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Quantum Fields and Strings: A Course for Mathematicians: Volume 2
 
Edited by: Pierre Deligne
Pavel Etingof
Daniel S. Freed
Lisa C. Jeffrey
David Kazhdan
John W. Morgan
David R. Morrison
Edward Witten
Quantum Fields and Strings: A Course for Mathematicians
Softcover ISBN:  978-0-8218-2013-1
Product Code:  QFT/2.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-7407-2
Product Code:  QFT/2.E
List Price: $48.00
MAA Member Price: $43.20
AMS Member Price: $38.40
Add to cart
Quantum Fields and Strings: A Course for Mathematicians
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Quantum Fields and Strings: A Course for Mathematicians: Volume 2
Edited by: Pierre Deligne
Pavel Etingof
Daniel S. Freed
Lisa C. Jeffrey
David Kazhdan
John W. Morgan
David R. Morrison
Edward Witten
Softcover ISBN:  978-0-8218-2013-1
Product Code:  QFT/2.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-7407-2
Product Code:  QFT/2.E
List Price: $48.00
MAA Member Price: $43.20
AMS Member Price: $38.40
Add to cart
  • Book Details
     
     
    1999; 778 pp
    MSC: Primary 81

    Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics.

    In 1996–97 the Institute for Advanced Study (Princeton, NJ) organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: “The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra.”

    These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures.

    These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory.
    Readership

    Graduate students and research mathematicians working in various areas of mathematics related to quantum field theory.
    This item is also available as part of a set:
  • Table of Contents
     
