliary Lie algebra......Page all_7369_to_00378.cpc0056.djvu
11.4. Components of constrained connections......Page all_7369_to_00378.cpc0059.djvu
References......Page all_7369_to_00378.cpc0062.djvu
2. Choices......Page all_7369_to_00441.cpc.djvu
4. Notation......Page all_7369_to_00441.cpc0002.djvu
5. Consequences of §2 on other signs......Page all_7369_to_00441.cpc0004.djvu
6. Differential forms......Page all_7369_to_00441.cpc0006.djvu
7. Miscellaneous signs......Page all_7369_to_00441.cpc0007.djvu
Note on Quantization (P. Deligne)......Page all_7369_to_00441.cpc0011.djvu
Introduction to QFT (D. Kazhdan)......Page all_7369_to_00441.cpc0021.djvu
1.0. Setup and notations......Page all_7369_to_00441.cpc0023.djvu
1.2. Wightman functions......Page all_7369_to_00441.cpc0024.djvu
1.3. Reconstruction of QFT from Wightman functions......Page all_7369_to_00441.cpc0025.djvu
1.4. Spin-statistics theorem......Page all_7369_to_00441.cpc0026.djvu
1.5. Mass spectrum of a theory......Page all_7369_to_00441.cpc0027.djvu
1.6. Asymptotics of Wightman functions......Page all_7369_to_00441.cpc0028.djvu
2.1. Analytic continuation of Wightman functions......Page all_7369_to_00441.cpc0031.djvu
2.2. Euclidean formulation of Wightman QFT......Page all_7369_to_00441.cpc0034.djvu
2.3. Schwinger functions and measures on the map-spaces......Page all_7369_to_00441.cpc0035.djvu
2.4. PCT theorem......Page all_7369_to_00441.cpc0036.djvu
2.5. Time-ordering......Page all_7369_to_00441.cpc0038.djvu
3.1. Some examples of free classical field theories......Page all_7369_to_00441.cpc0039.djvu
3.2. Clifford module......Page all_7369_to_00441.cpc0040.djvu
3.3. Examples of free QFT\'s......Page all_7369_to_00441.cpc0041.djvu
3.4. Free QFT of arbitrary spin......Page all_7369_to_00441.cpc0042.djvu
3.5. Wightman functions of a free field theory; truncated Wightman functions......Page all_7369_to_00441.cpc0045.djvu
3.7. Normal ordering......Page all_7369_to_00441.cpc0046.djvu
4.1. Introduction......Page all_7369_to_00441.cpc0049.djvu
4.2. System of n particles (potential scattering)......Page all_7369_to_00441.cpc0050.djvu
4.3. Haag-Ruelle theory......Page all_7369_to_00441.cpc0051.djvu
4.4. Scattering matrix......Page all_7369_to_00441.cpc0056.djvu
5.1. Feynman graph expansion......Page all_7369_to_00441.cpc0057.djvu
5.2. Quasi-classical (low-loop) approximations......Page all_7369_to_00441.cpc0060.djvu
5.3. Effective potential......Page all_7369_to_00441.cpc0062.djvu
Perturbative Quantum Field Theory (E. Witten)......Page all_7369_to_00441.cpc0063.djvu
1.1. Perturbative expansion of a two-point correlation function......Page all_7369_to_00504.cpc0002.djvu
1.2. The φ³-theory......Page all_7369_to_00504.cpc0004.djvu
1.3. Perturbative expansion of Feynman integrals......Page all_7369_to_00504.cpc0005.djvu
1.4. Computation of a Feynman integral over functions on a Minkowski space......Page all_7369_to_00504.cpc0006.djvu
1.5. Renormalization of divergent graphs......Page all_7369_to_00504.cpc0010.djvu
1.6. Renormalization in higher orders......Page all_7369_to_00504.cpc0012.djvu
2.1. Renormalizability of quantum field theories......Page all_7369_to_00504.cpc0016.djvu
2.2. Critical dimensions of some field theories......Page all_7369_to_00504.cpc0018.djvu
2.3. Perturbative renormalization of critical theories......Page all_7369_to_00504.cpc0021.djvu
3.1. Local functionals in a classical field theory......Page all_7369_to_00504.cpc0026.djvu
3.2. Quantization of local functionals in a free theory......Page all_7369_to_00504.cpc0027.djvu
3.3. Multiplication of composite operators......Page all_7369_to_00504.cpc0029.djvu
3.4. Operator product expansion (OPE) in the free theory......Page all_7369_to_00504.cpc0030.djvu
3.6. Composite operators in an interacting critical theory......Page all_7369_to_00504.cpc0033.djvu
3.7. Stability of the classical field equations under quantization......Page all_7369_to_00504.cpc0035.djvu
3.8. Operator product expansion in an interacting theory......Page all_7369_to_00504.cpc0037.djvu
4.1. Nonrelativistic scattering theory: the asymptotic conditions......Page all_7369_to_00504.cpc0042.djvu
4.2. Relation with experiments......Page all_7369_to_00504.cpc0043.djvu
4.3. The Lippmann-Schwinger equation......Page all_7369_to_00504.cpc0044.djvu
4.4. The Born approximation......Page all_7369_to_00504.cpc0045.djvu
4.5. Feynman diagrams......Page all_7369_to_00504.cpc0046.djvu
4.6. Relativistic versus non-relativistic scattering theory: propagation of particles......Page all_7369_to_00504.cpc0047.djvu
4.7. Relativistic versus non-relativistic scattering theory: propagation of signals......Page all_7369_to_00504.cpc0048.djvu
5.1. Ambiguity in operator products......Page all_7369_to_00504.cpc0050.djvu
