C-spaces, generalized geometry and double field theory

We construct a C-space associated with every closed 3-form on a spacetime $M$ and show that it depends on the class of the form in $H^3(M, Z)$. We also demonstrate that C-spaces have a relation to generalized geometry and to gerbes. C-spaces are constructed after introducing additional coordinates at the open sets and at their double overlaps of a spacetime generalizing the standard construction of Kaluza-Klein spaces for 2-forms. C-spaces may not be manifolds and satisfy the topological geometrization condition. Double spaces arise as local subspaces of C-spaces that cannot be globally extended. This indicates that for the global definition of double field theories additional coordinates are needed. We explore several other aspect of C-spaces like their topology and relation to Whitehead towers, and also describe the construction of C-spaces for closed k-forms.


Introduction
Double field theory (DFT) has been introduced to provide a geometric interpretation of the T-duality symmetries and to describe string theory in a T-duality covariant way, see [1,2,3] for early works and [4]- [10] for more recent developments. More general proposals 1 include E 11 [12,13] and exceptional field theories, see eg [14]- [24], and reviews [25,26,27] and references with. For the construction of DFTs, the spacetime M is enhanced with additional coordinates, leading to double spaces D M which have dimension twice that of spacetime. So far the construction of local actions relies on two ingredients. First, the use of infinitesimal transformations to prove invariance, and second the application of the strong section condition. These infinitesimal transformations combine the spacetime diffeomorphisms and the gauge transformations of the B-field that act on a generalized metric. This generalized metric is constructed from both the spacetime metric and the B-field. This is interpreted as a geometrization of B-field. The strong section condition in effect restricts the fields and their infinitesimal transformations to dependent on either the spacetime or dual coordinates. More recently several suggestions have been made to integrate the infinitesimal transformations of double field theory leading to the construction of finite transformations of the double spaces and those of the associated fields [28,27,29,30]. Another suggestion is to employ a non-trivial split metric on the extended spaces [31]. Similar results also hold for the exceptional field theories, however see also [32,33].
The global definition of double field theories remains an open problem. Using the solution of the strong section condition for the spacetime presented in [28,27], it has been shown in [34] that the patching of double spaces 2 constructed is consistent if and only if the 3-form field strength is exact. In section 4, we shall strengthen this statement. The C-spaces that we propose below resolve this global patching problem.
To identify the spaces which can implement the geometrization of the B-field in the context of DFT, it has also been argued in [34] that one necessary ingredient is the topological geometrization condition. This can be stated as follows: Given a manifold M, eg spacetime, and a closed k-form ω k , a space 3 C M satisfies the topological geometrization condition, if and only if there is a projection π : C M → M such that π * ω k represents the trivial class in H k (C M ).
Given M and ω k , this definition does not uniquely specify C M . There are several constructions of C-spaces via K-theory and homotopy theory. The latter applies for any manifold and for any form of any degree. The standard examples of C-spaces are circle bundles over M which satisfy the topological geometrization property for closed 2-forms, and implement the geometrization of the Maxwell fields in the context of Kaluza-Klein theory.