     
    • Preface
    • Brief Contents
    • Cross-Reference Codes
    • Contents
    • Part 3 Conformal Field Theory and Strings
    • Lectures on Conformal Field Theory
    • LECTURE 1 Simple Functional Integrals
    • §1.1. What is quantum field theory?
    • §1.2. Euclidean free field and Gaussian functional integrals
    • §1.3. Feynman-Kac formula
    • §1.4. Massless free field with values in S1
    • §1.5. Toroidal compactifications: the partition functions
    • §1.6. Toroidal compactifications: the correlation functions
    • LECTURE 2 Axiomatic Approaches to Conformal Field Theory
    • §2.1. Conformal field theory data
    • §2.2. Conformal Ward identities
    • §2.3. Physical positivity and Hilbert space picture
    • §2.4. Virasoro algebra and its primary fields
    • §2.5. Highest weight representations of Vir
    • §2.6. Segal's axioms and vertex operator algebras
    • LECTURE 3 Sigma Models
    • §3.1. lPI effective action and large deviations
    • §3.2. Geometric sigma models
    • §3.3. Regularization and renormalization
    • §3.4. Renormalization group effective actions
    • §3.5. Background field effective action
    • §3.6. Dimensional regularization
    • §3.7. Renormalization of the sigma models to 1 loop
    • §3.8. Renormalization group analysis of sigma models
    • LECTURE 4 Constructive Conformal Field Theory
    • §4.1. WZW model
    • §4.2. Gauge symmetry Ward identities
    • §4.3. Scalar product of non-abelian theta functions
    • §4.4. KZB connection
    • §4.5. Coset theories
    • §4.6. WZW factory
    • String Theory
    • LECTURE 1 Point Particles vs Strings
    • §1.1. Strings
    • §1.2. Interactions
    • §1.3. Loop expansion-topology of closed surfaces
    • §1.4. Transition amplitudes for strings
    • §1.5. Weyl invariance and vertex operator formulation
    • §1.6. More general actions
    • §1. 7. Transition amplitude for a single point particle
    • §1.8. Generalized point particle propagation
    • LECTURE 2 Spectrum of Free Bosonic Strings
    • §2.1. Basics of conformal field theory
    • §2.2. The free closed bosonic string conformal field theory
    • §2.3. The free open bosonic string conformal field theory
    • §2.4. Fock space, negative norm states
    • §2.5. Integration over Met(Σ)-Virasoro constraints
    • §2.6. Physical spectrum, no-ghost theorem
    • §2. 7. Spectrum of the critical bosonic string with D = 26, a= 1
    • §2.8. Lightcone gauge, density of states
    • §2.9. Primary fields and vertex operators for physical states
    • §2.10. Identifying the graviton: vertex operators from background fields
    • §2.11. Internal degrees of freedom of open strings: Chan-Paton rules
    • LECTURE 3 String Amplitudes and Moduli Space of Curves
    • §3.1. Finite-dimensional case
    • §3.2. Basic notation: tensors, derivatives
    • §3.3. Space of metrics - moduli space of Riemann surfaces
    • §3.4. Factorizing the integration measure
    • §3.5. Weyl rescalings of functional determinants
    • §3.6. Critical central charge-critical dimension
    • §3.7. Flat space-time manifold M
    • §3.8. Non-critical strings
    • §3.9. Tree level amplitudes
    • §3.10. One loop amplitudes
    • LECTURE 4 Faddeev-Popov Ghosts-BRST Quantization
    • §4.1. Determinants and b-c systems
    • §4.2. Ghost representation of the Faddeev-Popov determinant
    • §4.3. Conformal field theory of the b-c system
    • §4.4. Bosonization of the b-c system
    • §4.5. The b-c Fock space
    • §4.6. BRST quantization
    • LECTURE 5 Moduli Dependence of Determinants and Green Functions
    • §5.1. Worldsheets with constant curvature metric (h≥ 2)
    • §5.2. Holomorphicity in moduli
    • §5.3. The chiral splitting theorem
    • §5.4. Holomorphic and meromorphic differentials (a brief review of basics)
    • §5.5. Green functions, determinants and chiral splitting
    • LECTURE 6 Strings on General Manifolds
    • §6.1. Perturbation theory around general field configurations
    • §6.2. Renormalization of generalized non-linear sigma models
    • §6.3. General structure of Weyl dependence
    • §6.4. General structure of Weyl anomaly in low energy expansion
    • §6.5. Background field quantization method
    • §6.6. Covariant expansion methods
    • §6.7. Reformulation as an SO(l,D-1) gauge theory
    • §6.8. Weyl variation of the effective action
    • §6.9. Low energy string field equations and string effective action
    • §6.10. A first look at compactification
    • Appendix A
    • LECTURE 7 Free Superstrings
    • §7.1. Degrees of freedom of the RNS string
    • §7.2. Rarnond and Neveu-Schwarz Fock spaces
    • §7.3. Local supersymmetry on the worldsheet
    • §7.4. Functional integral representation of transition amplitudes
    • §7.5. Super-Virasoro algebra and physical spectrum
    • §7.6. The spectrum of physical states at low mass
    • §7.7. The GSO projection, space-time supersymmetry
    • §7.8. Type IIA, B superstrings and their spectra
    • §7.9. Type I superstring
    • LECTURE 8 Heterotic Strings
    • §8.1. Free fermion realization of internal degrees of freedom
    • §8.2. Free fermion realization of the Spin(32)/Z2 heterotic string
    • §8.3. Free fermion realization of the E8 x E8 heterotic string
    • §8.4. Bosonic realizations of the Spin(32)/Z2 and E8 x E8 strings
    • LECTURE 9 Superstring Perturbation Theory
