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Compactifications, fluxes and generalized geometry
The idea that the unification of the fundamental interactions may be related with the existence of supplementary spacetime dimensions is old, and dates back to the works of Kaluza and Klein [8], who derived Einstein’s general relativity together with Maxwell’s electromagnetism in four dimensions from a pure gravity action defined in five dimensions. Furthermore, many supergravity theories in diverse dimensions are related by compact-ification. In fact, since the early ages of supergravity, dimensional reductions have also served as a powerful tool to construct lower-dimensional supergravity theories starting from higher-dimensional ones, whose field content is simpler. A first, prominent example is the Cremmer-Julia derivation of the complete 4d maximal N = 8 supergravity [9], starting from eleven-dimensional supergravity [10].
In order to illustrate some general features of compactifications, we now briefly review the Kaluza-Klein model in arbitrary dimension, following [11].
We wish to reduce a (d + 1)-dimensional theory of pure gravity to d dimensions, by performing a compactification on a circle S1, of radius L. Calling y the coordinate along the circle, the (d+1)-dimensional coordinates zM split as zM = (xµ, y), where M = 0, . . . , d and µ = 0, . . . , d − 1. We start from the usual Einstein-Hilbert action for the metric field
gˆM N (x, y): 1
ˆ d+1 ˆ
S = 2ˆκ2 d z −gˆ R ,
where the hat symbol denotes quantities in d + 1 dimensions. Since the metric tensor has to be periodic along the circle, we can expand it in Fourier series as
gˆM N (x, y) = g(n) (x) einy/L .
M N
n
In principle we could simply substitute this into the action S, and integrate over the compact space S1 in order to define a d-dimensional action. However, in doing this we would obtain a theory containing an infinite number of fields, labeled by the Fourier mode number n. In order to define a lower-dimensional theory with a finite number of fields, we need to perform a truncation of the spectrum of gˆM N . The criterium to define the truncation in this case is readily available. Indeed, it turns out that the modes of gˆM N with n = 0 are massless from the d-dimensional viewpoint, while those with n = 0 are massive, with a mass of the order of | n /L. This can be seen by linearizing the Einstein equation ˆ | 1 : RM N = 0 for small fluctuations around the flat vacuum solution given by Minkowskid ×S gˆM N dzM dzN = ηµν dxµdxν + (dy)2 ,
where here we are choosing gdd = 1. If we choose a circle with a very small radius L, the states with n = 0 will be very massive, so that they can be neglected in a low energy approximation. We conclude that one can define a truncation to d-dimensional massless modes by taking all the fields independent of the compact coordinate y. Now, the most convenient way to identify the field content of the d-dimensional theory is to parameterize the generic higher-dimensional metric (independent of y) as gˆM N (x)dzM dzN = e2αφ(x)gµν (x)dxµdxν + e2βφ(x)(dy + A)2 , where A = Aµ(x)dxµ is a 1–form, and α and β = 0 are arbitrary constants. The vacuum solution written above is obviously recovered setting gµν = ηµν , A = 0 and eφ = 1 .
Substitution in the higher-dimensional action S and integration over S dimensional action
1 d √ 1 2 1 −2(d−1)αφ µν
S = d x −g R − (∂µφ) − e Fµν F ,
2κ2 2 4
where the d-dimensional gravitational coupling constant is κ2 = κˆ2/2πL, and we defined the field strength Fµν = 2∂[µAν]. Furthermore, in order to get canonically normalized kinetic terms the constants α and β have been chosen appropriately in terms of d [11]. We conclude that by reducing a theory of pure gravity on S1 one obtains a Maxwell-Einstein theory, also involving a scalar field φ. From the metric ansatz above we see that the gauge symmetry of the Maxwell field A → A + dλ(x) is inherited from the invariance under x-dependent reparameterizations of the circle coordinate y. Nowadays, the example illustrated here should be regarded just as a toy model, and typically the techniques required in supergravity and string theory compactifications are much more involved (both because the higher-dimensional action is not just pure gravity, and because the compactification manifold is not simply S1). However, in the following we highlight a couple of points, which have general validity and will be important in the remainder of this thesis.