In this paper, a construction of C-spaces, C M , is proposed for every closed 3-form, 1 See [11] for an early work on the geometrization of dualities. 2 Examples of double spaces have been investigated in [34] from the patching point of view and they have been found to depend on the choice of atlas. Therefore they are not general covariant. 3 The spaces that satisfy the topological geometrization property have been called charge spaces, or C-spaces for short, because the topological charge carried by ω k is stored in the topology of C, eg in the transition functions. Exploring the consistency of the patching conditions given in (2.6) at triple and 4-fold overlaps, it leads to the requirement that ω 3 must represent a class in H 3 (M, Z) as expected from the Dirac quantization condition. In addition, it is demonstrated that C M are not manifolds, in particular they do not have a well-defined dimension. Nevertheless they can be described in some detail using the transition functions and the additional coordinates. For example, one can show that C M satisfy the topological geometrization condition. This construction of C-spaces for closed 3-forms is related to gerbes. In particular, we explain how from C [ω 3 ] M one can construct the gerbe transition functions that arise in the approach of [35]. However the construction of C involves the open sets and their double overlaps, as well as the triple and 4-fold overlaps, in an essential way, and the emphasis is on the object itself rather than its transition functions on M. This is more close in spirit to the definition of gerbes in terms of sheafs [36] but without the complications of category theory. Furthermore, the construction of C leads to the emergence of generalized geometry on M as described by Hitchin and Gualtieri [37,38]. In particular we shall show that C M induces a bundle over M which is the extension of T M with T * M. As result one can define a generalized metric and carry out generalized differential geometry calculus on M.
To get some insight into the topological structure of C M , we consider the nerve of the good cover of M which provides a chain complex description of M. We find that every 2-simplex in the nerve of M together with the new angular coordinates give rise to a CP 2 in C M . We use this to raise the question whether this construction of C M is related to Whitehead towers. Furthermore, we construct, C T 3 , which is the C-space of 3-torus with a 3-form flux. We demonstrate that C T 3 resolves the patching problems of the double space construction of [27] for this model.
To elucidate the relation between C-spaces and double spaces, we revisit the global properties of the double spaces. We show that the mere use of the strong section condition, ie without invoking any information about the transformation of the generalized fields, together with the requirement of the general covariance of the spacetime imply that the double space must be diffeomorphic to T * M. Such a space cannot satisfy the topological geometrization property. Moreover if the transition functions of the B-field are related in a linear way to those of the dual coordinates, then the 3-form flux is exact.
On the other hand, the C-spaces, C The construction of C-spaces, C M , can be generalized to every k-form, ω k , which represents a class in H k (M, Z). This proceeds in a similar way to that of C Their construction also has a Kaluza-Klein interpretation. The extended space associated with a k-form, which is the generalization of a double space for k > 3, can be seen as local subspaces of C M . This again indicates that more coordinates are need for the global description of exceptional field theories.
There is a construction of C-spaces in the context of homotopy theory using Whitehead towers. Here we revisit the theory and point out that the Whitehead towers construction for 2-forms coincides, up to homotopy, with the standard circle bundle construction of Kaluza-Klein spaces. Then we review some of the properties of Whitehead towers construction for closed 3-forms and ask the question how these are related to C and investigate the 3-torus with 3-form flux C-space. In section 5, we explore the application of Cspaces to DFT. In section 6, we construct C-spaces for closed k-forms. In section 7, we explore the relation between C-spaces and Whitehead towers, and in section 8, we give our conclusions.

