The Puzzle of Global Double Field Theory: Open Problems and the Case for a Higher Kaluza‐Klein Perspective

The history of the geometry of Double Field Theory is the history of string theorists' effort to tame higher geometric structures. In this spirit, the first part of this paper will contain a brief overview on the literature of geometry of DFT, focusing on the attempts of a global description.


Introduction
Double Field Theory: The symmetry known as T-duality is one of the main features of String Theory, in comparison with classical field theories. Double Field Theory (DFT) is an attempt to make this symmetry manifest: in other words it is a T-duality covariant formulation of Type II supergravity. DFT was officially created in [56], but seminal work includes [82] and [83]. See [18] DOI: 10.1002/prop.202000102 for a review of the subject and [3] for a review in the broader context of extended field theories.
The Bundle Gerbe of Kalb-Ramond Field: Geometrically the Kalb-Ramond field is interpreted as the connection of a bundle gerbe ↠ M, a geometric object which possesses gauge transformations, but also gauge-of-gauge transformations. Bundle gerbes were originally introduced in [70] and the definition of their gauge transformation was defined later in [67]. See [71] for an introductory review. In [45] bundle gerbes were reformulated in terms ofČech cohomology. Given a good cover {U } of the base manifold M, the local 2-forms B ∈ Ω 2 (U ) are patched by local 1-form gauge transformations Λ ∈ Ω 1 (U ∩ U ) which are themselves patched by scalar gauge transformations G ∈  ∞ (U ∩ U ∩ U ) satisfying the cocycle condition on four-fold overlaps of patches. In other words the differential local data of the Kalb-Ramond field are patched on overlaps of patches by the conditions (1.0.1) More recently in [73] bundle gerbes were formalized as a specific case of principal ∞-bundle, which is a principal bundle where the ordinary Lie group fiber has been generalized to any L ∞ -group. Therefore any Kalb-Ramond field is the connection of a particular principal ∞-bundle.
Higher Geometry of T-Duality: Notice that T-duality has been naturally formulated in the context of higher geometry as an isomorphism of bundle gerbes between a string background and its dual in [13,[36][37][38][39] and [75]. Let us consider two T n -bundle spacetimes M ← ← ← ← ← → M 0 andM← ← ← ← ← → M 0 over a common base manifold M 0 .
Then the couple of bundle gerbes Π ← ← ← ← ← ← → M and̃Π ← ← ← ← ← ← →M, encoding two Kalb-Ramond fields respectively on M andM, are geometric T-dual if the following isomorphism exists Fortschr. Phys. 2021, 69,2000102 www.advancedsciencenews.com www.fp-journal.org (1.0.2) This diagram can be interpreted as the finite version of the one in [21] for the respective Courant algebroids. In this sense T-duality is a geometric property of bundle gerbes.
Higher Kaluza-Klein Theory: As argued in [6] and [3], DFT should be interpreted as a generalization of Kaluza-Klein Theory where it is the Kalb-Ramond field, and not a gauge field, that is unified with the pseudo-Riemannian metric in a bigger space. Since the Kalb-Ramond field is geometrized by a bundle gerbe, in [1] we proposed that DFT should be globally interpreted as a field theory on the total space of a bundle gerbe, just like ordinary Kaluza-Klein Theory lives on the total space of a principal bundle. In the reference we showed how to derive some known doubled spaces such as the ones describing T-folds, and how to interpret T-duality.
In this paper we want to clarify some aspects of Higher Kaluza-Klein geometry by comparing it to previous proposals of DFT geometry. In particular we will deal with the problem of equipping a bundle gerbe with suitable coordinates. Finally we will focus on the concept of tensor hierarchy and how this emerges from a bundle gerbe perspective.
Plan of The Paper: In Section 2 we will illustrate a concise review of the main proposals for a global geometry of DFT, together with a discussion of the main open problems. In Section 3 we will give a brief introduction to the Higher Kaluza-Klein proposal. In Section 4 we will introduce an atlas for the bundle gerbe and we will argue that the 2d local coordinates of the charts must be interpreted as the local coordinates of the doubled space of DFT. Finally, in Section 5, we will consider both an abelian T-fold and a Poisson-Lie T-fold and we will interpret them as particular cases of global tensor hierarchies. Thus we will propose a globalization for tensor hierarchies which relies on the dimensional reduction of the bundle gerbe.

Double Field Theory
As explained by [3], Double Field Theory (DFT) should be thought as a generalization of Kaluza-Klein Theory from gauge fields to the Kalb-Ramond field. In this subsection we will give a quick introduction to local DFT.
Let us consider an open simply connected 2d-dimensional patch  . We can introduce coordinates (x ,x ) :  → ℝ 2d , which we will call collectively x M := (x ,x ). We can define a tensor = MN dx M ⊗ dx N ∈ Ω 1 ( ) ⊗2 with matrix MN := ( 0 1 1 0 ). We want now to define a generalized Lie derivative which is compatible with the -tensor, i.e. such that it satisfies the property X = 0 for any vector X ∈ ( ). Thus for any couple of vectors X, Y ∈ ( ) we have the following definition: projecting the GL(2d)-valued function L X N into an (d, d)-valued one. We also define the C-bracket by the anti-symmetrization of the generalized Lie derivative, i.e. by X, Y C := 1 2 Thus, to assure the closure, we need to impose extra conditions. The weak and the strong constraint (also known collectively as section condition) are respectively the conditions We can define the -tensor by   ⊗  where  = ( 0 1 1 0 ) .
We can also define the generalized metric as a tensor    ⊗  on the group manifold, where the matrix   is constant. If we locally rewrite the generalized metric    ⊗  =  IJ dx I ⊗ dx J in the coordinate basis, we obtain metric g and Kalb-Ramond field B depending on the d-dimensional submanifold G. Finally, the structure constants C   can be naturally identified with the generalized fluxes by C    ∧  ∧  . See [5] and [43] for more details.

Review of Proposals for DFT Geometry
In this section we will give a brief overview on the main proposals for geometry of DFT. We will underline relations between the approaches and we will discuss some open problems.

