nLab Quantum Fields and Strings

This entry collects linked keywords for the book

• Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, David Morrison and Edward Witten, eds.

  Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

on quantum field theory and string theory.

Parts of this appear separately elsewhere, for instance

• Pierre Deligne, John Morgan, Notes on supersymmetry (pdf notes)

• Pierre Deligne, Daniel Freed, Supersolutions (arXiv:hep-th/9901094).

  (see also at signs in supergeometry)

on fundamental supergeometry needed for describing fermion particles (and superstrings). See also

• Daniel Freed, Five lectures on supersymmetry, AMS 1999

While advertized as “A course for mathematicians”, experience shows that it is not really suited for pure mathematicians without previous exposition to and tolerance for physics, particularly beyond the first chapters (which show strong ambition to be mathematically precise) towards the following lectures (which are mainly standard lectures of theoretical physicists). But it is much better than the average physics text.

More in detail: this is a long collection of (in parts) long lectures by many top string theorists and also by some genuine top mathematicians. Correspondingly it covers a lot of ground, while still being introductory. Especially towards the beginning there is a strong effort towards trying to formalize or at least systematize much of the standard lore. But one can see that eventually the task of doing that throughout had been overwhelming. Nevertheless, this is probably the best source that there is out there. If you only ever touch a single book on string theory, touch this one.

See also at string theory FAQ

Volume I

Part 1: Classical fields and Supersymmetry

• classical field theory

• supersymmetry

Classical field theory

Chapter 1. Classical mechanics

• classical mechanics

Chapter 2. Lagrangian theory of classical fields

• Lagrangian

• variational calculus

• Euler-Lagrange equations

Chapter 3. Free field theories

Chapter 4. Gauge theory

• gauge theory

• electromagnetism

• Yang-Mills theory

Chapter 5. $\sigma$-Models and coupled gauge theories

• sigma-model

Chapter 6. Topological terms

• Chern-Simons form

• Chern-Simons theory

• WZW model

• Deligne cohomology

Chapter 7. Wick rotation

• Wick rotation

Part 2: Formal Aspects of QFT

Volume II

Part 3: Conformal field theory and strings

Lectures on Conformal Field Theory

Lecture 1. Simple functional integrals

• path integral

Lecture 2. Axiomatic approaches to conformal field theory

• CFT

  • vertex operator algebra

  • FQFT

Lecture 3. $\sigma$-Models

• sigma-models

Lecture 4. Constructive conformal field theory

• WZW model

Perturbative String Theory

Lecture 1. Point varticles and strings

• particle

• string

Lecture 2. Spectrum of the free bosonic string

Lecture 3. String amplitudes and moduli space of curves

Lecture 4. Fadeev-Popov Ghost – BRST Quantization

• BV-BRST formalism

Lecture 5. Moduli dependence of determinants and Green functions

Lecture 6. Strings on general manifolds

• Killing spinor

• gravity, Kalb-Ramond field, dilaton

Lecture 7. Free superstrings

• type II string theory

• type I string theory

Lecture 8. Heterotic strings

• heterotic string theory

Lecture 9. Superstring perturbation theory

• perturbation theory

Lecture 10. Supersymmetry and supergravity

• supersymmetry

• supergravity

• Green-Schwarz superstring

Super Space Description of Super Gravity

• supergravity

Notes on 2d Conformal Field Theory and String Theory

Chpater 0. Introduction

• vertex operator algebra

• D-module

Chapter 1. Chiral algebra

• chiral algebra

• conformal block

• correlation function

Chapter 2. CFT data

• CFT

Chapter 3. Examples

• Kac-Moody algebra

• dilaton

• bc-system?

Chapter 4. BRST and string amplitudes

• BRST complex

Chapter 5. Further constructons

• Ran space

Chapter 6. The free bosonic theory

• canonical line bundle

Kaluza-Klein Compactifications, Supersymmetry, and Calabi Yau Spaces

Lecture 1. Compactifications to dimension four

• Kaluza-Klein mechanism

• spontaneous symmetry breaking

Lecture 2. Supersymmetry and Calabi-Yau manifolds

• supersymmetry and Calabi-Yau manifolds

Part 4: Dynamical Aspects of QFT

Dynamics of Quantum Field Theory

Lecture 1. Symmetry breaking

• spontaneous symmetry breaking

Lecture 2. Gauge symmetry breaking and more infrared behaviour

Lecture 3. BRST quantization of gauge theories

• gauge theory

• BRST complex

• BV-BRST complex

Lecture 4. Infrared behaviour of the S-matrix of the 2-dimensional $\sigma$-model with target space $S^{N-1}$

• sigma-model

• S-matrix

Lecture 5. The large $N$ limit of the $\sigma$-model into Grassmannians

Lecture 6. The Bose-Fermi correspondence and its applications

Lecture 7. Two-dimensional gauge theory of bosons, the Wilson line operator and confinement

• Wilson line

• confinement

Lecture 8. Abelian duality

Lecture 9. Solitons

• soliton

Lecture 10. Wilson loops, ‘t Hooft loops and ‘t Hooft’s picture of confinement

• Wilson loop

• confinement

Lecture 11. Quantum gauge theories in two dimensions and intersection theory on moduli space

• Landau-Ginzburg model

Lecture 12. Supersymmetric field theories

• supersymmetry

• BPS state

Lecture 13. $N=2$ SUSY theories in dimension two: part I

• N=2 D=4 super Yang-Mills theory

• topological string

Lecture 14. $N=2$ SUSY theories in dimension two: part II, Chiral rings and twisted theories

• chiral ring

Lecture 15. The Landau-Ginzburg description of $N = 2$ minimal models; Quantum cohomology and Kähler manifolds

• Landau-Ginzburg model

• quantum cohomology

Lecture 16. Four-dimensional gauge theories

• N=1 D=4 super Yang-Mills theory

Lecture 17. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 1

• N=2 D=4 super Yang-Mills theory

Lecture 18. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 2

Lecture 19. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 3, Topological applications

• topologically twisted D=4 super Yang-Mills theory

Dynamics of $N = 1$ Supersymmetric Field Theories in Four Dimensions

• N=1 D=4 super Yang-Mills theory

category: reference