    • §9.1. N = 1 supergeometry
    • §9.2. Functional integral representation of transition amplitudes
    • §9.3. Superconformal field theory (some basics)
    • §9.4. BRST quantization
    • §9.5. Vertex operators for physical states
    • §9.6. The chiral splitting theorem
    • §9.7. Tree-level amplitudes for NS-NS states
    • §9.8. One-loop amplitudes for NS-NS states in Type IIA, B
    • §9.9. One-loop amplitudes in the heterotic string
    • §9.10. The NS-NS 4-point function
    • LECTURE 10 Supersymmetry and Supergravity
    • §10.1. Global space-time supersymmetry in the RNS formulation
    • §10.2. The Green-Schwarz formulation
    • §10.3. Lightcone gauge quantization of the GS formulation
    • §10.4. Flat superspace GS formulation
    • §10.5. Supergravity and low energy superstrings
    • §10.6. Type IIA, D = 10, 'N = 2 and D = 11, '.N = 1 supergravities
    • §10.7. Type IIB, D = 10, N = 2 supergravity
    • §10.8. Type I - Heterotic, D = 10, '.N = 1 supergravities
    • §10.9. Superspace formulation of supergravities in D = ll and D=lO
    • §10.10. Local supersymmetric coupling of superstrings to supergravity
    • EXERCISES
    • Super Space Descriptions of Super Gravity
    • §1. General remarks
    • §2. Example: D = 2, N = 1 in the notations of §9.1 of D'Hoker's lectures
    • Notes on 2d Conformal Field Theory and String Theory
    • §0.1. Contents of these notes
    • §0.2. Some background on D-modules
    • CHAPTER 1 Chiral Algebras
    • § 1. 1. Definition of chiral algebras
    • §1.2. Lie-* algebras and construction of chiral algebras
    • §1.3. Conformal blocks, correlation functions
    • CHAPTER 2 CFT Data (Algebraic Version)
    • §2.1. Local O-modules on X
    • §2.2. A formulation of CFT (central charge 0)
    • §2.3. Introducing the central charge
    • CHAPTER 3 Examples
    • §3.1. Heisenberg and Kac-Moody algebras
    • §3.2. The linear dilaton
    • §3.3. The be-system
    • CHAPTER 4 BRST and String Amplitudes
    • §4.1. The BRST complex
    • §4.2. String amplitudes
    • CHAPTER 5 Further Constructions
    • §5.1. Chiral algebras via the Ran space
    • §5.2. Geometry of the affine Grassmannian
    • §5.3. Chiral algebra attached to the affine Grassmannian
    • CHAPTER 6 The Free Bosonic Theory
    • §6.1. The canonical line bundle
    • §6.2. Construction of the bosonic chiral algebra
    • BIBLIOGRAPHY
    • Kaluza-Klein Compactifications,Super symmetry, and Calabi-Yau Spaces
    • LECTURE 1 Compactification to Dimension Four
    • §1.1. Kaluza-Klein model
    • §1.2. Compactifying Einstein's equation from dimension ten to dimension four
    • §1.3. Adding Matter to the Mix
    • §1.4. The effective action from ten-dimensional heterotic string theory
    • LECTURE 2 Supersymmetry and Calabi-Yau Manifolds
    • §2.1. Review of material from the first lecture
    • §2.2. Partially breaking the supersymmetry by compactifying down to dimension four
    • §2.3. Geometric consequence of the unbroken supersymmetry
    • §2.4. Massless fields in the low energy effective Lagrangian
    • §2.5. Relation to Grand Unification Theory
    • Part 4 Dynamical Aspects of QFT
    • Dynamics of Quantum Field Theory
    • LECTURE 1 Symmetry Breaking
    • §1.0. Theories and realizations
    • §1.1. What is symmetry breaking, and why it does not happen in quantum mechanics
    • §1.2. Still no symmetry breaking in quantum field theory in finite volume
    • §1.3. Symmetry breaking in quantum field theory in infinite volume
    • §1.4. Infinite volume asymptotics of correlation functions
    • §1.5. Continuous symmetry breaking
    • §1.6. Goldstone's theorem
    • §1. 7. Infrared behavior of purely non-renormalizable field theories
    • §1.8. Effective action for Goldstone bosons
    • LECTURE 2 Gauge Symmetry Breaking and More on Infrared Behavior
    • §2.1. Gauge symmetry
    • §2.2. Breaking of gauge symmetry and charges at infinity
    • §2.3. Symmetry breaking and gauging
    • §2.4. No massless particles of higher spin
    • §2.5. Infrared limits
    • LECTURE 3 BRST Quantization of Gauge Theories
    • §3.1. The general setup
    • §3.2. The BRST differential
    • §3.3. The properties of the BRST derivation
    • §3.4. Operators in gauge theory and BRST cohomology
    • §3.5. Renormalization and BRST differential
    • §3.6. The Hamiltonian approach
    • §3.7. Anomalies
    • LECTURE 4 Infrared Behavior and the S-matrix of the Two-Dimensional Sigma Model with Target Space SN-1
    • §4. 1. Infrared behavior of certain two-dimensional sigma models
    • §4.2. Computation of the infrared behavior in the N→∞ limit
    • §4.3. Computation of the S-matrix
    • LECTURE 5 The Large N Limit of the a-model into Grassmannians
    • §5.1. The questions
    • §5.2. An equivalent formulation
    • §5.3. The large N effective theory
    • §5.4. Real Grassmannians
    • §5.5. Pure gauge theory
    • §5.6. Classical electromagnetism in two dimensions
    • §5.7. Quantum theory with matter
    • LECTURE 6 The Bose-Fermi Correspondence and its Applications
    • §6.1. Two-dimensional gauge theories with fermions
    • §6.2. Chiral symmetry
    • §6.3. Behavior of two-dimensional gauge theory with massive fermions
    • §6.4. Heavy fermions
    • §6.5. Bose-Fermi correspondence
    • §6.6. Bose-Fermi correspondence for nonlinear theories