5.3. An oversimplified version of experimental confirmation of asymptotic freedom......Page all_7369_to_00504.cpc0051.djvu
Index of Dirac Operators (E. Witten)......Page all_7369_to_00504.cpc0056.djvu
1.1. Introduction......Page all_7369_to_00504.cpc0058.djvu
1.2. The Dirac operator on a spin manifold......Page all_7369_to_00504.cpc0059.djvu
1.3. The case of a circle action......Page all_7369_to_00567.cpc0003.djvu
1.4. σ-models in 1 + 1 dimensions......Page all_7369_to_00567.cpc0010.djvu
2.2. The Lagrangian formulation: σ-models in two dimensions......Page all_7369_to_00567.cpc0015.djvu
2.3. Quantization......Page all_7369_to_00567.cpc0017.djvu
2.4. The index of Q+......Page all_7369_to_00567.cpc0018.djvu
2.5. The computation around the fixed points of the S¹-action......Page all_7369_to_00567.cpc0019.djvu
2.6. Path integral approach......Page all_7369_to_00567.cpc0023.djvu
2.7. Bundles whose coupled signature or Dirac operator has constant character......Page all_7369_to_00567.cpc0025.djvu
2.8. Generalization to vector bundles over the loop space......Page all_7369_to_00567.cpc0027.djvu
Elementary Introduction to Quantum Field Theory (L. Faddeev)......Page all_7369_to_00567.cpc0031.djvu
1.1. Observables and states......Page all_7369_to_00567.cpc0033.djvu
1.2. Dynamics......Page all_7369_to_00567.cpc0035.djvu
1.3. Quantization......Page all_7369_to_00567.cpc0037.djvu
2.1. The harmonic oscillator......Page all_7369_to_00567.cpc0041.djvu
2.2. Perturbations......Page all_7369_to_00567.cpc0042.djvu
2.3. Quantum field theory......Page all_7369_to_00567.cpc0044.djvu
2.4. S-matrix and Feynman diagrams......Page all_7369_to_00567.cpc0047.djvu
3.2. Mass renormalization......Page all_7369_to_00567.cpc0049.djvu
3.3. Charge renormalization......Page all_7369_to_00567.cpc0052.djvu
4.1. Lagrangian and Hamiltonian formalisms......Page all_7369_to_00567.cpc0055.djvu
4.2. Constraints......Page all_7369_to_00567.cpc0056.djvu
4.3. Examples......Page all_7369_to_00567.cpc0057.djvu
5.1. The physical variables......Page all_7369_to_00567.cpc0063.djvu
5.2. Gauge conditions in the functional integral......Page all_7369_to_00630.cpc0002.djvu
Renormalization Groups (D. Gross)......Page all_7369_to_00630.cpc0006.djvu
1.1. What is renormalization group?......Page all_7369_to_00630.cpc0008.djvu
1.2. The general scheme of the method of renormalization group......Page all_7369_to_00630.cpc0009.djvu
1.3. Wilsonian scheme for the theory of a scalar field: a mathematical description......Page all_7369_to_00630.cpc0010.djvu
1.5. Reminder of renormalization theory......Page all_7369_to_00630.cpc0014.djvu
1.6. Dimensional regularization......Page all_7369_to_00630.cpc0015.djvu
2.1. Finite renormalization......Page all_7369_to_00630.cpc0018.djvu
2.2. The dimensional regularization prescription of finite renormalization......Page all_7369_to_00630.cpc0019.djvu
2.3. Scale-dependence of finite renormalization prescriptions......Page all_7369_to_00630.cpc0020.djvu
2.4. The renormalization group flow corresponding to a scale-dependent renormalization prescription......Page all_7369_to_00630.cpc0021.djvu
2.6. Asymptotic freedom......Page all_7369_to_00630.cpc0025.djvu
3.1. Dynamical patterns of the renormalization group flow......Page all_7369_to_00630.cpc0028.djvu
3.2. Are there any asymptotically free theories without nonabelian gauge fields?......Page all_7369_to_00630.cpc0031.djvu
3.4. The renormalization group equation for composite operators......Page all_7369_to_00630.cpc0033.djvu
3.5. Anomalous dimension......Page all_7369_to_00630.cpc0035.djvu
3.6. The canonical part of the β-function......Page all_7369_to_00630.cpc0038.djvu
4.1. Dynamical mass generation......Page all_7369_to_00630.cpc0040.djvu
4.2. The Gross-Neveu model......Page all_7369_to_00630.cpc0042.djvu
4.3. The large N limit......Page all_7369_to_00630.cpc0043.djvu
5. The Wilsonian Renormalization Group Equation......Page all_7369_to_00630.cpc0050.djvu
1. The D-dimensional integral......Page all_7369_to_00630.cpc0052.djvu
2. D-dimensional integral with parameters......Page all_7369_to_00630.cpc0056.djvu
3. D-dimensional integral of functions arising from Feynman diagrams......Page all_7369_to_00630.cpc0060.djvu
4. Dimensional regularization of Feynman integrals......Page all_7369_to_00630.cpc0061.djvu
5. D-dimensional Stokes formula......Page all_7369_to_00630.cpc0062.djvu
Homework (E. Witten)......Page all_7369_to_00693.cpc.djvu
Problem Sets from fall term......Page all_7369_to_00693.cpc0003.djvu
Fall exam......Page all_7369_to_00693.cpc0021.djvu
Superhomework......Page all_7369_to_00693.cpc0024.djvu
Addendum to superhomework......Page all_7369_to_00693.cpc0034.djvu
2. Solutions to Selected Problems......Page all_7369_to_00693.cpc0039.djvu
Index......Page all_7369_to_00745.cpc0048.djvu