As a first thing, we observe that if we wish that the higher-dimensional theory be compactified to a lower-dimensional theory which has a finite number of fields (as it is always the case), a truncation of the modes of the higher-dimensional fields on the internal space is always required. We will call the prescription selecting the degrees of freedom to be kept the truncation ansatz. In the example above, the appropriate truncation ansatz was not hard to identify. However, in general, things are not that easy, and different approaches can be adopted.
A physically well motivated prescription, relevant if one is interested in describing the low-energy physics around a given ground state, is the so called Kaluza-Klein ansatz (see e.g. [12] for a review), whose identification proceeds through the following steps:
i) choose a ground state of the higher-dimensional theory displaying a ‘spontaneous compactification’, namely a solution which is a direct product of two independent spaces.
ii) linearize the higher-dimensional equations of motion by considering small field fluc-tuations around the chosen vacuum, and identify the contributions to be interpreted as mass terms from the lower-dimensional viewpoint. Generically, these mass terms will include wave operators on the compact manifold, arising from the splitting of the higher-dimensional kinetic terms in a 4d spacetime part and an internal part.
iii) expand the higher-dimensional fields in a basis of eigenmodes of the identified mass operators; then truncate the spectrum, keeping just the lightest modes (typically, the massless ones).
In our example, the truncation ansatz valid at linear order around the Minkowskid × S1 ground state has been easily extended to non-linear order simply by asking independence of the circle coordinates, but already considering slightly more complicated spaces, like e.g. higher-dimensional spheres, this step would become highly non-trivial, if not impossible. As a consequence, in general the outcome of a Kaluza-Klein analysis is limited to a chosen vacuum, and describes the physics of small field fluctuations around it.
A different approach to dimensional reductions would be to define a truncation ansatz by requiring the preserved degrees of freedom to be invariant under some given symmetry. For instance, one can demand invariance under the action of (a subgroup of) the isometry group of the internal manifold. A typical case in which this alternative approach can be pursued is a dimensional reduction on group manifolds or coset spaces. Notice that, since S1 ∼ U (1), actually the simple dimensional reduction described above is an example of this second approach as well: indeed, demanding independence of the internal coordinates corresponds to keep only the singlets under the action of U(1). This kind of truncation ansatz is not always physically motivated, in that the obtained lower-dimensional theory does not necessarily capture the complete low energy physics. On the other hand, it is independent of the choice of a vacuum, is perfectly well-defined from the mathematical viewpoint, and has the advantage of yielding consistent reductions, where by definition a reduction is called consistent if all solutions (not only the vacua) of the lower-dimensional theory lift to solutions of the original, higher-dimensional theory. In this thesis we will discuss an explicit example of this second approach in chapter 5.
As a second point concerning our example above, let us consider the role of the scalar field φ. If we restrict to linear order in the field fluctuations, this is a free, massless field, whose propagation describes the variation of the size of the circle S1 along the d-dimensional space. Its presence in the massless spectrum is not accidental, but is a consequence of the fact that its vev, describing the size of the circle in the Minkowskid × S1 solution, is arbi-trary. In fact, this is a characteristic feature of compactifications: it often happens that the compactification background displays a continuous degeneracy, whose parameters are called moduli; these include the allowed variations in shape and size of the compact mani-fold. In particular, typically the supersymmetric Calabi-Yau string backgrounds mentioned above come with a large number of moduli. In the lower-dimensional theory, the moduli always appear as massless scalar fields. Now, from a phenomelogical perspective these are unwelcome, and determine one of the major problems of compactifications, known as the moduli problem. Indeed, these massless scalar fields would carry long range interactions, which are not observed in the real world. Furthermore, the various couplings in the low energy 4d effective action depend on the vevs of the moduli; since these are undetermined, the theory loses most of its predictive power. It follows that string compactifications, and specifically Calabi-Yau compactifications, will not be fully satisfactory until this problem is solved.
