Seeking the balance: Patching double and exceptional field theories

We investigate the patching of double and exceptional field theories. In double field theory the patching conditions imposed on the spacetime after solving the strong section condition imply that the 3-form field strength $H$ is exact. A similar conclusion can be reached for the form field strengths of exceptional field theories after some plausive assumptions are made on the relation between the transition functions of the additional coordinates and the patching data of the form field strengths. We illustrate the issues that arise, and explore several alternative options which include the introduction of C-folds and of the topological geometrisation condition.


Introduction
It has been known for sometime that given a spacetime M and a 2-form gauge field, F , dF = 0, one can introduce an additional coordinate, a charge coordinate, and together with the Dirac quantisation condition, one can construct a U(1) bundle M F over the spacetime M. This bundle is a new manifold associated to the Maxwell field and the transition functions of the fibre are related to the patching conditions of the 1-form gauge potential of F . This construction is a manifestation of the isomorphism of U(1) bundles over M with the cohomology classes in H 2 (M, Z) and underpins Kaluza-Klein theory. In the latter, the U(1) gauge potential is a component of the metric in higher dimensions. This is sometimes referred as the geometrisation 1 of the U(1) field. A feature of the construction is that the charge of the U(1) field is replaced by information stored in the patching conditions of M F . Similar suggestions have emerged in the context of string theory and M-theory following the early works of [1,2,3]. These include double field theory (DFT) of [4,5,6] applied to string theory, the E 11 [7,8], and the exceptional field theory (EFT) [9,10,11,12] proposals, see also reviews [13,14,15] and references within. There are several reasons for this. One is to find a geometric realization of string and M-theory dualities and to describe duality covariant theories. Another is to explore the idea that string and M-theory dualities emerge as symmetries of the 10-and 11-dimensional theories rather than just their toroidal compactifications. The constructions are broadly based on a similar technology to the 2-form U(1) gauge field described above but now the metric in higher dimensions is replaced by a generalized metric which includes the form gauge potentials of string theory and M-theory, and the introduction of suitable new coordinates.
A related construction is that of generalized geometry [26,27]. In generalized geometry no new coordinates are introduced in addition to those of spacetime. Instead the tangent space of the spacetime is replaced by a vector bundle E which is an extension of T M equipped with an appropriate bracket. Such an approach has been used to explore some geometric properties of supergravities associated with strings and M-theory [16,17,18,19].
Although much work has been done to understand the geometry that underpins DFE and EFT several questions remain. One question is to unravel topological and differential structures of double and exceptional spaces, and another related question is to understand how information about string and brane charges is stored in their topology. In the context of the formalism developed so far, it is not possible to answer these questions because most of the computations have been made using infinitesimal transformations generated by generalized Lie derivatives. However for DFT a set of finite transformations have been proposed [20,15] by integrating the infinitesimal transformations that have been known before, and have been explored as transition functions for double spaces in [21]. Infinitesimal transformations for the coordinates of exceptional spaces underlying EFT have been proposed in eg [9,10,22,23,11].
In this paper, we shall review some of the properties of patching closed form field strengths on a spacetime and illustrate the issues that are involved. Then we shall demonstrate that the patching conditions proposed by [20] after solving the strong section condition imply that the NS-NS 3-form field strength H which arises in string theory is an exact 3-form.
We shall further argue that a similar conclusion can be reached in the context of EFT provided that the patching conditions of closed form field strengths are related in a linear way to the transition functions of the additional coordinates of the exceptional spaces and the combinatorial law follows the usual rules of tensor calculus. The latter point will be illustrated in the U(1) paradigm reviewed in the next section.
We shall explore several alternative ways to reconcile the transition functions of the double and exceptional spaces with the patching conditions of the closed form field strengths on the spacetime. We shall see that there are examples described in appendix A, where this can be done at a cost. In particular, these constructions do not exhibit some key properties of the U(1) paradigm. Nevertheless, they illustrate some of the issues involved and provide a local model of a consistent construction.
In the conclusions, we propose a general scheme with a minimum number of requirements that should be followed in order to construct double and exceptional spaces which exhibit the key properties of the U(1) paradigm and allow for a realization of duality groups. To distinguish them from previous constructions, we shall call these new spaces "C-folds", ie manifolds with charge coordinates. One of the key requirements that we propose is the "topological geometrisation condition". This states that the pull back on the C-fold of the closed form field strengths on a spacetime are exact. One of the consequences of this condition is that the additional coordinates of the C-fold must have a non-trivial topology.
This paper has been organized as follows. In section 2, we describe the U(1) field paradigm. In section 3, we show that for DFT the patching conditions allow for only exact form field strengths. In section 4, we show under certain assumptions a similar statement for the EFT. In section 5, we give our conclusions and explore some future directions. In appendix A, we give an example of an alternative construction for double and exceptional geometries consistent with closed form field strengths.