Non-Associative Proposal
The non-associative proposal was presented by [57] and further developed by [47]. Its aim is to realize the group of gauge transformations of DFT by diffeomorphisms of the doubled space. However, since the C-bracket structure on doubled vectors does not satisfy the Jacobi identity, its exponentiation will not give us a Lie group, but a geometric object which does not satisfy the associativity property.
Non-Trivial Three-Fold Overlaps: In the proposal by [47] the doubled space  is just a 2d-dimensional smooth manifold. This means that we can consider a cover { }, so that ⋃  = , and glue the coordinate patches on each two-fold overlap  ∩  of the doubled space by diffeomorphisms x ( ) = f ( ) (x ( ) ). Vectors of the tangent bundle T will be then glued on each T( ∩  ) by the GL(2d)-valued Jacobian matrix J ( ) := x ( ) ∕ x ( ) . However these transformations do not work for doubled vectors from DFT, thus [47] proposed that the doubled vectors should transform by the O(d, d)-valued matrix given by Notice that for the first time we see something resembling a gerbe-like structure spontaneously emerging in DFT geometry. Modified Exponential Map: The solution proposed by [47] consists, first of all, in a modified exponential map expΘ : ( ) → Diff ( ). This will map any vector X ∈ ( ) in the diffeomorphism given by , (2.1.3) where i and i are functions on x depending on the vector X in a way which guarantees that Θ(X) M M = X M M when applied to any field satisfying the strong constraint. This modified diffeomorphism crucially agrees with the gauge transformation V ′ (x) = e X V(x) of DFT, where X is the generalized Lie derivative defined by the D-bracket. Fortschr. Phys. 2021, 69,2000102 www.advancedsciencenews.com www.fp-journal.org ⋆-Product and Non-Associativity: In ordinary differential geometry the exponential map exp : ( ( ), [−, −]) → (Diff ( ), • ) maps a vector X  → e X into the diffeomorphism that it generates. The usual exponential map notoriously satisfies the property e X •e Y = e Z with Z ∈ ( ) given by the Baker-Campbell-Hausdorff series Z = X + Y + [X, Y]∕2 + … for any couple of vectors X, Y ∈ ( ). The idea by [47] consists in equipping the space of vector fields ( ) with another bracket structure ( ( ), −, − C ), where −, − C is the C-bracket of DFT. Now this algebra can be integrated by using the modified exponential map expΘ defined in (2.1.3) to a quasigroup (Diff ( ), ⋆ ) that satisfies It is possible to check that this ⋆-product is not associative: in other words the inequality , (2.1.5) where f, g, h ∈ Diff ( ) are diffeomorphisms, generally holds. Now let us call the diffeomorphisms f := e Θ(X) , g := e Θ(Y) and h := e Θ(Z) obtained by exponentiating three vectors X, Y, Z ∈ ( ). Then the obstruction of the ⋆-product from being associative is controlled by an element W which satisfies the equation and which is given by W = exp Θ(− 1 6  (X, Y, Z) + … ), where  (−, −, −) is the Jacobiator of the C-bracket. Even if it is wellknown that the Jacobiator is of the form  M = M  for a function  ∈  ∞ ( ), notice that the transformation W is nontrivial. Also if we consider diffeomorphisms on doubled space which satisfy f ( ) ⋆ f ( ) = f ( ) , we re-obtain the desired property  ( )  ( ) =  ( ) for doubled vectors.
We know that the diffeomorphisms group of the doubled space is not homeomorphic to the group G DFT of DFT gauge transformations e X . But now, by replacing the composition of diffeomorphisms with the ⋆-product, we can define a homomorphism : (

1.7)
which therefore satisfies the property (f ⋆ g) = (f ) (g). (2.1.8) This property determines the ⋆-product up to trivial gauge transformation. In the logic of [47] this will allow to geometrically realize DFT gauge transformation as diffeomorphisms of the doubled space.

Proposal with Gerbe-Like Local Transformations
The first paper in the literature explicitly recognizing the higher geometrical property of DFT is [4]. In the reference it is argued that we can overcome many of the difficulties of the nonassociative proposal by describing the geometry of DFT modulo local O(d, d)-transformations.
Gerbes Debut: Their proposal starts from the same problem (2.1.2), but proposes a different solution. We can rewrite the Cbracket of doubled vectors by X, Y C = [X, Y] Lie + (X, Y) where we called M (X, Y) := X N M Y N . This means that we can rewrite the algebra of DFT gauge transformations as [ X , Y ] = [X,Y] Lie + Δ(X, Y) where we defined Δ(X, Y) := (X,Y) . In [4] it is noticed that the extra Δ-transformation appearing in the DFT gauge algebra is non-translating, i.e. it involves no translation term if acting on tensors satisfying the strong constraint. Thus the diffeomorphism e X and the gauge transformation e X agree up to a local transformation e Δ = 1 + Δ. In fact, if we impose the strong constraint on fields and parameters , (2.2.1) wherẽ= X N Y N depends only on the d-dimensional physical subset U ⊂  of our doubled space patch. Hence the local Δ-transformation is just an infinitesimal gauge transformatioñ B = d̃of the Kalb-Ramond field. Further Discussion: As noticed by [55], Δ-transformations are integrated on a patch  to the group Ω 1 (U) of finite gauge transformations of the Kalb-Ramond field, while full gauge transformations generated by a strong constrained doubled vectors are integrated to Diff (U) ⋉ Ω 1 (U) ⊂ Diff ( ). Now we notice that the group of DFT gauge transformations effectively becomes the homotopy quotient G DFT = (Diff (U) ⋉ Ω 1 (U))∕∕Ω 1 (U), thus a 2group.
The doubled space is still a 2d-dimensional manifold  and then its coordinate patches on each two-fold overlap  ∩  are still glued by by diffeomorphisms x ( ) = f ( ) (x ( ) ). The doubled vectors are still glued by the O(d, d)-valued matrix  ( ) defined in (2.1.1), like in the non-associative proposal. Now, according to [4], on three-fold overlaps of patches  ∩  ∩  the transition functions of doubled vectors satisfy i.e. they satisfy the desired transitive property up to a local Δtransformation. In a more mathematical language we can say that doubled vectors would be sections of a stack on the 2d-dimensional manifold . This is not surprising since the algebra of G DFT is of the form ( (U) ⊕ Ω 1 (U))∕∕Ω 1 (U), which then must be glued on overlaps of patches by B-shifts d̃. Thus we could replace the concept of non-associative transformations with a gerbe-like structure.

