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Table of contents
1 Introduction
2 G-Structure and Torsion
2.1 G-structures
2.1.1 Structure group
2.1.2 G-structure
2.1.3 Almost complex structure and almost complex manifold
2.1.4 Hermitian metric and almost Hermitian manifold
2.1.5 Symplectic structures
2.1.6 Product structure
2.1.7 SU(3) and SU(2) structures in dimension
2.2 Holonomy group and torsion
2.2.1 Examples of integrable structures
2.2.2 Intrinsic torsion
2.2.3 Special holonomy
2.2.4 Torsion for SU(3)-structure in dimension
2.2.5 Torsion for SU(2)-structure in dimension
2.2.6 Calabi Yau manifold
3 Generalized Complex Geometry
3.1 The generalized tangent bundle
3.2 Generalized complex structure
3.2.1 Generalized complex structure and generalized complex manifolds
3.2.2 The Courant Bracket
3.3 Pure spinors
3.3.1 O(d, d) spinors
3.3.2 Pure spinors
3.3.3 Pure spinors and generalized complex structures
3.3.4 Type of a pure spinor
3.4 Generalized Calabi Yau manifolds
3.4.1 Differential structure of the manifold
3.5 Metric from pure spinors
4 Supersymmetry and N = 1 flux compactifications
4.1 Supersymmetric solutions
4.1.1 Supersymmetry equations
4.1.2 Bianchi identities and e.o.m for the forms
4.2 Pure geometry: Calabi-Yau compactifications
4.3 N = 1 flux compactifications: Generalized Calabi-Yau manifolds
4.3.1 Topological condition
4.3.2 Differential condition
5 Anti de Sitter vacua in type IIB SUGRA on cosets and group manifolds
5.1 Motivations
5.2 N = 1 SUSY AdS4 vacua in type IIB SUGRA
5.2.1 The SUSY equations for rigid SU(2) structure
5.2.2 Link to the SU(2) torsion classes
5.3 Restriction to parallelizable manifolds
5.3.1 Sources
5.3.2 O5 and O7 planes
5.4 Explicit examples 1: Cosets
5.5 Explicit examples 2: Group manifolds
5.5.1 Bianchi Identities
5.5.2 Intermediate SU(2) structures on group manifolds
5.5.3 Explicit vacua on group manifolds
5.6 Scales separation
5.6.1 Separation of scales without sources?
5.6.2 Study of scales separation on the explicit examples
6 Conclusion
A De Sitter Compactification
A.1 Motivations
A.2 The Manifold g p,−p,±1 5,17
A.3 Compactification
A.3.1 The metric
A.3.2 The fluxes
A.3.3 The 10-dimensional action
A.3.4 Bianchi Identities
A.3.5 The 4-dimensional action
A.4 Sources
A.4.1 Structure
A.4.2 SUSY source
A.4.3 SUSY breaking sources