A review of U (1) field paradigm
Before we proceed to investigate how to extend a spacetime with additional coordinates associated to a general closed k-form, it is instructive to revisit the construction for U(1) fields which is well known.
To begin, suppose M is a n-dimensional manifold equipped with a good cover {U α } α∈I , see eg [29], and a closed 2-form ω 2 . On each open set U α , the Poincaré theorem implies that there are 1-form potentials C 1 α such that Using repeatedly the Poincaré lemma at double and triple overlaps U α ∩ U β and U α ∩ U β ∩ U γ , one has that where n αβγ are constants. Now if ω 2 represents a class in H 2 (M, Z), then n αβγ ∈ 2πZ are integers.
The geometrisation of the U(1) field proceeds as follows. Starting from the good cover {U α } α∈I of M, one introduces a new space with charts U α × R, and transition functions where x α and f αβ are the coordinates and transition functions of M and θ α is an additional coordinate. At first sight it appears that the second transition function is consistent with (2.2) at triple overlaps iff n αβγ = 0. As we shall prove later, this implies that ω 2 is exact. However, for any closed ω 2 which represents a class in H 2 (M, Z), the transition functions are consistent at triple overlaps by taking the addition in the second transition function in (2.3) to be over mod 2πZ, ie the new transition functions are In such a case no further condition arises as the sum over a 0 's in (2.2) at triple overlaps vanishes mod 2πZ. The effect that this construction is to construct a circle bundle M ω 2 over M with transition functions φ αβ = exp(ia 0 αβ ). Some of the features and consequences of this construction are as follows.
• (i) An integral part of the construction is the modification of the combinatorial law that it is used to describe the transition functions of the additional coordinate. This is related to the requirement that ω 2 represents a class in H 2 (M, Z).
• (ii) The topological structure of M ω 2 is completely determined by the class [ , and vice versa, as there is a 1-1 correspondence between circle bundles and elements of H 2 (M, Z).
• (iii) The tangent bundle T M ω 2 of M ω 2 is an extension of T M with respect to a trivial real line bundle L, ie where π : M ω 2 → M is a projection.
• (iv) Another feature of the construction is that π * ω 2 is an exact form on M ω 2 as and (dθ − π * C 1 ) is a globally defined 1-form on M ω 2 .
The last property can be seen as the "topological geometrisation" of ω 2 . On M, ω 2 has charges which are given by the periods where (B i ) is a basis in H 2 (M, Z). Since π * ω 2 is exact on M ω 2 , all the periods of π * ω 2 on M ω 2 vanish. Instead all the information carried by the periods n i has been replaced by the topology of M ω 2 .
The above construction can be generalized to include more than one U(1) field strengths leading to toric fibrations over M. In addition this can be generalized to non-abelian gauge fields which in turn leads to to principal bundles with fibre the gauge group. In both cases all the properties mentioned above, with some minor modifications, apply to the more general set up.

.1 Transition functions of closed 3-forms
Before, we proceed to the patching of k-forms and the introduction of new coordinates, it is instructive to explain the patching of closed 3-forms ω 3 . For this let M be a n-dimensional manifold 2 with a good cover {U α } α∈I and a partition of unity {ρ α } α∈I subordinate to {U α } α∈I . This means that M admits functions ρ α ≥ 0 with support in U α such that at every point where the sum is taken over a finite collection. For a discussion on partitions of unity and the different kinds that exist see [29]. Here we shall use those partitions of unity that have the same index set I as that of the good cover and so they do not have necessarily compact support.
To continue, take ω 3 to be a closed 3-form on M. Using the Poincaré lemma at each open set U α , we can write ω 3 in terms of a gauge potential C as Using repeatedly the Poincaré lemma at the double, where n αβγδ are constants. Note that the patching data a 1 αβ , a 0 αβγ , n αβγδ are shew-symmetric in the interchange of any two of the open set labels, ie a 1 βγ = −a 1 γβ and similarly for the rest. Moreover n αβγδ are restricted to be integers if ω k represents a class in H 3 (M, Z). In particular, the patching condition of 2-form gauge potential in local coordinates reads Observe that the choice of gauge potentials is not unique. In particular, they are defined up to the gauge transformation Similarly, the patching data a 1 βγ of ω 3 at double overlaps are not uniquely defined either. In particular, they are defined up to a gauge transformation for a 0-form ψ 0 αβ defined on double overlaps. This is the only ambiguity that one has in determining the patching conditions on double overlaps of ω 3 . Anything else is inconsistent with the identification of ω 3 as a closed 3-form on M.