Doubled-Yet-Gauged Space Proposal
The idea of doubled-yet-gauged space was proposed by [78] as a solution for the discrepancy between finite gauge transformations and diffeomorphisms of the doubled space, in alternative to the non-associative proposal. Then it was further explored in [63], where a covariant action was obtained, and in [79], where it was generalized to the super-string case. Very intriguingly this formalism led to novel non-Riemannian backgrounds in [22,66] and [25]. Recently a BRST formulation for the action of a particle on the doubled-yet-gauged space has been proposed by [11] and related to the NQP-geometry involved by other proposals.
www.advancedsciencenews.com www.fp-journal.org The Coordinate Gauge Symmetry: In doubled-yet-gauged space proposal the doubled space  is, at least locally, a smooth manifold. A local 2d-dimensional coordinate patch  is characterized by coordinate symmetry, i.e. there exists a canonical gauge action on its local coordinates expressed by for any choice of functions i , i ∈  ∞ ( ). This observation is motivated by the fact that any strong constrained tensor satisfies the identity Let us choose coordinates for our doubled patch  such that the strong constraint is solved by letting all the fields and parameters depend only on the d-dimensional subpatch U ⊂  . Then the coordinate symmetry on the doubled space reduces to This coordinate symmetry, similarly to the Δ-transformations in the previous proposal, can be identified with the local gauge symmetry of the Kalb-Ramond field bỹB = d̃, where the parameter is exactlỹ:=̃dx . We can thus identify the physical d-dimensional patches with the quotients U ≅  ∕ ∼. Thus, as argued by [78], physical spacetime points must be identified with gauge orbits of the doubled-yetgauged space.
The coordinate gauge symmetry is also the key to solve the discrepancy between DFT gauge transformations e V and diffeomorphisms e V . Indeed, as argued by [78], the two exponentials induce two finite coordinate transformations x M  → x ′M and x M  → x ′′M whose ending points are coordinate gauge equivalent, i.e. x ′M ∼ x ′′M . Therefore, upon section constraint, they differ just by a Kalb-Ramond field gauge transformation.
How Can We Globalize The Doubled-Yet-Gauged Space? Now, the doubled-yet-gauged formalism encompasses the local geometry of the doubled space. However in this review we are interested in the global aspects of DFT, so we may try to understand how these doubled patches can be glued together. Let us first try a naïve approach, for pedagogical reasons: we will try to glue our doubled patches by diffeomorphisms that respect the section condition, i.e. on two-fold overlaps of patches  ∩  we will have This would imply the patching conditions B ( ) = f * ( ) B ( ) + dΛ ( ) where the local 1-forms Λ ( ) := Λ ( ) dx ( ) are given by the gluing conditions (2.3.3). But then, with these assumptions, the doubled space  would become just the total space (ℝ d ) * -bundle on the physical d-dimensional spacetime M. If we compose the transformations (2.3.3) on three-fold overlaps of patches  ∩  ∩  we immediately obtain the cocycle condition Λ ( ) + Λ ( ) + Λ ( ) = 0, which is the cocycle describing a topologically trivial gerbe bundle with [H] = 0 ∈ H 3 (M, ℤ) and not a general string background. Therefore this naïve attempt at gluing is not enough.
Further Discussion: The doubled-yet-gauged formalism gives us an unprecedented interpretation of the coordinates of DFT.
Upon choice of coordinates which are compatible with the section constraint, indeed, the coordinate gauge symmetry can be identified with the gauge transformations of the Kalb-Ramond field. This is a fundamental link between the geometry of the bundle gerbe formalizing the Kalb-Ramond field and the geometry of the doubled space. This also provides an interesting link with ordinary Kaluza-Klein geometry, where the points of the base manifold of a G-bundle are in bijection with the gauge Gorbits of the bundle. As we will see in Section 4, the local coordinate gauge symmetry which was discovered by [78] will be also recovered as fundamental property of the double space which arises from the Higher Kaluza-Klein perspective. We will see that the Higher Kaluza-Klein formalism recovers a globalized version of the doubled-yet-gauged space with gluing conditions which are a gerby version of the naïve patching conditions (2.3.3). Therefore the Higher Kaluza-Klein proposal can be seen also as a proposal of globalization of the doubled-yet-gauged space approach.

Finite Gauge Transformations Proposal
In [ [55], pag. 23] it was proposed that, given a geometric background M, the group of gauge transformations of DFT should be just , where we called j ( ) := x ( ) ∕ x ( ) the Jacobian matrix of the diffeomorphism. This way it is natural to recover equation (2.2.2), i.e.
where e Δ ( ) will generally be a non-trivial local B-shift. Further Discussion: This proposal clarifies the previous ones by illustrating that, whenever the strong constraint can be globally solved by letting the fields depend on a d-dimensional submanifold M, doubled vectors must be seen as sections of a Then the extra coordinates (y 1 ( ) , ( ) ), which have "the degree" of a 1-form and of a scalar, must be then glued on two-fold and threefold overlaps of patches of M by using the transition functions of the gerbe, i.e. by With this identification, a change of coordinates (y 1 ( ) , ( ) )  → (y 1 ( ) + ( ) , ( ) + ( ) ) induces a gauge transformation for the Kalb-Ramond field given by (2.5.3) in analogy with the extra coordinate of ordinary Kaluza-Klein Theory.
Moreover, if we take the differential of the first patching condition in (2.5.2), we obtain the condition −dy 1 ( ) + dy 1 ( ) = dΛ ( ) for the differentials. This means that if we rewrite in components y 1 ( ) = y 1 ( ) dx , we can also rewrite −dy 1 ( ) + dy 1 ( ) = dΛ ( ) . If we define the dual vectors ∕ y 1 ( ) to the 1-forms dy 1 ( ) as vectors satisfying ⟨ ∕ y 1 ( ) , dy 1 ( ) ⟩ = , we obtain doubled vector of the following form: (2.5.4) which are exactly the same as the ones in (2.4.2). Therefore the analogue of the tangent bundle of the C-space can be identified with a Courant algebroid E ↠ M twisted by the gerbe (2.5.1). Further Discussion: The proposal seems to capture something quite fundamental of the geometry of DFT, by suggesting that the doubled space should be the total space of the gerbe itself. This looks consistent with the existing idea that doubled vectors should belong to a Courant algebroid twisted by a gerbe, which is the analogous to the tangent bundle for a gerbe. However this intuition is still waiting for a proper formalization: for example it is not clear how to construct coordinates that are 1-forms on M. Moreover it is still not clear what is the relation with the new extra coordinates and the T-dual spacetime.