    • LECTURE 7 Two-Dimensional Gauge Theory of bosons, the Wilson Line Operator, and Confinement
    • §7.1. Infrared behavior of U(l) gauge theories with bosons in two dimensions
    • §7.2. The vacuum energy density
    • §7.3. Instantons
    • §7.4. lnstanton gas
    • §7.5. Summing over instantons
    • §7.6. The Wilson line operator
    • § 7. 7. The path integral representation of the Wilson line operator
    • §7.8. The Higgs and the confinement regimes
    • §7.9. The confinement conjecture
    • LECTURE 8 Abelian Duality
    • §8.1. Introduction
    • §8.2. Duality in two dimensions
    • §8.3. Duality in three dimensions
    • §8.4. Application to the Polyakov model
    • §8.5. Duality in four dimensions and SL(2, Z)
    • §8.6. The Hamiltonian formalism
    • LECTURE 9 Solitons
    • §9.1. What is a soliton?
    • §9.2. Solitons and components of the space of classical solutions
    • §9.3. Solitons and quantization
    • §9.4. Solitons in theories with fermions
    • §9.5. Solitons in 2+1 and 3+1 dimensions
    • §9.6. The 3+1-dimensional theory with the 0-angle
    • LECTURE 10 Wilson Loops, 't Hooft Loops, and 't Hooft's Picture of Confinement
    • 10.1. 't Hooft loop operator
    • 10.2. Hilbert space interpretation of the 't Hooft loop operator
    • 10.3. The picture of confinement
    • LECTURE 11 Quantum Gauge Theories in Two Dimensions and Intersection Theory on Moduli Spaces
    • §11.1. The partition function in two dimensional Yang-Mills theory
    • §11.2. A finite dimensional analogue: the Cartan model
    • §11.3. Infinite dimensional Cartan: the descent equations
    • §11.4. Equivariant integration and localization
    • §11.5. Equivariant integration: the infinite dimensional case
    • §11.6. The partition function of Yang-Mills theory
    • LECTURE 12 Supersymmetric Field Theories
    • §12.1. Supersymmetric Configurations
    • §12.2. Supersymmetric solitons {BPS states)
    • §12.3. The role of BPS states in quantum theory
    • §12.4. N = 2 supersymmetry in 2 dimensions
    • §12.5. N = 2 BPS states
    • §12.6. N = 1 Supersymmetry in 4 dimensions
    • §12. 7. N = 2 Supersymmetry in 4 dimensions
    • LECTURE 13 N = 2 SUSY theories in Dimension Two: Part I
    • §13.1. Introduction
    • §13.2. Generalities on N = 2 SUSY theories in dimension two
    • §13.3. The U(l) Theories
    • §13.4. One Example
    • §13.5. Another example: flops
    • §13.6. Cases in which c1≠0
    • LECTURE 14 N = 2 SUSY Theories in Dimension Two, Part II: Chiral Rings and Twisted Theories
    • §14.1. R-symmetry revisited
    • §14.2. Q-cohomology of operators
    • §14.3. Twisting the theory to give it global meaning
    • §14.4. A Gauge Theory Example
    • §14.5. A σ-model example
    • LECTURE 15 The Landau-Ginzburg Description of N = 2 Minimal Models; Quantum Cohomology of Kahler Manifolds
    • §15.1. Landau-Ginzburg models
    • §15.2. The elliptic genus
    • §15.3. Introduction to quantum cohomology
    • §15.4. The space of zero-energy states
    • §15.5. Generalities on the chiral ring
    • §15.6. More Details on the Ring Structure
    • §15. 7. Calculations for CPn-1
    • §15.8. Calculations for Fano Hypersurfaces
    • LECTURE 16 Four-dimensional gauge theories
    • §16.1. Gauge theory without supersymmetry
    • §16.2. N = 1 supersymmetric pure gauge theory
    • §16.3. N = l theories with chiral superfields
    • LECTURE 17 N = 2 Supersymmetric Yang-Mills Theories in Dimension Four: Part 1
    • §17.1. Introduction
    • §17.2. Low energy U(l) N = 2 super Yang-Mills theories
    • LECTURE 18 N = 2 Supersymmetric Yang-Mills Theories in Dimension Four: Part 2
    • §18.1. Review of material from the last lecture
    • §18.2. First results about the moduli space M of quantum vacua
    • §18.3. The nature of infinity in M
    • §18.4. BPS states and singularities in M
    • §18.5. The number of singularities in M
    • §18.6. The new massless particles
    • §18.7. Explicit nature of the family of elliptic curves
    • §18.8. Description of the BPS spectrum
    • LECTURE 19 N = 2 Super-symmetric Yang-Mills Theories in Dimension Four: Part 3, Topological Applications
    • §19.1. A survey of N = 2 supersymmetric gauge theories in dimension four
    • §19.2. From Minkowski space to a compact Riemannian four-manifold
    • §19.3. The general form of the high energy computations
    • §19.4. Low Energy Computations for Donaldson theory
    • EXERCISES
    • SOLUTIONS TO SELECTED EXERCISES
    • Dynamics of N = 1 Supersymmetric Field Theories in Four Dimensions
    • LECTURE 1 Basic Aspects of N = 1 QCD
    • §1.1. Wess-Zumino model
    • §1.2. Pure supersymmetric gauge theory
    • §1.3. Supersymmetric QCD
    • LECTURE 2 Quantum Behavior of Super QCD: Nf Small
    • §2.1. Nf = Nc - 1
    • §2.2. Nf < Nc - 1
    • §2.3. Nf = Nc
    • §2.4. Nf= Nc+ l
    • §2.5. 't Hooft anomaly matching condition
    • LECTURE 3 Quantum Behavior of Super QCD: Nf Large
    • §3.1. 3Nc/2 < Nf < 3Nc: non-trivial infrared fixed points
    • §3.2. Nc+ 2 < Nf ≤ 3Nc/2: infrared free magnetic theory
    • §3.3. Further tests of duality
    • EXERCISES
    • SOLUTIONS TO EXERCISES (BY SIYE WU) General Remarks
    • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 1-2
    • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 3-4
    • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 5-6
    • Index
  • Additional Material
     