C-spaces for closed 3-forms 2.1 C-spaces for closed 2-forms
Before, we proceed to give the patching conditions of C-spaces associated with closed 3-forms, let us briefly review the standard Kaluza-Klein space, C M , for 2-forms. Let M be a manifold and {U α } α∈I be a good cover 5 of M, for the precise definition see eg [39]. Moreover suppose that ω 2 represents a class in H 2 (M, R). Then within theČech-de Rham theory applying the Poincaré lemma on the open sets U α as well as their U αβ and U αβγ intersections 6 M is constructed from M by introducing a new coordinate τ α at each open set U α with patching conditions which is consistent at triple overlaps U αβγ if and only if n αβγ ∈ Z and so 1 2π [ω 2 ] ∈ H 2 (M, Z). Taking the exterior derivative of patching condition, one finds that M is a circle bundle on M with first Chern class given by 1 2π [ω 2 ].

Patching C-spaces for closed 3-forms
To begin the construction of C and similarly the transition functions are defined up to gauge transformations as These gauge transformations are the only ones compatible with the closure of ω 3 .
The construction of C proceeds with the introduction of new coordinates y 1 α and θ αβ associated with the open sets U α and the double overlaps U αβ , respectively. These are new coordinates in addition those of the spacetime. They should be thought in the same way as the Kaluza-Klein coordinate τ we have introduced for the description of C in the previous section. Though y 1 is assigned the degree of a 1-form. In addition, one imposes the gluing transformations on U αβ and U αβγ . Using the second condition in (2.3), one finds that consistency of the first condition on triple overlaps yields This is implied from the second condition in (2.6). Next investigating the consistency of the second condition of (2.6) are 4-fold overlaps and after using the last condition in (2.3), one finds that This is satisfied provided that 1 2π ω represents a class in H 3 (M, Z). One of the questions that arises in imposing (2.6) is how one is supposed to think about these new coordinates and their gluing transformations. The coordinates should be thought in the same way as in the usual construction of a circle bundle over a manifold utilizing the patching conditions of a manifold together with those of a closed 2-form. For the gluing transformations, this particularly applies to the second patching condition which involves triple overlaps and three coordinates rather than double overlaps and two coordinates which is the usual patching conditions for manifolds. To give some insight into this question, one can view the usual patching of manifolds as follows. Given two charts, ie open sets and coordinates adapted to each one of the sets, the patching condition at the double intersection relates the coordinates of first chart to the coordinates of the second chart, and vice versa. Now if we introduce additional coordinates θ αβ at each double overlap, the second patching condition in (2.6) specifies how the three coordinates, each one associated with one of the three double overlaps that contribute to the triple overlap, are related.
The C-spaces C M constructed with the above patching conditions are not manifolds. To see this, first observe that by construction there is a projection π : C M does not have a well-defined dimension.

Dependence on ω 3
Here we shall investigate whether or not C There is additional gauge redundancy in the definition of b α and that of the transition transition functions given in (2.4) and (2.5), respectively. This is eliminated we perform the compensating transformations In addition to these, one should also investigate a more subtle choice in the construction of C M . This will require a better understanding of the class of objects that contain C M so their notion of equivalence can be established. We shall not duel on this question. The expectation is that for every choice of a good cover, or at least for a large class of good covers, all C

Topological geometrization condition
It has been argued in [34] that any spaces which geometrizes a k-form flux has to be a C-space, ie there must be a projection from the C-space on the spacetime and that the pull back of the k-form flux must represent the trivial class in the C-space.
Here we shall demonstrate that C M is a C-space. As we have mentioned, there is a projection π from C [ω 3 ] M onto M. Next taking the differential of the first patching condition in (2.6), one finds that Using the second condition in (2.3), this can be rewritten as M . Therefore C 3 Relation to gerbes and generalized geometry 3

.1 Gerbes
In the definition of [35], a gerbe is the object which represents a class in H 3 (M, Z) in the same way that a circle bundle represents a class in H 2 (M, Z). It is expected that given a manifold M and a class in H 3 (M, Z), in a certain sense, the topology of the gerbe is specified uniquely. Then the gerbes are investigated via their transition functions. To relate the transition functions of a gerbe as defined in [35] to the transition functions we use here, we note that Then the second condition in (2.3) reads as which can be recognized as the patching condition for gerbes on 4-fold overlaps.
One of the aims of this article is the construction of C-spaces that can apply to double field theory. So the emphasis is not only on the transition functions but rather in the description of a particular object that represents a class in H 3 (M, Z). It is not expected that each class in H 3 (M, Z) is represented with a unique such object unless additional requirements are put in place. In fact, this is not the case even for the elements of H 2 (M, Z). In particular these can be represented with complex line bundles L as well. Furthermore L and the direct sum L⊕I, where I is the trivial I line bundle, represent the same class in H 2 (M, Z). However they have different geometric properties which can be essential in certain applications. Furthermore, in the construction of C

Generalized geometry
The M induces naturally a generalized geometry structure on M. This can be seen from the transition functions of dy 1 , ie dy 1 α patches as a 1-form on M accompanied with a shift with the transition function of B α . This assertion regarding the degree of dy 1 α requires some explanation. We have assigned the degree of a 1-form on y 1 from the start. So dy 1 has degree 2. However y 1 is an independent coordinate. So, from the perspective of M, dy 1 transforms as an 1-form. For the rest of the section, we shall neglect the grading of y. Therefore dy span T * M. In fact, the patching condition (3.3) defines an extension of the tangent bundle with respect to the cotangent bundle as This is the first step required to define a generalized structure on M as it is described in [37,38]. For example, one can define a generalized metric G from a metric g on M and the B form as follows. We have seen that dy − B is globally defined on M. Using this, we can write where dy i = i ∂ ∂x i dy. This is the expected form of a generalized metric written in the basis (dx i , dy i ) of vectors and 1-forms.