Transition functions and exact 3-forms
Before we proceed to compare the above patching conditions of closed 3-forms with those that arise in DFT, we shall prove a technical lemma which arises in the context ofČechde Rham theory. In particular, if there is a choice of patching data, up to (3.6) gauge transformations, such that a 1 αβ on triple overlaps U α ∩ U β ∩ U γ satisfies the cocycle 3 condition then ω 3 is exact.
For this suffices to show that Observe that dC α = dC α = ω 3 and where to establish the last two equalities we have used (3.7) and (3.1), respectively. Thus if (3.7) holds, ω 3 is exact and represents the trivial class in H 3 (M, R).

Patching DFT
After integrating the infinitesimal transformations generated by generalized Lie derivatives, generalized finite tensor transformations have been proposed for DFT 4 in [20] and further explored as transition functions in [21]. According to these generalized tensor transformations, 1-forms are transformed as and where X M are the coordinates of the double space and indices are raised and lowered with the split metric η M N and X ′M = X ′M (X N ). All the fields and coordinate transformations satisfy the strong section condition. Viewing the above transformations as patching conditions and putting them into the language of the previous section, we have An extensive investigation of the transformations induced on the fields after solving the strong section condition has been made in [15]. Interpreting these transformations as patching conditions, one can show that if one considers either or shift transformations of the double coordinates, then the 2-form gauge potential b of the NS-NS 3-form field strength transforms either as a 2-form, b α = b β or as b α = b β + dζ αβ , where in the latter case the x transformation is the identity Clearly, none of these two patching conditions are satisfactory. If b transforms as a 2-form, then one has to restrict H to be exact. The shift transformation is also not sufficient as we know that string theory admits solutions which require that M patches with non-trivial transition functions.
The authors of [15] have also investigated the combined transformation which includes diffeomorphisms of the spacetime M combined with shift transformations of the y coordinates which as a patching condition reads In turn these give rise to the patching condition for the 2-form gauge potential b.
It is clear from (3.16) that consistency on triple overlaps on the spacetime requires that If this is the case, a consequence of the lemma proven in section 3.2 implies that there are 1-forms {u α } defined on the open sets {U α } such that where Thus (3.17) can be rewritten as 21) or equivalently, As a resultb Thus H is exact. One therefore concludes that the patching conditions induced on the spacetime from the generalized coordinate transformations of DFE after solving the strong section condition imply that H is an exact form.
This conclusion cannot be satisfactory as we know string theory has as solutions spacetimes that admit non-trivial patching and a closed but not exact form H. The restriction that H is exact is a direct consequence of the introduction of the new coordinates y, their transition functions and their relation to the patching conditions of the 2-form gauge potentials as this is implied from the generalized coordinate transformations.
It is worth pointing out that H is not restricted to be exact in the context of generalized geometry 5 . In generalized geometry, there are no additional coordinates that have to be patched. The analogous consistency condition which arises on triple overlaps reads and it does not impose additional restrictions on the transition functions of H. The above condition is always satisfied as it can be seen from the analysis of the section 3.1.

Seeking a consistent patching
It is not apparent how to reconcile the transition functions of the double space with those of the spacetime, and the patching conditions of the 3-form field strength H without imposing additional conditions on the fluxes. Any choice of transition functions for the double space of the type where the patching conditions a 1 αβ of H are linear combination of ζ 1 αβ 's and the combinatorial rules denoted are as those of tensor calculus, will lead to the conclusion that H is exact. Another indication that such a direct approach may not be fruitful is that there is nowhere use of the quantisation condition [H] ∈ H 3 (M, Z) which has a central role in the exploration of the U(1) paradigm.
In appendix A, a modification of the patching conditions of the y coordinates was proposed and it was shown that there are no additional restrictions at triple overlaps. If this modification is considered, the double space patches consistently and it has some attractive features like its tangent bundle is an extension of the tangent bundle of the spacetime. However, the topological geometrisation of H fails, ie when H is pulled back on the double space is not exact which is in conflict with the U(1) field paradigm. Perhaps this is not surprising as to be able to topologically geometrize H, the additional coordinates have to exhibit non-trivial topology related to the class of H in H 3 (M, Z). In the conclusions, we propose a general framework where all these questions may be addressed.