A way out to the moduli problem would be to generate a non-trivial scalar potential in the 4d action, having the twofold effect of stabilizing the vevs and of providing mass terms for the moduli. In the last years, it has been realized that a promising mechanism to generate a potential for the moduli can be obtained by considering string theory compacti-fications with fluxes [13] (see [14, 15] for previous work). Since then, flux compactifications have been the object of an intense research activity. Some very nice recent reviews on the subject are [16, 17, 18].
Fluxes are associated with a nonvanishing background value of the p-form field-strengths which are contained in the higher-dimensional supergravity theories (notice that, as far as one takes the typical length of the compactification manifold well below the string scale ℓs, it is justified to work in the supergravity approximation to string theory). More precisely, let Fp be a p-form field strength, satisfying the Bianchi identity dFp = 0, and let Σp be a non-trivial p-cycle of the compact manifold. Then one has a flux of Fp threading Σp if
Fp = n = 0 .
Σp
As for Dirac’s magnetic monopole, fluxes are subject to quantization conditions, so that in a quantum mechanical picture n can take just discrete values.
The reason why fluxes generate a 4d potential for the parameters controlling the com-pact geometry is apparent by considering the kinetic term for the internal field strength Fp in the higher-dimensional action. This reads
S= … F∧∗F,
M4 M6
where M6 is the compact manifold, and V is a function of the geometric moduli, since Fp couples with the metric on the compact manifold through the Hodge-∗. In particular, the potential V will depend on the parameters controlling the size of the cycles threaded by the flux.
Moduli stabilization is not the only motivation for studying flux compactifications. Indeed, fluxes are also naturally sourced by the spacetime-filling D-branes which are con-sidered in the modern approach to realistic compactifications. Furthermore, non-vanishing background values of the supergravity field strengths open new perspectives in the study of the geometry of string theory vacua. Indeed, they yield a nonvanishing contribution to the energy-momentum tensor of the higher-dimensional Einstein equation; it follows that the empty-space Ricci-flatness condition is removed. For instance, Calabi-Yau manifolds are no more available solutions (indeed, SU(3) holonomy implies Ricci-flatness). In some cases the backreaction due to the fluxes is mild, and one can still work with an underlying Calabi-Yau geometry [13], while in other situations it can be more drastic, and strongly deform the compact geometry (see e.g. [19] for an early example). In order to achieve a deeper understanding of the structure of string theory, and its relations with the lower-dimensional world, it becomes therefore very interesting to explore, and eventually classify, the possible compactifications with fluxes.
A systematic study of compactifications with fluxes is a challenging goal which requires new mathematical techniques. For this task, supersymmetry is again a powerful ally. Indeed, even though in the presence of fluxes the differential conditions for supersymmetric backgrounds become more involved, still one finds interesting underlying structures. In particular, one is led [20] to consider six-dimensional manifolds whose structure group lies in SU(3) [21]. Manifolds with SU(3) structure share with Calabi-Yau manifolds the existence of a globally defined and nowhere vanishing spinor, but are more general since the latter needs not being covariantly constant in the Levi-Civita connection. This global spinor is required in order to properly decompose the spinorial generators of the higher-dimensional supersymmetry transformations.
Actually, type II theories – on which this thesis focuses – involve two spinorial pa-rameters, so that there is also the possibility to employ a pair of internal spinors in the decomposition associated with the compactification. This yields an enhanced freedom in the study of compactifications preserving a minimal fraction of supersymmetry. As we will discuss in detail in chapter 2, a suitable formalism for studying these compactifications is provided by generalized geometry, introduced in the mathematical literature by Hitchin in 2002 [22], and further developed in [23, 24, 25]. Generalized geometry deals with structures defined on T ⊕ T ∗, the sum of the tangent and cotangent bundle of the compact manifold.