Pre-NQP Manifold Proposal
The pre-NQP manifold proposal was developed by [30], generalized to Heterotic DFT by [27] and then applied to the particular example of nilmanifolds by [31]. This approach to DFT is based on the fact that n-algebroids can be equivalently described by differential-graded manifolds, including the Courant algebroid, which describes the local symmetries of the bundle gerbe of the Kalb-Ramond field. The idea is thus that we can describe the geometry of DFT by considering the differential graded manifold which geometrizes the Courant algebroid and by relaxing some of the conditions. L ∞ -Algebroids as NQ-Manifolds: Given a L ∞ -algebroid ↠ M on some base manifold M, we can always associate to its Chevalley-Eilenberg algebra CE( ), which is essentially the differential graded algebra of its sections. This is defined by where the underlying complex is defined by where the k for any k ∈ ℕ are the ordinary vector bundles underlying the L ∞ -algebroid. In the definition d CE is a degree 1 differential operator on the graded complex ∧ • Γ(M, * • ) which encodes the L ∞ -bracket structure of the original L ∞ -algebroid . Now a NQ-manifold is defined as a graded manifold  equipped with a degree 1 vector field Q satisfying Q 2 = 0. The fundamental feature of NQ-manifolds is that the algebra of functions of any NQ-manifold  is itself a differential graded algebra ( ∞ (), Q) where the role of the differential operator is played by the vector Q, which is thus called cohomological.
Crucially, there exists an equivalence between L ∞ -algebroids and NQ-manifolds given by where Q H is the cohomological vector twisted by the curvature H ∈ Ω 3 cl (M) of the gerbe. To show this, notice first that in this case the differential graded algebra of functions on our NQ-manifold will be truncated at degree < 2. The degree 1 sections will be sums of a vector and a 1-form X + ∈ Γ(M, TM ⊕ T * M) and the degree 0 sections will be just functions f ∈  ∞ (M) on the base manifold. Now we can explicitly rewrite the underlying chain complexes of the two differential graded algebras (2.6.4) by ) , ) , (2.6.5) moreover the derived bracket structure (see [30] for details) defined by the cohomological vector Q H on  ∞ (T * [2]T[1]M) is exactly the bracket structure of the Courant 2-algebroid: A Pre-NQP-Manifold for DFT: By following [30], we choose as 2d-dimensional base manifold M = T * U the cotangent bundle of some d-dimensional local patch. This is because we are interested in the local geometry of the doubled space and we have still no information about how to patch together these local 2ddimensional T * U manifolds. Thus the Courant algebroid on T * U will be given by the NQP-manifold T * [2]T[1](T * U), as we have seen. This will have coordinates (x M , e M ,ē M , p M ) still respectively in degrees 0, 1, 1 and 2, but with M = 1, … , 2d. We must then think the local coordinates x M = (x ,x ) to be the doubled coordinates of DFT.
Since T * U is canonically equipped with the tensor MN , we can make a change of degree 1 coordinates by (2.6.10) and the degree 0 functions will be just ordinary functions of the form f ∈  ∞ (T * U). The symplectic form restricted to the submanifold  will now be (2.6.11) The new Hamiltonian function will be |  = E M p M + H MNL E M E N E L , where H MNL now is the curvature of a bundle gerbe on the 2d-dimensional base T * U, which we should think as the extended fluxes of DFT. Crucially our  will still be a symplectic graded manifold, however it will not be a NQPmanifold since the new restricted vector Q is not nilpotent on , i.e. we have that Q 2 ≠ 0. This is exactly the reason why [30] named  pre-NQP manifold and therefore this cannot be seen an L ∞ -algebroid. However this pre-NQP manifold satisfies a very interesting property: the pre-NQP-manifold has a number of sub-manifold which are proper NQP-manifolds and thus well-defined sub-2algebroids. Schematically we have where is one of these sub-2-algebroids. On any of these, the bracket of doubled vectors in degree 1 will be exactly the D-bracket of DFT, which will be given by X, Y D := {QX, Y}. For instance we can choose the differential graded algebra of functions which are pullbacks from the submanifold  := {x =p = 0} ⊂ , which is exactly the Courant 2-algebroid  = T * [2]T [1]U. This corresponds to choosing a sub-2-algebroid which satisfies the strong constraint and therefore this restriction Fortschr. Phys. 2021, 69,2000102 reduces the pre-NQP-geometry to bare Generalized Geometry on the manifold U. Any other solution of the strong constraint will correspond to a viable choice of sub-2-algebroid.
We can also introduce tensors of the form  MN E M ⊗ E N on  and use the Poisson bracket to define a natural notion of D-and Cbracket on tensors. This allows to define a notion of generalized metric, curvature and torsion in analogy with Riemannian geometry.
An Example of Global Pre-NQP Manifold: It is well-known that higher geometry is the natural framework for geometric Tduality, see the formalization by [13,[36][37][38][39] and [75]. Assume formalizing two Kalb-Ramond fields respectively on M andM, are geometric T-dual if the following isomorphism exists (2.6.13) This picture is nothing but the finite version of T-duality between Courant algebroids illustrated by [21]. Now, in [31] it is proposed that we should consider the fiber product of the pull-back of both the gerbes and̃to the correspondence space M × M 0M of the T-duality, which will be itself a gerbe of the form (2.6.14) Now, as previously explained, we can take the algebroid of infinitesimal gauge transformations of this gerbe ⊗̃← ↠ M × M 0M and express it as a differential graded manifold (T * andĒ I := (ē I + IJ e J )∕ √ 2 and setĒ I = 0 to zero so that we obtain a new differential graded manifold . This new manifold will be locally isomorphic to (T * [2] ⊕ T [1])T 2n ⊕ T * [2]T[1]U on each patch U ⊂ M 0 of the base manifold, but which is globally welldefined. In [31] this machinery is applied for fiber dimension n = 1 to the particular case where M andM are nilmanifolds on a common base torus M 0 = T 2 .
Further Discussion: This is the first proposal to interpret strong constrained doubled vectors as sections of the 2-algebroid of the local symmetries of a gerbe: the Courant 2-algebroid. This suggests that it could be a complementary approach to the ones attempting to realize the doubled space as a geometrization the bundle gerbe itself.
However there are still some open problems. The only nontrivial global case that was constructed in this framework was, as we saw, on the correspondence space M × M 0M equipped with the pullback of both the gerbe and its dual̃. But, for this construction, the correspondence space of the T-duality is not derived from the pre-NQP manifold theory, but it must be assumed and prepared by using the machinery of topological T-duality. Besides, the total gerbe (2.6.14) has "repeated" information: for example, if we start from a gerbe i,j with Dixmier-Douady number i on a nilmanifold with 1st Chern number j, its dual will be a gerbẽj ,i on a nilmanifold with inverted Dixmier-Douady and 1st Chern number. Now the total gerbe i,j ⊗̃j ,i contain each number twice: as 1st Chern number and as Dixmier-Douady number. Moreover, in literature, a globally defined pre-NQP manifold for a non-trivially fibrated spacetime M was proposed only for the case of geometric T-duality. Recently [26] applied pre-NQP geometry to the case of DFT on group manifolds. However the extension of this formalism to general T-dualizable backgrounds is not immediate.