     
  • Reviews
     
     

    • An immense amount of valuable material on recent developments. The development of classical supersymmetry by Deligne and collaborators is careful and systematic ... masterful treatment ...the book is a magnificent achievement.
      SIAM Review

    • A concise introduction to the quantum field theory and perturbative string theory, with as much emphasis on a mathematically satisfying exposition and clarity as possible ... will be helpful to all mathematicians and mathematical physicists who wish to learn about the beautiful subject of quantum field theory.
      European Mathematical Society Newsletter
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
1999; 778 pp
MSC: Primary 81

Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics.

In 1996–97 the Institute for Advanced Study (Princeton, NJ) organized a special year-long program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: “The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra.”

These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures.

These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory.
Readership

Graduate students and research mathematicians working in various areas of mathematics related to quantum field theory.
This item is also available as part of a set:
  • Preface
  • Brief Contents
  • Cross-Reference Codes
  • Contents
  • Part 3 Conformal Field Theory and Strings
  • Lectures on Conformal Field Theory
  • LECTURE 1 Simple Functional Integrals
  • §1.1. What is quantum field theory?
  • §1.2. Euclidean free field and Gaussian functional integrals
  • §1.3. Feynman-Kac formula
  • §1.4. Massless free field with values in S1
  • §1.5. Toroidal compactifications: the partition functions
  • §1.6. Toroidal compactifications: the correlation functions
  • LECTURE 2 Axiomatic Approaches to Conformal Field Theory
  • §2.1. Conformal field theory data
  • §2.2. Conformal Ward identities
  • §2.3. Physical positivity and Hilbert space picture
  • §2.4. Virasoro algebra and its primary fields
  • §2.5. Highest weight representations of Vir
  • §2.6. Segal's axioms and vertex operator algebras
  • LECTURE 3 Sigma Models
  • §3.1. lPI effective action and large deviations
  • §3.2. Geometric sigma models
  • §3.3. Regularization and renormalization
  • §3.4. Renormalization group effective actions
  • §3.5. Background field effective action
  • §3.6. Dimensional regularization
  • §3.7. Renormalization of the sigma models to 1 loop
  • §3.8. Renormalization group analysis of sigma models
  • LECTURE 4 Constructive Conformal Field Theory
  • §4.1. WZW model
  • §4.2. Gauge symmetry Ward identities
  • §4.3. Scalar product of non-abelian theta functions
  • §4.4. KZB connection
  • §4.5. Coset theories
  • §4.6. WZW factory
  • String Theory
  • LECTURE 1 Point Particles vs Strings
  • §1.1. Strings
  • §1.2. Interactions
  • §1.3. Loop expansion-topology of closed surfaces
  • §1.4. Transition amplitudes for strings
  • §1.5. Weyl invariance and vertex operator formulation
  • §1.6. More general actions
  • §1. 7. Transition amplitude for a single point particle
  • §1.8. Generalized point particle propagation
  • LECTURE 2 Spectrum of Free Bosonic Strings
  • §2.1. Basics of conformal field theory
  • §2.2. The free closed bosonic string conformal field theory
  • §2.3. The free open bosonic string conformal field theory
  • §2.4. Fock space, negative norm states
  • §2.5. Integration over Met(Σ)-Virasoro constraints
  • §2.6. Physical spectrum, no-ghost theorem
  • §2. 7. Spectrum of the critical bosonic string with D = 26, a= 1
  • §2.8. Lightcone gauge, density of states
  • §2.9. Primary fields and vertex operators for physical states
  • §2.10. Identifying the graviton: vertex operators from background fields
  • §2.11. Internal degrees of freedom of open strings: Chan-Paton rules
  • LECTURE 3 String Amplitudes and Moduli Space of Curves
  • §3.1. Finite-dimensional case
  • §3.2. Basic notation: tensors, derivatives
  • §3.3. Space of metrics - moduli space of Riemann surfaces
  • §3.4. Factorizing the integration measure
  • §3.5. Weyl rescalings of functional determinants
  • §3.6. Critical central charge-critical dimension
  • §3.7. Flat space-time manifold M
  • §3.8. Non-critical strings
  • §3.9. Tree level amplitudes
  • §3.10. One loop amplitudes
