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Book Details1999; 778 ppMSC: 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 yearlong 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.ReadershipGraduate 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

CrossReference 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. FeynmanKac 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 nonabelian 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 expansiontopology 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, noghost 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: ChanPaton rules

LECTURE 3 String Amplitudes and Moduli Space of Curves

§3.1. Finitedimensional 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 chargecritical dimension

§3.7. Flat spacetime manifold M

§3.8. Noncritical strings

§3.9. Tree level amplitudes

§3.10. One loop amplitudes

LECTURE 4 FaddeevPopov GhostsBRST Quantization

§4.1. Determinants and bc systems

§4.2. Ghost representation of the FaddeevPopov determinant

§4.3. Conformal field theory of the bc system

§4.4. Bosonization of the bc system

§4.5. The bc 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 nonlinear 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,D1) 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 NeveuSchwarz Fock spaces

§7.3. Local supersymmetry on the worldsheet

§7.4. Functional integral representation of transition amplitudes

§7.5. SuperVirasoro algebra and physical spectrum

§7.6. The spectrum of physical states at low mass

§7.7. The GSO projection, spacetime 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. Treelevel amplitudes for NSNS states

§9.8. Oneloop amplitudes for NSNS states in Type IIA, B

§9.9. Oneloop amplitudes in the heterotic string

§9.10. The NSNS 4point function

LECTURE 10 Supersymmetry and Supergravity

§10.1. Global spacetime supersymmetry in the RNS formulation

§10.2. The GreenSchwarz 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 Dmodules

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 Omodules on X

§2.2. A formulation of CFT (central charge 0)

§2.3. Introducing the central charge

CHAPTER 3 Examples

§3.1. Heisenberg and KacMoody algebras

§3.2. The linear dilaton

§3.3. The besystem

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

KaluzaKlein Compactifications,Super symmetry, and CalabiYau Spaces

LECTURE 1 Compactification to Dimension Four

§1.1. KaluzaKlein 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 tendimensional heterotic string theory

LECTURE 2 Supersymmetry and CalabiYau 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 nonrenormalizable 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 Smatrix of the TwoDimensional Sigma Model with Target Space SN1

§4. 1. Infrared behavior of certain twodimensional sigma models

§4.2. Computation of the infrared behavior in the N→∞ limit

§4.3. Computation of the Smatrix

LECTURE 5 The Large N Limit of the amodel 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 BoseFermi Correspondence and its Applications

§6.1. Twodimensional gauge theories with fermions

§6.2. Chiral symmetry

§6.3. Behavior of twodimensional gauge theory with massive fermions

§6.4. Heavy fermions

§6.5. BoseFermi correspondence

§6.6. BoseFermi correspondence for nonlinear theories

LECTURE 7 TwoDimensional 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+1dimensional theory with the 0angle

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 YangMills 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 YangMills 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. Rsymmetry revisited

§14.2. Qcohomology of operators

§14.3. Twisting the theory to give it global meaning

§14.4. A Gauge Theory Example

§14.5. A σmodel example

LECTURE 15 The LandauGinzburg Description of N = 2 Minimal Models; Quantum Cohomology of Kahler Manifolds

§15.1. LandauGinzburg models

§15.2. The elliptic genus

§15.3. Introduction to quantum cohomology

§15.4. The space of zeroenergy states

§15.5. Generalities on the chiral ring

§15.6. More Details on the Ring Structure

§15. 7. Calculations for CPn1

§15.8. Calculations for Fano Hypersurfaces

LECTURE 16 Fourdimensional 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 YangMills Theories in Dimension Four: Part 1

§17.1. Introduction

§17.2. Low energy U(l) N = 2 super YangMills theories

LECTURE 18 N = 2 Supersymmetric YangMills 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 Supersymmetric YangMills 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 fourmanifold

§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. WessZumino 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: nontrivial 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 12

SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 34

SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 56

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


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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 yearlong 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.
Graduate students and research mathematicians working in various areas of mathematics related to quantum field theory.

