M-Theory
Prerequisite: Quantum Field Theory, General Relativity
AdS/CFT correspondense Lecture note
String theory and M-theory
1. Introduction
2. The bosonic string
3. Conformal field theory and string interactions
4. Strings with wold-sheet supersymmetry
5. Strings with space-time supsersymmetry
6. T-duality and D-branes
7. The heterotic string
8. M-theory and string duality
9. String geometry
10. Flux compactifications
11. Black holes in string theory
12. Gauga theory/string theory dualities
String theory I
1. A first look at strings
1.1 Why strings?
1.2 Action principles
1.3 The open string spectrum
1.4 Closed and unoriented strings
2. Conformal field theory
2.1 Massless scalars in two dimensions
2.2 The operator product expansion
2.3 Ward identities and Noether's theorem
2.4 Conformal invariance
2.5 Free CFTs
2.6 The Virasoro algebra
2.7 Mode expansions
2.8 Vertex operators
2.9 More on states and operators
3. The Polyakov path integral
3.1 Sums over world-sheets
3.2 The Polyakov path integral
3.3 Gauge fixing
3.4 The Weyl anomaly
3.5 Scattering amplitudes
3.6 Vertex operators
3.7 String in curved spactime
4. The string spectrum
4.1 Old covariant quantization
4.2 BRST quantization
4.3 BRST quantization of the string
4.4 The no-ghost theorem
5. The string S-matrix
5.1 The circle and the torus
5.2 Moduli and Riemann surfaces
5.3 The measure for moduli
5.4 More about the measure
6. Tree-level amplitudes
6.1 Riemann surfaces
6.2 Scalar expectation values
6.3 The $bc$ CFT
6.4 The Veneziano amplitude
6.5 Chan-Paton factors and gauge interactions
6.6 Closed string tree amplitudes
6.7 General results
7. One-loop amplitudes
7.1 Riemann surfaces
7.2 CFT on the torus
7.3 The torus amplitude
7.4 Open and unoriented one-loop graphs
8. Toroidal compactification and $T$-duality
8.1 Toroidal compactification in field theory
8.2 Toroidal compactification in CFT
8.3 Closed strings and $T$-duality
8.4 Compactification of several dimensions
8.5 Orbifiolds
8.6 Open strings
8.7 D-branes
8.8 $T$-duality of unoriented theoreis
9. Higher order amplitudes
9.1 General tree-level amplitudes
9.2 Higher genus Riemann surfaces
9.3 Sewing and cutting world-sheets
9.4 Sewing and cutting CFTs
9.5 General amplitudes
9.6 String field theory
9.7 Large order behavior
9.8 High energy and high temperature
9.9 Low dimensions and noncritical strings
String theory II
10. Type I and type II superstrings
10.1 The superconformal algebra
10.2 Ramond and Neveu-Schwarz sectors
10.3 Vertex operators and bosonization
10.4 The superconformal ghosts
10.5 Physical states
10.6 Superstring theories in ten dimensions
10.7 Modular invariance
10.8 Divergences of type I theory
11. The heterotic string
11.1 World-sheet supersymmetries
11.2 The $SO(32)$ and $E_8\times E_8$ heterotic strings
11.3 Other ten-dimensional heterotic strings
11.4 A little Lie algebra
11.5 Current algebras
11.6 The bosonic construction and toroidal compactification
12. Superstring interactions
12.1 Low energy supergravity
12.2 Anomalies
12.3 Superspace and superfields
12.4 Tree-level amplitudes
12.5 General amplitudes
12.6 One-loop amplitudes
13. D-branes
13.1 $T$-duality of type II strings
13.2 $T$-duality of type I strings
13.3 The D-brane charge and action
13.4 D-brane interactions: statics
13.5 D-brane interactions: dynamics
13.6 D-brane interactions: bound states
14. Strings at strong coupling
14.1 Type IIB string and $SL(2,Z)$ duality
14.2 $U$-duality