Some topological aspects of C [ω 3 ]
M and an example 4

.1 Topological aspects
One way to get an insight into the topological structure of the C-space is to investigate C M in a chain complex approximation of the spacetime. Given a good cover {U α } α∈I on M, one can associate a chain complex with M the nerve N of the cover, see eg [39]. N is constructed as follows. One introduces a vertex for each open set U α of the cover. This is because of the second patching condition in (2.6) as the three angular coordinates associated to each side are restricted to two.
Therefore one can describe this construction at a face of N as follows. The 2-tori of the face degenerate to circles at each of the three sides, and in turn, the circles at the sides and the tori of the face degenerate to a point at they approach the vertices. Such a structure is reminiscent 7 to that of CP 2 . To see this consider the algebraic equation of S 5 , w 1w1 + w 2w2 + w 3w3 = 1 . (4.1) Setting t 1 = w 1w1 , t 2 = w 2w2 and t 3 = w 3w3 , this can be seen as the defining equation of a 2-simplex. The three phases of the complex numbers w 1 , w 2 and w 3 associate a circle at every vertex, a 2-torus at every point of a side, and a 3-torus at every point of the face. As CP 2 is the base space of the fibration, S 1 → S 5 → CP 2 , where S 1 acts from the right on the triplet (w 1 , w 2 , w 3 ), a circle in removed from every point of the simplex leading to the picture describe above for N . As a result the topology of C is different from that of spacetime M. As we shall see CP 2 , or rather CP ∞ , appears also in the homotopy approach to C-spaces using Whitehead towers.

The C-space of 3-torus with a 3-form flux
The construction of C [ω 3 ] M described in section 2 is general and applies to every manifold with a good cover equipped with a closed 3-form which represents a class in H 3 (M, Z).

As good covers exist on manifolds, one can construct C
for all smooth solutions of supergravity theories including the NS5-brane solution 8 .
Here we construct the C-space of a 3-torus with a 3-form flux. This example was initially investigated from the perspective of double spaces in [27]. Later it was explored from the patching point of view in [34] where it was found that the construction depends on the choice of the atlas on T 3 . Another feature of the construction was that a quantization condition was imposed at the triple overlaps rather than the 4-fold overlaps which are involved in the Dirac quantization condition of 3-forms field strengths. We shall follow the notation of [34] where all the data regarding the patching conditions of the 3-form flux can be found 9 . The patching conditions of the C-space are where we have set α 1 = i 1 j 1 k 1 and so on. In the atlas we have chosen on T 3 , the components of a 1 α 1 α 2 and a 0 α 1 α 2 α 3 are linear in the coordinates of T 3 . However the above patching conditions do not depend on this choice. This particularly applies to the second condition in (4.2) as the consistency required for it deals to n α 1 α 2 α 3 α 4 ∈ Z on 4-fold overlaps. Since n α 1 α 2 α 3 α 4 are constant for any choice of an atlas, the quantization condition is atlas independent. This should be contrasted with the double field theory computation which arises after taking all the angular coordinates to zero. As a result of (2.7), consistency in this case requires that the components of da 0 α 1 α 2 α 3 are constant and should vanish up to some period. As da 0 α 1 α 2 α 3 is a local 1-form, the constancy of its components is an atlas dependent statement [34].