Patching k-forms and exceptional coordinates 4.1 Transition functions of closed k-forms
The patching conditions of any closed k-form ω k , k > 3, on a manifold M can be found in a similar way as those for closed 3-forms in section 3.1. For this, it is convenient to use the difference operator δ of theČech-de Rham theory defined as where χ is a q-form defined at p + 1-overlaps and the caret denotes omission. δχ is a q-form defined at (p+2)-overlaps and it is understood that in the right-hand-side of the above equation χ is restricted on U α 0 ∩ · · · ∩ U α p+1 . Observe that δ 2 = 0 and dδ = δd. For more details on the properties of δ see eg [29]. Given now a globally defined closed k-form on M, k ≥ 3, we use the Poincaré lemma to write ω k α = dC k−1 α . Then we obtain the patching conditions and so on till where n α 0 ...α k are constants. If n α 0 ...α k ∈ 2πZ, then ω k represents a class in H k (M, Z). The (k-1)-form potentials {C k−1 α } are not uniquely defined. In particular, there are defined up to a gauge transformation as

Exact k-forms and patching conditions
As in the 3-form case utilizing the ambiguity in the definition of the patching conditions (4.5), it is possible to show that if a p , p = 0, . . . , k − 2 satisfies the cocycle condition then ω k is exact.
In what follows, it suffices to prove this for the a k−2 α 0 α 1 as those are responsible for the patching a k-form at double overlaps, and potentially can be used to construct the exceptional generalized spaces. The proof is similar to that we have given for 3-forms. In particular, suppose that (δa k−2 ) α 0 α 1 α 2 = 0 . (4.7) Then defineC Clearly dC k−1 α 0 = dC k−1 α 0 = ω k α 0 asC k−1 and C k−1 are related up to a gauge transformation. MoreoverC k−1 is a globally defined (k-1)-form as where we have used (4.7) and that γ ρ γ = 1.

Seeking a consistent patching for EFT
For EFT there is not an analogue of the patching conditions of [20] available for DFT. Instead the constructions have been based on using infinitesimal symmetries generated by generalized Lie derivatives, see eg [9,10,22,21,23,11,12] for detailed descriptions. One excepts that whatever the final form of the finite transformations are for EFT, these will generate both the transition functions of the underlying spacetime and the patching conditions of the form field strengths of the theory. After solving the strong section condition, the two must be related. Following the analogous analysis for DFT, one may hypothesize that the transition functions of the exceptional space read as where x and y k−2 are the spacetime and additional coordinates, respectively, and the patching data a k−2 αβ of the k-form field strength are a linear combination of ζ k−2 α 0 α 1 . If this is the case, then again consistency at triple overlaps will require that ω k is exact.
In appendix A, we propose a modification of the patching conditions (A.1) which resolves the restriction at triple overlaps. However, it does not topologically geometrize the k-form field strength. Similar constructions can be made in theories that we include the dual fields as demonstrated in appendix A for 11-dimensional supergravity.
The above result does not hold for exceptional generalized geometries, ie those that no new coordinates are introduce in addition to those of spacetime. This is because they do not require the condition (4.7) but instead dδ(a k−2 ) α 0 α 1 α 2 . (4.11) This does not introduce a restriction on the patching conditions of form field strengths.