14.3 $SO(32)$ type I-heterotic duality
14.4 Type IIA string and M-theory
14.5 The $E_8\times E_8$ heterotic string
14.6 What is string theory?
14.7 Is M for matrix?
14.8 Black hole quantum mechanics
15. Advanced CFT
15.1 Representations of the Virasoro algebra
15.2 The conformal bootstrap
15.3 Minimal modles
15.4 Current algebras
15.5 Coset models
15.6 Representations of the $N=1$ superconformal algebra
15.7 Rational CFT
15.8 Renormalization group flows
15.9 Statistical mechanics
16. Orbifolds
16.1 Orbifolds of the heterotic string
16.2 Spacetime supersymmetry
16.3 Examples
16.4 Low energy field theory
17. Calabi-Yau compactification
17.1 Conditions for $N=1$ supersymmetry
17.2 Calabi-Yau manifolds
17.3 Massless spectrum
17.4 Low energy field theory
17.5 Higher corrections
17.6 Generalizations
18. Physics in four dimensions
18.1 Continuous and discrete symmetries
18.2 Gauge symmetries
18.3 Mass scales
18.4 More on unification
18.5 Conditions for spactime supersymmetry
18.6 Low energy actions
18.7 Supersymmetry breaking in perturbation theory
18.8 Supersymmetry beyond perturbation theory
19. Advanced topics
19.1 The $N=2$ superconformal algebra
19.2 Type II string on Calabi-Yau manifolds
19.3 Heterotic string theoreis with (2,2) SCFT
19.4 $N=2$ minimal models
19.5 Gepner models
19.6 Mirror symmetry and applications
19.7 The conifold
19.8 String theoreis on K3
19.9 String duality below ten dimensions
19.10 Conclusion
Mirror symmetry
Part 1. Mathematical Preliminaries
1. DIfferential Geometry
2. Algebraic Geoemtry
3. Differential and Algebraic Topology
4. Equivariant Cohomology and Fixed-Point Theorems
5. Complex and Kahler Geometry
6. Calabi-Yau Manifolds and Their Moduli
7. Toric Geometry for String Theory
Part 2. Physics Preliminaries
8. What Is a QFT?
9. QFT in $d=0$
10. QFT in Dimension 1: Quantum Mechanics
11. Free Quantum Field Theories in 1+1 Dimensions
12. $\mathcal{N}=(2,2)$ Supersymmetry
13. Non-linear Sigma Models and Landau-Ginzburg Models
14. Renormalization Group Flow
15. Linear Sigma Models
16. Chiral Rings and Topological Field Theory
17. Chiral Rings and the Geometry of the Vacuum Bundle
18. BPS Solitons in $\mathcal{N}=2$ Landau-Ginzburg Theories
19. D-branes
Part 3. Mirror Symmetry: Physics Proof
20. Proof of Mirror Symmetry
Part 4. Mirror Symmetry: Mathematics Proof
21. Introduction and Overview
22. Complex Curves (Non-singular and Nodal)
23. Moduli Spaces of Curves
24. Moduli Spaces $\bar{\mathcal{M}}_{g,n}(X,\beta)$ of Stable Maps
25. Cohomology Classes on $\bar{\mathcal{M}}_{g,n}$ and $\bar{\mathcal{M}}_{g,n}(X,\beta)$
26. The Virtual Fundamental Class, Gromov-Witten Invariants, and Descendant Invariants
27. Localization on the Moduli Space of Maps
28. The Fundamental Solution of the Quantum Differential Equation
29. The Mirror Conjecture for Hypersurfaces I: The Fano Case
30. The Mirror Conjecture for Hypersurfaces II: The Calabi-Yau Case
Part 5. Advanced Topics
31. Topological Strings
32. Topological Strings and Target Space Physics
33. Mathematical Formulation of Gopakumar-Vafa Invariants
34. Multiple Covers, Integrality, and Gopakumar-Vafa Invariants
35. Mirror Symmetry at Higher Genus
36. Some Applications of Mirror Symmetry
37. Aspects of Mirror Symmetry and D-branes
38. More on the Mathematics of D-branes: Bundles, Derived Categories, and Lagrangians
39. Boundary $\mathcal{N}=2$ Theories
Reference
String theory and M-theory
String theory
Mirror symmetry