Preface

Brief Contents

CrossReference 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. FeynmanKac 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 nonabelian 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 expansiontopology 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, noghost 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: ChanPaton rules

LECTURE 3 String Amplitudes and Moduli Space of Curves

§3.1. Finitedimensional 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 chargecritical dimension

§3.7. Flat spacetime manifold M

§3.8. Noncritical strings

§3.9. Tree level amplitudes

§3.10. One loop amplitudes

LECTURE 4 FaddeevPopov GhostsBRST Quantization

§4.1. Determinants and bc systems

§4.2. Ghost representation of the FaddeevPopov determinant

§4.3. Conformal field theory of the bc system

§4.4. Bosonization of the bc system

§4.5. The bc 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 nonlinear 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,D1) 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 NeveuSchwarz Fock spaces

§7.3. Local supersymmetry on the worldsheet

§7.4. Functional integral representation of transition amplitudes

§7.5. SuperVirasoro algebra and physical spectrum

§7.6. The spectrum of physical states at low mass

§7.7. The GSO projection, spacetime 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. Treelevel amplitudes for NSNS states

§9.8. Oneloop amplitudes for NSNS states in Type IIA, B

§9.9. Oneloop amplitudes in the heterotic string

§9.10. The NSNS 4point function

LECTURE 10 Supersymmetry and Supergravity

§10.1. Global spacetime supersymmetry in the RNS formulation

§10.2. The GreenSchwarz 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 Dmodules

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 Omodules on X

§2.2. A formulation of CFT (central charge 0)

§2.3. Introducing the central charge

CHAPTER 3 Examples

§3.1. Heisenberg and KacMoody algebras

§3.2. The linear dilaton

§3.3. The besystem

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

KaluzaKlein Compactifications,Super symmetry, and CalabiYau Spaces

LECTURE 1 Compactification to Dimension Four

§1.1. KaluzaKlein 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 tendimensional heterotic string theory

LECTURE 2 Supersymmetry and CalabiYau 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 nonrenormalizable 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 Smatrix of the TwoDimensional Sigma Model with Target Space SN1

§4. 1. Infrared behavior of certain twodimensional sigma models

§4.2. Computation of the infrared behavior in the N→∞ limit

§4.3. Computation of the Smatrix

LECTURE 5 The Large N Limit of the amodel 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 BoseFermi Correspondence and its Applications

§6.1. Twodimensional gauge theories with fermions

§6.2. Chiral symmetry

§6.3. Behavior of twodimensional gauge theory with massive fermions

§6.4. Heavy fermions

§6.5. BoseFermi correspondence

§6.6. BoseFermi correspondence for nonlinear theories

LECTURE 7 TwoDimensional 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+1dimensional theory with the 0angle

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 YangMills 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 YangMills 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. Rsymmetry revisited

§14.2. Qcohomology of operators

§14.3. Twisting the theory to give it global meaning

§14.4. A Gauge Theory Example

§14.5. A σmodel example

LECTURE 15 The LandauGinzburg Description of N = 2 Minimal Models; Quantum Cohomology of Kahler Manifolds

§15.1. LandauGinzburg models

§15.2. The elliptic genus

§15.3. Introduction to quantum cohomology

§15.4. The space of zeroenergy states

§15.5. Generalities on the chiral ring

§15.6. More Details on the Ring Structure

§15. 7. Calculations for CPn1

§15.8. Calculations for Fano Hypersurfaces

LECTURE 16 Fourdimensional 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 YangMills Theories in Dimension Four: Part 1

§17.1. Introduction

§17.2. Low energy U(l) N = 2 super YangMills theories

LECTURE 18 N = 2 Supersymmetric YangMills 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 Supersymmetric YangMills 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 fourmanifold

§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. WessZumino 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: nontrivial 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 12

SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 34

SOLUTIONS TO EXERCISES (BY SIYE WU) Problems 56

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.
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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.
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