Tensor Hierarchies Proposal
The idea of tensor hierarchy was introduced in [50] in the context of the dimensional reduction of DFT, then further formalized in [7,51] and [8] as a higher gauge structure. See also work by [23] and [24].
Embedding Tensor and Leibniz-Loday Algebra: Let : (d, d) ⊗ R → R be the fundamental representation of the Lie algebra (d, d) of the Lie group O(d, d). The vector space underlying the fundamental representation of O(d, d) is nothing but R ≅ ℝ 2d . Let us use the notation x ⊗ Y  → x Y ∈ R. The embedding tensor of DFT is defined as a linear map Θ : (ℝ 2d ) →  ∞ (ℝ 2d , (d, d)) which satisfies the following compatibility condition, usually called quadratic constraint:  )). Now the embedding tensor defines a natural action of (ℝ 2d ) on itself by This is exactly the D-bracket of DFT. Thus the anti-symmetric part will be the C-bracket On the other hand the symmetric part of the D-bracket is given by is defined by f  → M f and the metric is defined by the contraction ⟨X, Y⟩ := MN X M Y N . Therefore the D-bracket can be expressed in terms of these operators by An interesting consequence is that the couple ( (ℝ 2n ), • ) is not a Lie algebra, since the D-bracket is not anti-symmetric, but it is a Leibniz-Loday algebra, since it satisfies the Leibniz for any triple of vectors X, Y, Z ∈ (ℝ 2n ). Now something remarkable happens: the Leibniz-Loday algebra ( (ℝ 2n ), • ) of infinitesimal DFT gauge transformations naturally defines a Lie 2-algebra ( (ℝ 2n ), i ) of infinitesimal DFT gauge transformations. This is given by the underlying cochain complex , (2.7.6) equipped with the following L ∞ -bracket structure: for any f ∈  ∞ (ℝ 2d ) and X, Y, Z ∈ (ℝ 2n ). Now notice that the quadratic constraint, which is the condition controlling the closure of the Leibniz bracket X•Y, requires to impose an additional constraint: this condition is nothing but the strong constraint. This makes the underlying complex of sheaves reduce to the one of sections of the standard Courant 2-algebroid ) . (2.7.11) Hence if we want ( (ℝ 2n ), • ) to be a well-defined Leibniz-Loday algebra we need to restrict to Generalized Geometry and the Dbracket • must reduce to the Dorfman bracket of Generalized Geometry, not twisted by any flux. At the present time no ways to generalize this construction beyond the strong constraint have been found.
Tensor Hierarchies: Now that we have our well-defined L ∞algebra ( (ℝ 2n ), n ), we can ask ourselves what happens if we use it to construct an higher gauge field theory on a (d − n)dimensional manifold M. The answer is that the theory resulting from this gauging process is exactly a tensor hierarchy, which is supposed to describe DFT truncated at codimension n.
Luckily for our gauging purposes, there exists a well-defined notion of the tensor product of a differential graded algebra with an L ∞ -algebra (see [59] for the formal definition). Thus we can define the prestack of local tensor hierarchies Ω(U, (ℝ 2n )) by the tensor product of the differential graded algebra of the de Rham complex (Ω • (U), d) with the L ∞ -algebra ( (ℝ 2n ), i ). In other words we define Ω(U, (ℝ 2n )) := Ω • (U) ⊗ (ℝ 2n ) for any contractible open set U ⊂ M. Its underlying complex of sheaves of this prestack will be and the bracket structure is found by applying the definition by [59]. Explicitly, for any elements  p ∈ Ω • (U) ⊗ (ℝ 2n ) and  p ∈ Ω • (U) ⊗  ∞ (ℝ 2n ), we have the following bracket structure: where we introduced the following compact notation for (ℝ 2n )valued differential forms: • − ∧ , − C is a wedge product on Ω • (U) and a C-bracket on (ℝ 2n ), • ⟨− ∧ , −⟩ is a wedge product on Ω • (U) and a contraction ⟨X, Y⟩ = IJ X I Y J on (ℝ 2n ).
The prestack Ω(U, (ℝ 2n )) encodes the local fields of a tensor hierarchy on a local doubled space of the form U × ℝ 2n with base manifold dim(U) = d − n. In our degree convention the connection data of a tensor hierarchy is given by a degree 2 multiplet while its curvature is given by the degree 3 multiplet Notice that all the fields of the hierarchy depend not just on the coordinates x of the base manifold U, but also on the coordinates (y,ỹ) of the vector space ℝ 2n . The curvature of the tensor hierarchy can be expressed in terms of the connection, as it is found in [50], by where we introduced the covariant derivative D := d − •∧ defined by the 1-form connection A, which acts explicitly by D = d +  ∧ ,  C and D = d + ⟨A ∧ , ⟩. Notice the characteristic C-bracket Chern-Simons term in the expression of 3-form curvature. We will call CS 3 (), so we will be able to write the curvature of the tensor hierarchy in a compact fashion: (2.7.17) By calculating the differential of the field curvature multiplet, this immediately gives the Bianchi identities of the tensor hierarchy: The infinitesimal gauge transformations of a tensor hierarchy are given by degree 1 multiplets where the covariant derivative explicitly acts as D = d + , C and DΞ = dΞ + ⟨ ∧ , Ξ⟩. Notice the extraordinary similarity of these equations to the ones defining a principal String-bundle. (This similarity will be discussed in Section 5).
Recovering Doubled Torus Bundles: Notice that, in the particular case of a tensor hierarchy where all the fields do not depend on the internal space ℝ 2n , the curvature reduces to the familiar equations of a doubled torus bundle, i.e.