  • LECTURE 4 Faddeev-Popov Ghosts-BRST Quantization
  • §4.1. Determinants and b-c systems
  • §4.2. Ghost representation of the Faddeev-Popov determinant
  • §4.3. Conformal field theory of the b-c system
  • §4.4. Bosonization of the b-c system
  • §4.5. The b-c Fock space
  • §4.6. BRST quantization
  • LECTURE 5 Moduli Dependence of Determinants and Green Functions
  • §5.1. Worldsheets with constant curvature metric (h≥ 2)
  • §5.2. Holomorphicity in moduli
  • §5.3. The chiral splitting theorem
  • §5.4. Holomorphic and meromorphic differentials (a brief review of basics)
  • §5.5. Green functions, determinants and chiral splitting
  • LECTURE 6 Strings on General Manifolds
  • §6.1. Perturbation theory around general field configurations
  • §6.2. Renormalization of generalized non-linear sigma models
  • §6.3. General structure of Weyl dependence
  • §6.4. General structure of Weyl anomaly in low energy expansion
  • §6.5. Background field quantization method
  • §6.6. Covariant expansion methods
  • §6.7. Reformulation as an SO(l,D-1) gauge theory
  • §6.8. Weyl variation of the effective action
  • §6.9. Low energy string field equations and string effective action
  • §6.10. A first look at compactification
  • Appendix A
  • LECTURE 7 Free Superstrings
  • §7.1. Degrees of freedom of the RNS string
  • §7.2. Rarnond and Neveu-Schwarz Fock spaces
  • §7.3. Local supersymmetry on the worldsheet
  • §7.4. Functional integral representation of transition amplitudes
  • §7.5. Super-Virasoro algebra and physical spectrum
  • §7.6. The spectrum of physical states at low mass
  • §7.7. The GSO projection, space-time supersymmetry
  • §7.8. Type IIA, B superstrings and their spectra
  • §7.9. Type I superstring
  • LECTURE 8 Heterotic Strings
  • §8.1. Free fermion realization of internal degrees of freedom
  • §8.2. Free fermion realization of the Spin(32)/Z2 heterotic string
  • §8.3. Free fermion realization of the E8 x E8 heterotic string
  • §8.4. Bosonic realizations of the Spin(32)/Z2 and E8 x E8 strings
  • LECTURE 9 Superstring Perturbation Theory
  • §9.1. N = 1 supergeometry
  • §9.2. Functional integral representation of transition amplitudes
  • §9.3. Superconformal field theory (some basics)
  • §9.4. BRST quantization
  • §9.5. Vertex operators for physical states
  • §9.6. The chiral splitting theorem
  • §9.7. Tree-level amplitudes for NS-NS states
  • §9.8. One-loop amplitudes for NS-NS states in Type IIA, B
  • §9.9. One-loop amplitudes in the heterotic string
  • §9.10. The NS-NS 4-point function
  • LECTURE 10 Supersymmetry and Supergravity
  • §10.1. Global space-time supersymmetry in the RNS formulation
  • §10.2. The Green-Schwarz formulation
  • §10.3. Lightcone gauge quantization of the GS formulation
  • §10.4. Flat superspace GS formulation
  • §10.5. Supergravity and low energy superstrings
  • §10.6. Type IIA, D = 10, 'N = 2 and D = 11, '.N = 1 supergravities
  • §10.7. Type IIB, D = 10, N = 2 supergravity
  • §10.8. Type I - Heterotic, D = 10, '.N = 1 supergravities
  • §10.9. Superspace formulation of supergravities in D = ll and D=lO
  • §10.10. Local supersymmetric coupling of superstrings to supergravity
  • EXERCISES
  • Super Space Descriptions of Super Gravity
  • §1. General remarks
  • §2. Example: D = 2, N = 1 in the notations of §9.1 of D'Hoker's lectures
  • Notes on 2d Conformal Field Theory and String Theory
  • §0.1. Contents of these notes
  • §0.2. Some background on D-modules
  • CHAPTER 1 Chiral Algebras
  • § 1. 1. Definition of chiral algebras
  • §1.2. Lie-* algebras and construction of chiral algebras
  • §1.3. Conformal blocks, correlation functions
  • CHAPTER 2 CFT Data (Algebraic Version)
  • §2.1. Local O-modules on X
  • §2.2. A formulation of CFT (central charge 0)
  • §2.3. Introducing the central charge
  • CHAPTER 3 Examples
  • §3.1. Heisenberg and Kac-Moody algebras
  • §3.2. The linear dilaton
  • §3.3. The be-system
  • CHAPTER 4 BRST and String Amplitudes
  • §4.1. The BRST complex
  • §4.2. String amplitudes
  • CHAPTER 5 Further Constructions
  • §5.1. Chiral algebras via the Ran space
  • §5.2. Geometry of the affine Grassmannian
  • §5.3. Chiral algebra attached to the affine Grassmannian
  • CHAPTER 6 The Free Bosonic Theory
  • §6.1. The canonical line bundle
  • §6.2. Construction of the bosonic chiral algebra