Discussion and outlook
We have shown that the patching conditions of DFT as arise from generalize coordinate transformations after solving the strong section condition imply that the NS-NS 3-form field is an exact 3-form. A similar conclusion may hold in the context of EFT under some plausive assumptions regarding the relation between the transition functions of additional exceptional coordinates and the patching data of the form field strengths. We have also explored some alternative possibilities. These resolve some of the difficulties, like the restriction on the form field strengths to be exact, but they do not obey the topological geometrisation condition which is one of the key properties that the U(1) field paradigm.
To introduce new coordinates that extend the spacetime in a consistent way without any further conditions on the fields, like exactness of the form field strengths, one may try to generalize some aspects of the 2-form paradigm reviewed in section 2. Some of the directions that can be pursued are the following.
• To modify the combinatorial law of transition functions.
• To modify the transition functions.
• To introduce topology on the generalized spaces.
It is clear that the 2-form paradigm is consistent because it has been possible to modify the combinatorial law of the transition functions in a consistent way. Of course, this has been achieved under the additional requirement that ω 2 represents a class in H 2 (M, Z). Although this is imposes a restriction on the transition functions of ω 2 , this restriction is required by the Dirac quantisation condition. In the context of DFTs and EFTs, this is an indication that non-commutative geometry has a role as it is not possible to alter the combinatorial law of transition functions in a straight forward way, see also [28].
As we has seen a mild modification of the patching conditions of the form field strengths, eg as linear functions of the transition functions of the additional coordinates, leads to the conclusion that consistency at triple overlaps requires that the form field strengths are exact. However as we explain in appendix A, there is a modification of the transition functions of the additional coordinates such that the double and exceptional spaces are consistent at triple overlaps without any restrictions on the patching conditions of the form field strengths. However, as such a modification has its problems, like for example the topological geometrisation condition does not hold. Moreover there is no use of the cohomological analogue the Dirac quantisation condition [ω k ] ∈ H k (M, Z) in the construction of the double of exceptional space above which has a central role in the U(1) field paradigm.
It is clear from the above that whatever the construction of these extended spaces is the additional coordinates have to have a non-trivial topology. This is the only way that both the topological geometrisation and the cohomological analogue of the Dirac quantisation [ω k ] ∈ H k (M, Z) conditions can be utilized to construct these spaces. It is not a priori apparent how this can be done or whether a consistent construction is possible for all cases of interest, beyond those of toroidal compactifications, but a way to proceed is as follows.
One of the difficulties in adapting the U(1) field paradigm in the context of string theory and M-theory is that after applying a duality transformation the spacetime may change as a manifold at the same time as the fields of the theory. One way to incorporate this into the construction of extended spaces is as follows. Suppose that (M I , F I ) I∈I be a family of spacetimes M I with field content F I such that any two pairs (M I , F I ) and (M J , F J ) are related by a duality transformation D IJ , D IJ : (M I , F I ) → (M J , F J ). One way to geometrize the data (M I , F I ) and D IJ is to assume that there is a space CM, a C-fold, and maps π I : CM → M I such that The last property is the implementation of the topological geometrisation condition which should appear as a weak restriction for carrying out the geometrisation programme. It would be necessary to impose additional conditions on CM but the above three conditions can serve as a minimal requirement. It is not apparent whether the construction of C-folds CM would be possible for all backgrounds in string theory and M-theory within classical or even non-commutative geometry. However, there are constructions which satisfy all three conditions mentioned above. For this consider the trivial case, where the duality transformation D IJ is the identity and let us view CM as a fibration over M. There are many ways that the construction of a CM can be achieved which satisfies the topological geometrisation property. One way is K-theory which generalizes the 2-form paradigm. Suppose that one tries to construct a C-fold, CM ω 4 , of a spacetime that has field content a closed 4-form ω 4 . If ω 4 is represented by the first Pontryagin class of a vector bundle E, then as a CM can be taken as the principal bundle P that E is associated to. In such a case ω 4 will represent an integral class and the pull-back of ω 4 on P will represent the trivial class. It may be possible to find amongst the K-theory class which resolves the topological geometrisation condition a representative which will also exhibit the local geometric requirements as expected from DFTs and EFTs. K-theory has appeared before in string theory and M-theory [30], see also [31] and references within, and provides a topological description of D-brane charges. Therefore, it may not be a surprise that it could be also applicable to this context. Moreover the tangent space of all fibrations are extensions of that of the base space, and so T (CM ω 4 ) → π * T M → 0. This is one of the properties expected for generalized manifolds.
It is tantalizing for example that the fibration SO(4) → SO(5) → S 4 related to M5branes via the solution AdS 7 × S 4 has the above properties. Observe also that T e SO(4) = so(4) = Λ 2 (R 4 ). So the tangent space of the fibres provide a local model for the 2-forms on S 4 . This is in line with the expectation that the additional coordinates of this example are locally modeled by 2-forms on S 4 .
Although there are some suggestions how to construct generalized manifolds which allow for form field strengths to represent non-trivial cohomology classes, it is not apparent which will be the most fruitful way to proceed. The expectation is that the full theory at the end will combine many of the local computations that have been done so far with the global aspects that many of the backgrounds have in a consistent way.