 = d
∈ Ω 2 cl (U, ℝ 2n ), which is exactly the curvature of the String(T n × T n )-bundle araising in the case of a globally geometric T-duality, as explained in [1]. Also the gauge transformations reduce to This particular example of tensor hierarchy allows a globalization to a principal 2-bundle with gauge 2-group String(T n × T n ). Moreover, if we forget the higher form field, we stay with a welldefined T 2n -bundle on the (d − n)-dimensional base manifold M. This leads to the question about how to geometrically globalize and interpret general tensor hierarchies.
Further Discussion: The Doubled Space as A Higher Object: This proposal is the first to understand that the doubled connections  I , which we have also for the doubled torus bundles, are just a part of the full connection of the prestack Ω(−, (ℝ 2n )) including also  . Thus the doubled space is intrinsically a higher geometric object.
Further Discussion: What Global Picture? In [7] it was proposed that the global higher gauge theory of tensor hierarchies on a (d − n)-dimensional manifold M should consist in the L ∞ -algebra of (ℝ 2n )-valued differential forms on M, i.e. the L ∞ -algebra we called Ω(M, (ℝ 2n )) in our notation. However this must be taken as a local statement, since we know that gauge and p-form fields are not generally global differential forms on M, unless their underlying principal bundles are topologically trivial. Exactly like gauge fields, the global stack of tensor hierarchies must be instead given by the stackification of the prestack of local tensor hierarchies Ω(−, (ℝ 2n )). For a formal definition of stackification see [65]. This is true, at least, if we want to formalize tensor hierarchies as higher gauge theories. (In Section 5 we will discuss a different perspective).
Let us thus define the stack of DFT tensor hierarchies ℋ(−) as stackification of the prestack Ω(−, (ℝ 2n )) of local tensor hierarchies. By construction this means that on any set U ⊂ M of a good cover of our (d − n)-dimensional manifold M we will have the isomorphism This conveys the intuition that ℋ(−) is a globalization of Ω(−, (ℝ 2n )), but not necessarily a topologically-trivial one. By construction ℋ(−) maps any (d − n)-dimensional smooth manifold M to the 2-groupoid ℋ(M) whose objects are tensor hierarchies and whose morphisms are gauge transformations of tensor hierarchies on M.
In more concrete terms a global tensor hierarchy, which is an object of the 2-groupoid ℋ(M), can be expressed in a local trivialization by aČech cocycle. Given any good cover {U } for the www.advancedsciencenews.com www.fp-journal.org (d − n)-dimensional manifold M, such a cocycle will be of the form where the fields are of the following differential forms: and they are glued on two-fold, three-fold and four-fold overlap of patches as it follows: where the covariant derivatives are D = d −  ( ) •∧. Notice the similarity, at least locally, of the potential  ( ) ∈ Ω 1 (U × ℝ 2n , U × Tℝ 2n ) with the non-principal connection defined for instance in [ [61] , pag. 77] for a general bundle. In Section 5 we discuss the possibility of a more general definition of global tensor hierarchy, which can be obtained by directly dimensionally reducing the bundle gerbe and not as a higher gauge theory.
Further Discussion: However, if we accept that the global picture of tensor hierarchies is a higher gauge algebra, we would still have some open questions. From [50] we know that a tensor hierarchy is supposed to be a split version of DFT with a (d − n)dimensional base manifold M and 2n-dimensional fibers for an arbitrary n. But since tensor hierarchies are higher gauge theories, this hints that the full 2d-dimensional doubled space should carry a bundle gerbe structure. Such structure, as we have seen for previous proposals, still needs to be clarified.

Born Geometry
The first proposal of interpretation of DFT geometry as para-Kähler manifold is developed by [88] and then generalized to para-Hermitian manifolds by [89]. These ideas were further elaborated by [34,35,68,69,84] and [85]. This proposal sees the doubled space as a 2d-dimensional smooth manifold, whose tangent bundle is canonically split in two rank d lagrangian subbundles. The fluxes of DFT are then interpreted as the obstruction for the integrability of this structure.
Doubled Space as An Almost Para-Hermitian Manifold: An almost para-complex manifold (, K) is a 2d-dimensional manifold which is equipped with a (1,1)-tensor field K ∈ End(T) such that K 2 = id T , called almost para-complex structure, and such that the ±1-eigenbundles L ± ⊂ T of K have both rank(L ± ) = d. Thus, since the para-complex structure defines a splitting T = L + ⊕ L − , the structure group of the tangent bundle T of the almost para-complex manifold is The ±-integrability of K implies that there exists a foliation  ± on the manifold  such that L ± = T ± . An almost para-complex manifold (, K) is a proper para-complex manifold if and only if K is both +-integrable and −-integrable at the same time.
Our almost para-complex manifold (, K) becomes an almost para-Hermitian manifold if we equip it with a metric ∈ ⨀ 2 T *  of Lorentzian signature (d, d) which is compatible with the almost para-complex structure by Now we have a natural 2-form which is defined by (−, −) := (K−, −) ∈ Ω 2 (), called fundamental 2-form. Notice that the subbundles L ± are both maximal isotropic subbundles respect to and Lagrangian subbundles respect to . An almost para-Hermitian manifold (, K, ) becomes a para-Kähler manifold if the fundamental 2-form is closed, i.e. d = 0.
The closed 3-form  ∈ Ω 3 cl () defined by  := d is interpreted as encoding the fluxes of DFT.
Born Geometry: A Born Geometry is the datum of an almost para-Hermitian manifold (, K, ) equipped with a Riemannian metric  ∈ ⨀ 2 T *  which is compatible with both the metric and the fundamental 2-form by This Riemannian metric must be identified with the generalized metric of DFT.
Generalized T-Dualities: The generalized diffeomorphisms of DFT are now identified with diffeomorphisms of  preserving the metric , i.e isometries Iso(, ). Notice that the pushforward of a generalized diffeomorphism f ∈ Iso(, ) is noth- , as expected. This group of symmetries can be further extended to the group of generalized T-dualities, which are general bundle automorphisms of T preserving the metric .

