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Generalized complex structure and generalized complex manifolds
Just as for an ordinary almost complex structure, it is possible to give an integrability condition for a generalized almost complex structure. We will define integrability as the requirement that the “holomorphic” part of the complexified TM ⊕T∗M is integrable with respect to a bracket, the Courant bracket, that generalize the Lie bracket toTM ⊕ T∗M. The definition of the Courant bracket will be given in the next section.
A generalized almost complex structure J is integrable if its i eigenbundle LJ is closed under the Courant bracket ¯ [(v + ξ), (w + η)]C = 0 , (3.20).
where is the projector on LJ ⊂ TM ⊕ T∗M. In this case, J is called a generalized complex structure. A manifold on which such a tensor exists is called a generalized complex manifold.
Twisted Courant bracket
The main feature of a derived bracket is that it contains a differential. For both Lie and Courant the differential is d, but one can generalize it to other differentials. A natural generalization is the inclusion a closed three–form H, to form the differential d−H∧. The bracket will of course be modified in a way that is very natural in string theory. Indeed, if B is not a globally defined form, but is actually a field, like it is the case for the (e.g. the NS two-form, a B-transform is not be a symmetry of Courant. We can can use it to modify the definition of the Courant bracket, and introduce the twisted Courant bracket [v + ξ,w + η]H = [v,w] + Lvη − Lwξ − 1 2d (ιvη − ιwξ) + ιvιwH .
Pure spinors and generalized complex structures
There is a one-to-one correspondence between generalized complex structures and pure spinors. The isomorphism holds both at the topological and differential level.
At the topological level, the correspondence is based on the fact that, given an almost complex structure, J , one can always built a pure spinor that has the +i eigenspace of J as annihilator annihilator of = i − eigenspace ofJ . (3.55) For instance, for an SU(3) structure we have J1 ←→ J2 ←→ e−iJ . (3.56) Given a pure spinor , we can also define the corresponding generalized almost complex structure J as J± = hRe(±)), Re(±)i , (3.57) where , are indices on TM ⊕ T∗M, and are Cliff(d, d) gamma matrices.
Since rescaling the pure spinor does not change its annihilator L, to each almost complex structure we can associate a line bundle of pure spinors. In general this line bundle does not have a global section. When this is the case, the structure group on TM ⊕ T∗M is further reduced from U(n, n) to SU(n, n).
At the differential level, the correspondence is a relation between the integrability of J and some differential properties of the associated pure spinor. J integrable ⇔ d = (ιv + ξ∧).
Supersymmetric solutions
We will actually restrict even more the form of the solutions we are after, by imposing that they have minimal supersymmetry, namely N = 1 in four dimensions.
From a technical point of view, looking for supersymmetric solutions simplifies life a lot. Indeed, it can be shown that, under some hypothesis1 that are verified for the type of solutions we are interested in [15], instead of solving the equations of motion, which are second order differential equations, one can solve a set of first order equations:
• vanishing of supersymmetry variations for the gravitino and the dilatino.
• Bianchi identities and equations of motion for the fluxes.
Minimal supersymmetry is a phenomenological requirement, since extended supersymmetries do not admit chiral fermions and thus do not give rise to a physically relevant spectrum of particles in the low energy actions obtained compactifying around one such vacuum. We do not address here the issue of how supersymmetry is broken. We simply assume that it is spontaneously broken at low enough energies.
Supersymmetry equations
For a purely bosonic background, the conditions for unbroken supersymmetry is that the variations of the fermionic fields vanish. Indeed, an unbroken supersymmetry is such that its generator (the conserved supercharge) annihilates the vacuum state Q|0 >= 0 , (4.7) which is equivalent to the condition that for all operators O in the theory < 0|Q,O|0 >= 0 .
Pure geometry: Calabi-Yau compactifications
We start by considering the case of purely geometric compactifications, where the only non trivial field is the metric. Since all the fluxes are set to zero, in order to find a solution it is enough to solve the supersymmetry variations.
When all fluxes are set to zero, using the metric ansazt (4.3) and the splitting (4.15) and (4.16) for the spinors, the dilatino variation (4.11) reduces to the six-dimensional equations2 /∂φ η1,2 = 0 , (4.23) where /∂φ = γm∂mφ. Since ||/∂φ η1,2||2 = (∂φ)2||η1,2||2, it follows that the dilaton must be constant. The gravitino variations reduce to the requirement that the supersymmetry parameters must be covariantly constant δψ1M = ∇Mǫ1 = 0 , δψ2M = ∇Mǫ2 = 0 , (4.24) where ∇M = ∂M + 1 4ωM ABAB is the standard spinorial covariant derivative.
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
B R´esum´e long en fran¸cais
B.1 Structures sur les vari´et´es
B.1.1 Groupes de structures
B.1.2 Les cas SU(2) et SU(3)
B.2 G´eom´etrie complexe g´en´eralis´ee
B.2.1 Structure complexe g´en´eralis´ee
B.2.2 Spineurs purs
B.3 Supersym´etrie
B.3.1 G´en´eralit´es
B.3.2 G´eom´etrie complexe g´en´eralis´ee et compactifications N = 1
B.4 Etudes des vides AdS sur les vari´et´es parall´elisable
B.4.1 Recherche sur les groupes de Lie
B.4.2 Quotients
B.4.3 S´eparation d’´echelle