One of the main applications of the generalized geometry formalism in string theory has been an elegant reformulation of the supersymmetry conditions for type II flux vacua [26, 27]. These take the form of differential equations for the differential forms which characterize the generalized geometry, and can in part be understood as an integrabil-ity condition for structures on T ⊕ T ∗. Subsequent related work, employing generalized geometry for the study of flux backgrounds, can be found in [28]–[40]. In physics, general-ized geometry has also been applied to the study of supersymmetric worldsheet σ-models, starting with [42, 43]; see e.g. [44] for a review of this topic.
In compactifications, the study of the ground state is usually understood as a first (essential) step towards the determination of the lower-dimensional theory. At the level of the action, the minimal amount of supersymmetry preserved by type II pure supergravity compactifications is N = 2 in four dimensions. Reductions preserving just N = 1 are possible if one adds further ingredients, like certain projections induced by localized sources, relating the two ten-dimensional supersymmetry parameters. On the same footing as for the background, the best studied case of compactification leading to an N = 2 supergravity action in 4d again involves Calabi-Yau manifolds. As we will see, possible deformations of the Calabi-Yau dimensional reductions can be studied using the tools of generalized geometry. A program in this direction was started in [45] and pursued in [46], as a natural consequence of the previous studies of reductions on SU(3) structure manifolds, pionereed in [47].
In this thesis, we will build on this line of research to further study a general procedure for truncating type II theories to N = 2 supergravity in four dimensions. As it is generically the case when considering compactifications with fluxes, we will be led to study gauged N = 2 supergravities, which as an essential feature involve a scalar potential.
Outline of the thesis
The main aim of this thesis is to study dimensional reductions of type II theories leading to N = 2 supergravity in four dimensions. In doing this, we allow for a general set of background fluxes. The principal tools that are employed throughout the work are generalized geometry and gauged N = 2 supergravity.
The thesis is structured as follows.
We start in chapter 2 with an introduction to the mathematical notions necessary for the analysis of the compactification. The spinor ansatz for dimensional reductions preserving N = 2 in 4d leads us to discuss G-structures on the 6d compact manifold. Then, by extending the notion of G-structure to the generalized tangent bundle T ⊕ T ∗, we introduce Hitchin’s generalized geometry. In particular, we focus on the key notion of SU(3)×SU(3) structure, which encodes all the NSNS degrees of freedom on the compact manifold. In view of the study of the scalar kinetic terms in 4d, we consider the metric on the parameter space of the internal NSNS fields. The latter can be reformulated in terms of deformations of the O(6,6) pure spinors which characterize the SU(3)×SU(3) structure, and this allows to highlight an underlying special K¨ahler geometry, defined locally on the internal manifold. This result, to which we give original contributions, parallels to some extent the structure of the Calabi-Yau moduli space, which we also review.
In chapter 3 we turn to the study of type II compactifications on flux backgrounds admitting SU(3)×SU(3) structure. We start with a brief review of the ‘democratic’ ver-sion of type II supergravities [48], which is particularly suitable for generalized geometry applications. We also discuss dimensional reductions on Calabi-Yau manifolds, which rep-resent the model of reference for the subsequent developements. We illustrate the relation between fluxes and gaugings.
The truncation of type II supergravity is implemented on general SU(3)×SU(3) struc-ture backgrounds via the expansion of the higher-dimensional fields in a finite basis of differential forms on the compact manifold. Compatibility with N = 2 supergravity in 4d requires this basis to respect a restrictive set of geometrical constraints, which have been identified in [45, 46, 49], and which we revisit. Then we fill a gap existing in the literature by deriving via dimensional reduction the complete four-dimensional bosonic action. In particular, we focus on the way its data are determined by generalized geometry, and we establish various results.