www.advancedsciencenews.com www.fp-journal.org
A generalized T-duality f ∈ Aut(T) induces a morphism of Born geometries on  by which implies also the transformation  → f * of the fundamental 2-form.
Particularly interesting is the case of b-transformations which we can see as a bundle morphisms e b : T → T covering the identity id  of the base manifold. K  → K + b. Therefore a btransformation maps the splitting T = L + ⊕ L − to a new one T = L b + ⊕ L − which preserves L − , but does not preserve L + and +-integrability. This also implies  → + 2b.
Further Discussion: We can notice that Born Geometry is not (at least immediately) related to bundle gerbes, even if theory of foliations is closely related to higher structures as seen by [90]. In the next subsection we will mostly discuss the relation between Born Geometry and the bundle gerbe of the Kalb-Ramond field, trying to clarify it.

Can DFT Actually Recover Bosonic Supergravity?
Recovering Physical Spacetime: We will now try to recover a general bosonic string background, consisting in a pseudo-Riemannian manifold (M, g) equipped with a non-trivial H-flux [H] ∈ H 3 (M, ℤ), from Born Geometry as prescribed by [69] and [85].
Let us start from the almost para-Hermitian manifold (, K, ). The para-complex structure K splits the tangent bundle T = L + ⊕ L − where L ± are its ±1-eigenbundles. Since we want to recover a conventional supergravity background let us firstly assume that L − is integrable (physically this corresponds to set the R-flux to zero, see [69]). This implies that there exists a foliation  − of  such that L − = T − . Secondly, since we want to recover a conventional supergravity background, let us require that the leaf space M := ∕ − of this foliation is a smooth manifold. Indeed, according to [69] and [85], physical spacetime must be identified with the leaf space M. Thus the foliation  − is simple and the canonical quotient map :  ↠ M = ∕ − is a surjective submersion, making  a fibered manifold. Now we can use adapted (or fibered) coordinates (x ( ) ,x ( ) ) on each patch  of a good cover of the manifold  = ⋃  . Thus there exist a frame {Z ,Z } and a dual coframe {e ,ẽ }, given on local patches  as follows such that they diagonalize the tensor K and such that {Z } is a local completion of the holonomic frame for Γ(L − ). Notice that the N ( ) ∈  ∞ ( ) are local functions. In this frame we can express the global O(d, d)-metric =ẽ ⊙ e and the fundamental 2-form =ẽ ∧ e . In local coordinates (x ( ) ,x ( ) ) the latter can be written on each patch  as Now, by following [69], we can define a local 2-form B ( ) ∈ Ω 2 ( ) by the second term of the 2-form |  , i.e.
Now we must ask: what is the condition to make the local 2-form B ( ) on  descend to a proper local 2-form on the leaf space M = ∕ − (which is the physical spacetime)? By following [69] we can impose the condition that N ( ) are basic functions, i.e.
which assures exactly this. In local coordinates on  this condition can be rewritten as Papadopoulos' Puzzle Revised: In the adapted (or fibered) coordinates the transition functions of  on two-fold overlaps of patches  ∩  will have the simple following form: An adapted atlas will be also provided with the property that the sets U := ( ), where :  ↠ M = ∕ − is the quotient map, are patches of the leaf space M with local coordinates (q ( ) ) defined by the equation x ( ) = q ( ) • . These charts (U , q ( ) ) are uniquely defined. The local 2-form B ( ) will then descend to the where we suppressed the patch indices on the 1-forms {dx }: this is because {e } are global 1-forms on  and thus we can slightly abuse the notation by calling e ≡ dx . Thus we have Since the local 2-forms B ( ) descend to local 2-forms on patches U ⊂ M = ∕ − of the leaf space and these, according to [69] and [85], must be physically identified with the local data of the Kalb-Ramond Field, we must have bundle gerbe local data of the following form: (2.9.9) Fortschr. Phys. 2021, 69,2000102 Now, from the patching relations (2.9.6) of the adapted coordinates we obtain This, combined with (2.9.8) and (2.9.9), implies the following equations We can immediately solve the first equation by decomposing where d ( ) ∈ * Ω 1 ex (U ∩ U ) are local exact basic 1-forms on overlaps of patches. The cocycle condition for transition functions of a manifold on three-fold overlaps of patches implies then (2.9.13) Since Λ ( ) + Λ ( ) + Λ ( ) = dG from (2.9.9), then we must have the trivialization where c ( ) ∈ ℝ are local constants which must satisfy the following cocycle condition Open Problem: Therefore it does not seem possible to recover a general geometric string background made of a smooth manifold M equipped with a non-trivial Kalb-Ramond field [H] ∈ H 3 (M, ℤ). And, since DFT was introduced to extend supergravity, the impossibility of recovering supergravity poses a problem. This means that the original argument by [76] is still relevant whenever we try to construct the doubled space as a manifold.
Digression: Para-Hermitian Geometry for Group Manifolds: However, as we will see, Born geometry is still extremely efficient in dealing with doubled group manifold and, in particular, Drinfel'd doubles. Remarkable results from the application of para-Hermitian geometry to group manifolds, e.g. Drinfel'd doubles, can be found in [15,46,68,69,81] and [16]. For example, it has been shown in [46], that we can choose  := G × G, equipped with para-Hermitian metric for any x L,R , x ′ L,R ∈ , and  − := G diag , so that for G = SU(2) we have a spacetime M = ∕ − ≅ S 3 . In this example, it is clear that, by using a well-defined para-Hermitian manifold , we can geometrize a Kalb-Ramond field with non-trivial Dixmier-Douady class [ Moreover, as shown by [68], Drinfel'd doubles are naturally para-Hermitian manifolds. However, these group manifolds, where fluxes are constant, allow a simple geometrization of the bundle gerbe which is not possible in the most general case, where the higher geometric nature of the Kalb-Ramond field is fully manifest. Notice that a link between Drinfel'd doubles and bundle gerbes was firstly found by [93]. (In Subsection 4.4 we will briefly show that the para-Hermitian geometry of group manifolds can be recovered from the more general formalism we propose).
Higher Kaluza-Klein Perspective on The Problem: The Higher Kaluza-Klein proposal is an attempt to attack this problem and allow the geometrization of general bundle gerbes. In the Higher Kaluza-Klein perspective the doubled space would be identified with the total space of a gerbe M ↠ M. Thus the quotient :  ↠ M = ∕ − is reinterpreted as a local version of the projector of the gerbe as a principal ∞-bundle, i.e.
which is the higher geometric version of the statement : P ↠ M ≅ P∕G for any G-bundle P. See Section 3 for an introduction.
In Section 5 we will also use this higher Kaluza-Klein perspective to derive a globally well-defined notion of Tensor Hierarchy. As a particular example, we will derive the usual geometry of doubled group manifolds. Such a geometry, in the higher higher Kaluza-Klein perspective, will be equivalent to the natural para-Hermitian geometry of doubled group manifolds we mentioned.