  • BIBLIOGRAPHY
  • Kaluza-Klein Compactifications,Super symmetry, and Calabi-Yau Spaces
  • LECTURE 1 Compactification to Dimension Four
  • §1.1. Kaluza-Klein model
  • §1.2. Compactifying Einstein's equation from dimension ten to dimension four
  • §1.3. Adding Matter to the Mix
  • §1.4. The effective action from ten-dimensional heterotic string theory
  • LECTURE 2 Supersymmetry and Calabi-Yau Manifolds
  • §2.1. Review of material from the first lecture
  • §2.2. Partially breaking the supersymmetry by compactifying down to dimension four
  • §2.3. Geometric consequence of the unbroken supersymmetry
  • §2.4. Massless fields in the low energy effective Lagrangian
  • §2.5. Relation to Grand Unification Theory
  • Part 4 Dynamical Aspects of QFT
  • Dynamics of Quantum Field Theory
  • LECTURE 1 Symmetry Breaking
  • §1.0. Theories and realizations
  • §1.1. What is symmetry breaking, and why it does not happen in quantum mechanics
  • §1.2. Still no symmetry breaking in quantum field theory in finite volume
  • §1.3. Symmetry breaking in quantum field theory in infinite volume
  • §1.4. Infinite volume asymptotics of correlation functions
  • §1.5. Continuous symmetry breaking
  • §1.6. Goldstone's theorem
  • §1. 7. Infrared behavior of purely non-renormalizable field theories
  • §1.8. Effective action for Goldstone bosons
  • LECTURE 2 Gauge Symmetry Breaking and More on Infrared Behavior
  • §2.1. Gauge symmetry
  • §2.2. Breaking of gauge symmetry and charges at infinity
  • §2.3. Symmetry breaking and gauging
  • §2.4. No massless particles of higher spin
  • §2.5. Infrared limits
  • LECTURE 3 BRST Quantization of Gauge Theories
  • §3.1. The general setup
  • §3.2. The BRST differential
  • §3.3. The properties of the BRST derivation
  • §3.4. Operators in gauge theory and BRST cohomology
  • §3.5. Renormalization and BRST differential
  • §3.6. The Hamiltonian approach
  • §3.7. Anomalies
  • LECTURE 4 Infrared Behavior and the S-matrix of the Two-Dimensional Sigma Model with Target Space SN-1
  • §4. 1. Infrared behavior of certain two-dimensional sigma models
  • §4.2. Computation of the infrared behavior in the N→∞ limit
  • §4.3. Computation of the S-matrix
  • LECTURE 5 The Large N Limit of the a-model into Grassmannians
  • §5.1. The questions
  • §5.2. An equivalent formulation
  • §5.3. The large N effective theory
  • §5.4. Real Grassmannians
  • §5.5. Pure gauge theory
  • §5.6. Classical electromagnetism in two dimensions
  • §5.7. Quantum theory with matter
  • LECTURE 6 The Bose-Fermi Correspondence and its Applications
  • §6.1. Two-dimensional gauge theories with fermions
  • §6.2. Chiral symmetry
  • §6.3. Behavior of two-dimensional gauge theory with massive fermions
  • §6.4. Heavy fermions
  • §6.5. Bose-Fermi correspondence
  • §6.6. Bose-Fermi correspondence for nonlinear theories
  • LECTURE 7 Two-Dimensional Gauge Theory of bosons, the Wilson Line Operator, and Confinement
  • §7.1. Infrared behavior of U(l) gauge theories with bosons in two dimensions
  • §7.2. The vacuum energy density
  • §7.3. Instantons
  • §7.4. lnstanton gas
  • §7.5. Summing over instantons
  • §7.6. The Wilson line operator
  • § 7. 7. The path integral representation of the Wilson line operator
  • §7.8. The Higgs and the confinement regimes
  • §7.9. The confinement conjecture
  • LECTURE 8 Abelian Duality
  • §8.1. Introduction
  • §8.2. Duality in two dimensions
  • §8.3. Duality in three dimensions
  • §8.4. Application to the Polyakov model
  • §8.5. Duality in four dimensions and SL(2, Z)
  • §8.6. The Hamiltonian formalism
  • LECTURE 9 Solitons
  • §9.1. What is a soliton?
  • §9.2. Solitons and components of the space of classical solutions
  • §9.3. Solitons and quantization
  • §9.4. Solitons in theories with fermions
  • §9.5. Solitons in 2+1 and 3+1 dimensions
  • §9.6. The 3+1-dimensional theory with the 0-angle
  • LECTURE 10 Wilson Loops, 't Hooft Loops, and 't Hooft's Picture of Confinement
  • 10.1. 't Hooft loop operator
  • 10.2. Hilbert space interpretation of the 't Hooft loop operator
  • 10.3. The picture of confinement
  • LECTURE 11 Quantum Gauge Theories in Two Dimensions and Intersection Theory on Moduli Spaces
  • §11.1. The partition function in two dimensional Yang-Mills theory
  • §11.2. A finite dimensional analogue: the Cartan model