Revisiting the patching of double manifolds
In the formulation of double field theory so far, one introduces a new set of coordinates 10 z in addition to those of the spacetime x and imposes on all fields and their transformations the strong section condition which reads Setting for A and B the infinitesimal local transformations δx i and δz i of x i and z i , respectively, and assuming that δx i must be arbitrary functions of x, which is required in 8 The dilaton singularity does not affect the construction. 9 Strictly speaking one should introduce a third open set on S 1 , U 3 = (− π 4 , π 4 ), so that the cover is a good cover. As the transition functions between U 1 and U 3 , and U 2 and U 3 are the identity, there is no change in the computations on [34] and the effects of U 3 have already been taken into account via the choice of n x . 10 Usually the dual coordinates z are denoted with y. Here we denote them with z to distinguish them from those of the C-space as they have different transformation properties. order to account for all reparameterizations the spacetime 11 , one concludes that the most general solutions to the above conditions are In particular, the second equation in (5.1) implies that δx i can depend only on x. Then the first equation for A = δx i and B = δz i implies that δz i can dependent only on x as well. These infinitesimal transformations can be integrated to give Moreover in [28,27], u i is related linearly to the gauge transformations of the b field.
To investigate the global properties of double field theories, these transformations are interpreted as patching conditions, where we have introduced a good cover {U α } α∈I on the spacetime M.
The strong section condition has another solution where z and x exchange places, this is the solution for the dual space. It also has many more solutions 12 provided that one weakens the requirement that δx i must be an arbitrary function of x and does not allow for general reparametrization of M but this breaks general covariance.
So in order to allow for reparametrization of spacetime, one is forced to patch the theory with transformations of the type (5.4). If this is the case, then κ αβ + κ βγ + κ γα = 0 . (5.5) Using the results of [34], one concludes that this is possible if and only if the double space is diffeomorphic to D M = T * M. This result is independent from the form of finite transformations on the fields and other geometric considerations. It is a consequence of the application of the strong section condition. Thus if one uses the strong section condition to describe the double theory and allows for general reparameterizations of the spacetime coordinates, then one is led to the conclusion that the double space is T * M.
This has immediate consequences. T * M is contractible to M, so π * ω 3 is not trivial in T * M. Thus this space does not satisfy the topological geometrization condition. Furthermore, if the transition of functions of ω 3 at double overlaps are related via a linear transformations to κ, then ω 3 is exact [34].

Relation of double spaces to C-spaces
Now let us compare the results of the previous section with those we have obtained for the C  [27] from the patching point of view in [34] and in section 4.2 supports this assertion. However, it is not apparent how the additional coordinates θ can be inserted in the description of DFTs.
The construction of C M is independent from the choice of the transition functions in (6.4) provided we allow the new coordinates to transform as In addition one can show that C obeys the topological geometrization condition. In particular, it is easy to see from the construction above that dy k−2

Applications
Most of the properties and applications we have explored for C which extends the generalized geometry considerations beyond the co-tangent bundle and has applications in exceptional field theories. As dy k−2 M , one can write a generalized metric in a way similar to that of C M provides also a model for a k-gerbe. In the context of exceptional field theories, the strong section condition, under similar assumptions to the DFT case, will lead to a patching condition for the (k-2)-form coordinates. Again this implies that the exceptional spaces are diffeomorphic to Λ k−2 (M). Such a space cannot satisfy the topological geometrization condition. Furthemore if κ k−2 αβ are related to the transition functions of ω k at double overlaps with a linear map, then ω k represents the trivial class in cohomology. The exceptional spaces are local subspaces of C where all coordinates of the latter apart from x and y k−2 are set to zero. These topological considerations lead to the conclusion that for the global definition of exceptional field theories many more coordinates are needed in analogy with DFTs.

Concluding remarks
We have proposed a C-space, C M . An interpretation of this is that for the global definition of DFTs additional coordinates are required. We argue that these new coordinates are necessary on topological grounds and this should not depend of the details of geometry. However how these can enter in the existing local description of DFTs remains an open problem.
We have also generalized the construction of C-spaces for any closed k-form on M, and we have established that C M . It is expected that these spaces are required for the global definition of exceptional field theories.
The construction of C M can be done starting from any spacetime with a good cover and a closed 3-form. As a result such spaces can be found for all relevant supergravity backgrounds including those of the NS5-branes. Here we have explored in detail the 3torus with a 3-form flux model of [27]. We demonstrate how several puzzles associated with the construction of double spaces for this model [34] are resolved via the use of C-spaces.
Another method to topologically geometrize k-forms in the context of homotopy theory is that of Whitehead towers. It was emphasized that for simply connected manifolds, the Whitehead construction coincides with the construction of C M can be also related to the Whitehead construction and in particular whether the former provide a model for the latter. Such a relation will elucidate the topological structure of C-spaces.
Although C-spaces resolve the global patching problem of double spaces, the additional coordinates which are non-linear, are still too special to allow for a full covariance under all required symmetries, diffeomorphisms and dualities, without any further assumptions on the structure of spacetime. Nevertheless, they may prove to be useful way to proceed. In addition, the understanding how to incorporate the additional coordinates in DFT may lead to some new insights into the structure of these theories.