First we deal with the reduction of the NSNS sector of type II supergravity. We make the link with the space of deformations studied in the previous chapter. Then we study the role of a B-twisted Hodge star operator, and in particular we show how its action on the basis of forms generalizes to the SU(3)×SU(3) context the well-known expression for the usual Hodge-∗ acting on the harmonic three-forms of a Calabi-Yau manifold. This allows to derive a formula for the period matrices of the N = 2 special K¨ahler geometry. Next we focus on the 4d scalar potential: we prove a formula expressing the internal Ricci curvature in terms of the generalized geometry data, and we apply it to deduce a geometric expression for the scalar potential. Once restated in terms of 4d variables, this gives back the symplectically invariant and mirror-symmetric expression found in [50] by means of purely 4d gauged supergravity methods.
In the last part of the chapter we move to the RR sector, with a focus on type IIA. Instead of directly reducing the action, we choose to reduce the equations of motion. As a consequence of a self-duality constraint, these can also be read as Bianchi identities. The expansion of the democratic RR field on the internal basis automatically introduces forms of all possible degrees in the 4d spacetime. We interpret a subset of the reduced RR equations as 4d Bianchi identities; by their solution we define the 4d fundamental fields. The remaining equations are seen as 4d equations of motion, from which we reconstruct the reduced action.
In chapter 4 we illustrate further the consistency between the outcome of the dimen-sional reduction and the formalism of gauged N = 2 supergravity. A consistent formulation in the presence of a complete set of fluxes requires the introduction of tensor multiplets. We focus on the quantities determining the gauging, whose associated charges are gener-ated by the NSNS, RR and geometric fluxes. Then, starting from the expression of the N = 2 Killing prepotentials, and using some general results about N = 2 supergravity with tensor multiplets, we deduce the fermionic shifts in the 4d supersymmetry variations.
In the second part of the chapter we confront the 4d and 10d approaches to the N = 1 flux backgrounds. At the 10d level, we adopt the generalized geometry reformulation of the equations for an N = 1 vacuum found in [26, 27]. In order to perform a comparison with the 4d supersymmetry conditions, we rephrase these equations in a 4d framework performing the integral over the internal manifold. Next we derive the N = 1 vacuum conditions within the 4d N = 2 theory, by imposing the vanishing of the fermionic shifts under a single susy transformation. Exploiting the properties of special K¨ahler geometry, we establish a precise matching with the integrated version of the pure spinor equations. We also perform a similar study by considering the 4d N = 1 supergravity which arises as a truncation of the previously analyzed N = 2 theory: we derive the expressions for the superpotential and D-terms, and we impose the F-flatness and D-flatness conditions; again we find precise correspondence with the 10d equations.
Finally, in chapter 5 we present some concrete examples of N = 2 compactifications, based on coset spaces with SU(3) structure. To some extent, these can be seen as an appli-cation of the general study done in the previous chapters. Thanks to the full control on the geometry allowed by the coset structure, we can perform an explicit analysis. In particular, we establish the consistency of the dimensional reduction based on a left-invariant trun-cation ansatz. This gives a solid justification to the choice of the expansion forms. Then we explore the supersymmetric and non-supersymmetric backgrounds associated with the compactification, parameterizing the solutions in terms of the fluxes. Exploiting the con-straints imposed by N = 2 supersymmetry, we study the string loop corrections to the 4d scalar potential, and we perform a preliminary search of de Sitter extrema.
In chapter 6 we draw our final considerations.
We relegate some technical discussions to the appendix. Appendix A summarizes our conventions. Appendix B gives some details about the Mukai pairing and the Clifford map used in the generalized geometry computations. Appendix C discusses the relation between the democratic and the standard formulation of type IIA supergravity, clarifying some sub-tleties related to the presence of fluxes. Appendices D and E give the definition and some properties respectively of special K¨ahler manifolds and of quaternionic-K¨ahler manifolds, which play a central role in N = 2 supergravity. Finally, appendix F collects some details of the coset space dimensional reductions discussed in chapter 5, and appendix G derives the string loop corrections to the associated 4d, N = 1 vacua.