  • §11.3. Infinite dimensional Cartan: the descent equations
  • §11.4. Equivariant integration and localization
  • §11.5. Equivariant integration: the infinite dimensional case
  • §11.6. The partition function of Yang-Mills theory
  • LECTURE 12 Supersymmetric Field Theories
  • §12.1. Supersymmetric Configurations
  • §12.2. Supersymmetric solitons {BPS states)
  • §12.3. The role of BPS states in quantum theory
  • §12.4. N = 2 supersymmetry in 2 dimensions
  • §12.5. N = 2 BPS states
  • §12.6. N = 1 Supersymmetry in 4 dimensions
  • §12. 7. N = 2 Supersymmetry in 4 dimensions
  • LECTURE 13 N = 2 SUSY theories in Dimension Two: Part I
  • §13.1. Introduction
  • §13.2. Generalities on N = 2 SUSY theories in dimension two
  • §13.3. The U(l) Theories
  • §13.4. One Example
  • §13.5. Another example: flops
  • §13.6. Cases in which c1≠0
  • LECTURE 14 N = 2 SUSY Theories in Dimension Two, Part II: Chiral Rings and Twisted Theories
  • §14.1. R-symmetry revisited
  • §14.2. Q-cohomology of operators
  • §14.3. Twisting the theory to give it global meaning
  • §14.4. A Gauge Theory Example
  • §14.5. A σ-model example
  • LECTURE 15 The Landau-Ginzburg Description of N = 2 Minimal Models; Quantum Cohomology of Kahler Manifolds
  • §15.1. Landau-Ginzburg models
  • §15.2. The elliptic genus
  • §15.3. Introduction to quantum cohomology
  • §15.4. The space of zero-energy states
  • §15.5. Generalities on the chiral ring
  • §15.6. More Details on the Ring Structure
  • §15. 7. Calculations for CPn-1
  • §15.8. Calculations for Fano Hypersurfaces
  • LECTURE 16 Four-dimensional gauge theories
  • §16.1. Gauge theory without supersymmetry
  • §16.2. N = 1 supersymmetric pure gauge theory
  • §16.3. N = l theories with chiral superfields
  • LECTURE 17 N = 2 Supersymmetric Yang-Mills Theories in Dimension Four: Part 1
  • §17.1. Introduction
  • §17.2. Low energy U(l) N = 2 super Yang-Mills theories
  • LECTURE 18 N = 2 Supersymmetric Yang-Mills Theories in Dimension Four: Part 2
  • §18.1. Review of material from the last lecture
  • §18.2. First results about the moduli space M of quantum vacua
  • §18.3. The nature of infinity in M
  • §18.4. BPS states and singularities in M
  • §18.5. The number of singularities in M
  • §18.6. The new massless particles
  • §18.7. Explicit nature of the family of elliptic curves
  • §18.8. Description of the BPS spectrum
  • LECTURE 19 N = 2 Super-symmetric Yang-Mills Theories in Dimension Four: Part 3, Topological Applications
  • §19.1. A survey of N = 2 supersymmetric gauge theories in dimension four
  • §19.2. From Minkowski space to a compact Riemannian four-manifold
  • §19.3. The general form of the high energy computations
  • §19.4. Low Energy Computations for Donaldson theory
  • EXERCISES
  • SOLUTIONS TO SELECTED EXERCISES
  • Dynamics of N = 1 Supersymmetric Field Theories in Four Dimensions
  • LECTURE 1 Basic Aspects of N = 1 QCD
  • §1.1. Wess-Zumino model
  • §1.2. Pure supersymmetric gauge theory
  • §1.3. Supersymmetric QCD
  • LECTURE 2 Quantum Behavior of Super QCD: Nf Small
  • §2.1. Nf = Nc - 1
  • §2.2. Nf < Nc - 1
  • §2.3. Nf = Nc
  • §2.4. Nf= Nc+ l
  • §2.5. 't Hooft anomaly matching condition
  • LECTURE 3 Quantum Behavior of Super QCD: Nf Large
  • §3.1. 3Nc/2 < Nf < 3Nc: non-trivial infrared fixed points
  • §3.2. Nc+ 2 < Nf ≤ 3Nc/2: infrared free magnetic theory
  • §3.3. Further tests of duality
  • EXERCISES
  • SOLUTIONS TO EXERCISES (BY SIYE WU) General Remarks
  • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 1-2
  • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 3-4
  • SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 5-6
  • Index

  • An immense amount of valuable material on recent developments. The development of classical supersymmetry by Deligne and collaborators is careful and systematic ... masterful treatment ...the book is a magnificent achievement.
    SIAM Review

  • A concise introduction to the quantum field theory and perturbative string theory, with as much emphasis on a mathematically satisfying exposition and clarity as possible ... will be helpful to all mathematicians and mathematical physicists who wish to learn about the beautiful subject of quantum field theory.
    European Mathematical Society Newsletter
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