Appendix A Modifying transition functions A.1 New transition functions and k-forms
Let ω k be closed k-form on M, k ≥ 3, with transition functions as those defined in section 4. To defineM ω k , we introduce coordinates (x, y k−2 ), where x are the coordinates of M, and impose the transition functions Since δ 2 = 0, consistency at triple overlaps requires that which is satisfied as it can be shown after a calculation using (4.2). It is clear that there is a projection π fromM ω k onto M and so it can be thought that M ω k is a bundle over M. However as in the case of 3-forms there is no natural global section of M inM ω k , eg the zero section is not preserved by the transition functions.
SinceM ω k is a manifold, one can investigate the tangent as well as all the other tensor bundles in the standard way. We shall do this forM ω 3 as the generalization toM ω k is straightforward. To do this let us write the patching conditions (A.1) explicitly as Note that the second patching condition in (A.3) does not satisfy the strong section condition. However one can do a coordinate redefinition and set simply (y 1 α ) i = (y 1 β ) i + (ã 1 αβ ) i but this less attractive in calculations.
To find the patching conditions of the tangent bundle consider the vector field X α = A i α ∂ ∂x i α + B α,i ∂ ∂(yα) i and demanding X α = X β , we find that It is clear from this that TM ω 3 is an extension of π * T M with respect to the cotangent bundle π * Λ 1 M of M, ie In particular, π * Λ 1 M is a subbundle of TM ω 3 . This is reminiscent of generalized geometry where the bundle E over M which is an extension of T M is now replaced with TM ω 3 . Furthermore observe that the pairing of T M and Λ 1 M is naturally extended to TM ω 3 as the transition functions of TM ω 3 preserve it. As a result, one can define an O(n, n) structure on TM ω 3 as expected from string theory considerations which also arises in the context of generalized geometry. But of course this O(n, n) is related to the modified transition functions rather than those associated to the original patching conditions a 1 αβ of of the 2-form gauge potential.
In the more general case ofM ω k , one finds that A.2 Testing for other properties One of the properties of the construction of C-folds for 2-forms ω 2 is that when one pulls back the 2-form on the C-fold, ω 2 becomes exact. It is not expected that this property holds onM ω k because by construction the fibres have trivial topology, ie by constructioñ It is clear from the above equation that ω k cannot be written as the exterior derivative of a (k − 1)-form onM ω k . For the latter, one would have expected that dy k−2 α − C k−1 α should have patched globally as a (k-1)-form onM ω k . But as we have shown this is not the case and the (k-1)-form that patches globally receives a correction that depends on the derivative of the functions ρ α which appear in the partition of unity. This correction is like a source term. Assuming that the partition of unity functions have compact support and that the good cover is very fine, ρ α resemble delta-functions. Therefore the source term is like the derivative of a delta function. Note that in the construction of some exceptional field theories additional field are required in the form of tensor hierarchies, see [11,12] which is reminiscent to these additional terms.

A.3M for M-theory
It would be of interest in view of applications in strings and M-theory to generalize the construction of the previous appendix from single k-forms, ω k , to differential algebras A.
We shall not give a general treatment of this. Instead we shall focus on the differential algebra of M-theory generated by the 4-form field strength F and its dual G, where now A : dF = 0 , dG = F ∧ F , (A. 10) and G is treated as an independent field. Suppose that M now has a good cover, the above equations can be solved at each open set of the cover as Next on double overlaps, we have δC 3 αβ = da 2 αβ , δC 6 αβ = db 5 αβ − a 2 αβ ∧ F (A. 12) and at triple overlaps δa 2 αβγ = da 1 αβγ , δb 5 αβγ = db 4 αβγ + a 1 αβγ ∧ F (A.13) Next to construct the C-fold, we introduce coordinates (x, y 2 , w 5 ) and introduce the transition functions (A.14) After performing a computation similar to that we have explained in previous case, the transition functions are consistent at triple overlaps. This proves that one can define a manifoldM A for the M-theory differential algebra A in (A.10).
Most of the properties of theM ω k spaces constructed for single k-forms can be generalized to this case. First there is a projection fromM A onto M, and soM A can be thought as a bundle over M. The tangent space TM A is an extension of T M with respect to the space of 2-and 5-forms on M, ie 0 → π * Λ 2 M ⊕ π * Λ 5 M → TM A → π * T M → 0 (A.15) As this construction is intended as an application to M-theory, M is 11-dimensional. However a similar construction can be applied to compactifications.