Table of contents :
1 Introduction
1.1 Compactifications, fluxes and generalized geometry
1.2 Outline of the thesis
2 Generalized structures in type II supergravity
2.1 Motivation
2.2 G-structures
2.2.1 Integrability
2.2.2 SU(3) structures and spinors
2.3 Calabi-Yau manifolds and their moduli space
2.3.1 Definition and properties
2.3.2 The moduli space of Calabi-Yau 3-folds
2.4 Generalized structures
2.4.1 Making up generalized structures
2.4.2 Extracting data from generalized structures
2.4.3 Description via pure Spin(6,6) spinors: the polyform picture
2.4.4 Spin(6,6) pure spinors and Spin(6) bispinors
2.5 Deformations of SU(3)×SU(3) structures
2.5.1 The generalized diamond
2.5.2 The space of deformations
2.5.3 Metric deformations
2.6 Discussion
3 The dimensional reduction
3.1 Democratic formulation of type II supergravity
3.2 The archetype: type II on Calabi-Yau 3-folds
3.2.1 The Kaluza-Klein (on-shell) approach
3.2.2 Going off-shell, and gauging by fluxes
3.3 Defining the truncation ansatz
3.3.1 The philosophy
3.3.2 The basis forms
3.3.3 Special K¨ahler geometry on the truncated space of pure spinors
3.3.4 The twisted Hodge star ∗b
3.3.5 Differential conditions on the basis forms
3.4 Reduction of the NSNS sector
3.4.1 Scalar kinetic terms
3.4.2 Variations of √g6 and the dilaton
3.4.3 The scalar potential
3.5 Reduction of the RR sector
3.5.1 Reduction of the RR self-duality constraint
3.5.2 Reduction of the equations of motion / Bianchi identities
3.5.3 pAI = 0 = qIA case. SU(3) structure
3.5.4 General case
3.6 Summary and discussion
4 The 4d N = 2 supergravity picture, and the N = 1 vacuum conditions
4.1 N = 2 structure of the 4d theory
4.1.1 N = 2 Killing prepotentials from the dimensional reduction
4.1.2 Gauging the quaternionic isometries
4.1.3 N = 2 fermionic shifts with electric and magnetic charges
4.2 The N = 1 vacuum conditions
4.2.1 N = 1 equations from the ten dimensional analysis
4.2.2 N = 1 conditions from the effective action, and matching
4.3 Aspects of N = 2 → N = 1 theories
4.3.1 N = 1 superpotential
4.3.2 D-terms from N = 2 → N = 1 truncations
4.3.3 Supersymmetric vacua from O6-induced truncations
4.4 Summary and discussion
5 Consistent reductions on cosets with SU(3) structure
5.1 Introduction and overview
5.2 Introducing the internal geometries
5.2.1 Coset spaces and expansion forms
5.2.2 The SU(3) structure
5.2.3 An alternative basis?
5.3 Supersymmetric 10d solutions parametrized by fluxes
5.3.1 Flux quantization and K-theory
5.3.2 The solution
5.4 The dimensional reduction
5.4.1 The truncation scheme
5.4.2 The 4d action
5.4.3 Consistency of the truncation
5.5 The 4d potential via N = 2
5.5.1 Tree level
5.5.2 All string loop
5.6 Non-supersymmetric vacua
5.6.1 Tree level
5.6.2 Non-supersymmetric Nearly K¨ahler companions
5.6.3 de Sitter vacua at all string loop order?
6 Conclusions
A Notation and conventions
A.1 Indices
A.2 Differential forms and the Hodge dual
A.3 Clifford algebra and spinors
A.3.1 Gamma matrices
A.3.2 Weyl spinors and Majorana spinors
A.4 SU(3) structure conventions
B Mukai pairing and Clifford map
C Type IIA action with fluxes 161
D Geometry of N = 2 supergravity I : Special K¨ahler manifolds
E Geometry of N = 2 supergravity II : Quaternionic-K¨ahler manifolds
E.1 Definition
E.2 Properties
F Details of the dimensional reduction on coset spaces
F.1 Special K¨ahler geometry from the NSNS sector
F.2 The RR sector
G String loop corrections to the N = 1 coset vacua
Bibliography