Introduction to Higher Kaluza-Klein Theory
In this section we will give a very brief introduction to the Higher Kaluza-Klein perspective on the geometry of DFT we started to develop in [1]. For a brief glossary of the fundamental notions in higher geometry, see Appendix A.

The Doubled Space as A Bundle Gerbe
In the Higher Kaluza-Klein proposal (see [1] for details) the doubled space of DFT is identified with the total space of a bundle gerbe with connection. In this section we will mostly describe the geometry of the bundle gerbe. Let us now give a concrete geometric characterization to the concept of bundle gerbe.
The Bundle Gerbe: Let {U } be a cover for a smooth manifold M. We define (see [45] for details) a bundle gerbe π :  ←↠ M on the base manifold M by a collection of circle bundles {P ↠ U ∩ U } on each overlap of patches U ∩ U ⊂ M such that: • there exists an isomorphism P ≅ P −1 on any two-fold overlap of patches U ∩ U , • there exists an isomorphism P ⊗ P ≅ P on any three-fold overlap of patches U ∩ U ∩ U given by the gauge transfor- where for a given circle bundle P we denote with P −1 the circle bundle with opposite 1st Chern class, i.e. with c 1 (P −1 ) = −c 1 (P).
Notice that the trivialization we introduced defines aČech cocycle corresponding to an element [g (  More recently, in [73], the bundle gerbe has been reformulated as a principal ∞-bundle, where the gauge 2-group is G = BU(1), i.e. the group-stack of circle bundles. To see that the set of circle bundles on any manifold M carries a group-stack structure, notice that we have the isomorphisms where thus the tensor product ⊗ plays the role of the group multiplication, while the trivial bundle M × U(1) plays the role of the identity element and P −1 plays the role of the inverse element of P.
Automorphisms of The Bundle Gerbe: As seen also in [20], the 2-group of symmetries of a bundle gerbe  ↠ M is where ( ) ∈ Ω 1 (U ) are local 1-forms and ( ) ∈  ∞ (U ∩ U ) local scalars, whose patching conditions are the following: In [73] it was understood that the sections of a bundle gerbe are equivalently twisted U(1)-bundles. Then the local connection data of the twisted bundle can be expressed by local 1- Then locally on each patch, a section of the connective bundle gerbe looks like a section of T * U , which is a property expected by the doubled space. Geometric Strong Constraint: The bundle gerbe  ↠ M, being a particular example of principal ∞-bundle, will come equipped with a natural principal action. Crucially, the principal action : BU(1) ×  →  reproduces exactly the 2-group of gauge transformations and gauge-of-gauge transformations of the Kalb-Ramond field. Moreover the homotopy quotient of the bundle gerbe by the principal action is just the base manifold. This is totally analogous to an ordinary In [1] we show that we can define a generalized metric as a principal action-invariant structure on our bundle gerbe (see also Section 4 for more details).
where the curvature of the gerbe and the harmonic function are respectively with m ∈ ℤ and r 2 := ij y i y j radius in the four dimensional transverse space. Here, we called {x , y i ,x ,ỹ i } the external and internal doubled coordinates. This is a direct generalization of the Gross-Perry monopole in ordinary Kaluza-Klein Theory, globally realizing Berman-Rudolph monopole [17] .

Doubled Geometry from Reduction of Bundle Gerbes
We will use this subsection for a a deeper discussion of generalized correspondence spaces and of how they emerge from the bundle gerbe picture. Let us start from the simpler example of the dimensional reduction of an electromagnetic field. This example will facilitate the introduction of the dimensional reduction of the bundle gerbe.
Toy Example: Dimensional Reduction of An Electromagnetic Field: Let us consider a spacetime M which is a principal T n -bundle on some base manifold M 0 . Let us consider an electromagnetic field on M, which is globally given by a principal U(1)-bundle P ↠ M. Now, we want to study the dimensional reduction of this electromagnetic field, from the total space of the bundle M to the base manifold M 0 . Let dim(M 0 ) = d and dim(M) = d + n.
In local coordinates of M, the operation of dimensional reduction is nothing but the coordinate split ,…n are the local coordinates of the T n -fiber. Notice that we are not truncating the dependence of the electromagnetic field on the fiber coordinates { i }. The stack formalism will allow us to deal with the global geometric picture of such a coordinate split.
Generally, P ↠ M 0 is not a principal (U(1) × T n )-bundle on the base manifold M 0 , since the principal T n -action of M cannot generally be lifted to a T n -action on M. Therefore, generally, an electromagnetic field P ↠ M is not dimensional reduced to a welldefined electromagnetic field on the base manifold M 0 . In this general case, a dimensional reduction of a principal U(1)-bundle is given as it follows: Therefore, in this particular case, the electromagnetic field P ↠ M is dimensionally reduced to a globally well-defined electromagnetic field on the base manifold M 0 . Now, we will review the generalisation of this argument to the dimensional reduction of bundle gerbes. This notion will be the key for obtaining a globally well-defined notion of T-duality in the higher Kaluza-Klein perspective.
Correspondence Space from Reduction of The Gerbe: In [1] we applied to DFT the dimensional reduction of bundle gerbes, which was defined in terms of L ∞ -algebras by [37]. The statement is that, if M is the total space of a G-bundle on some base manifold M 0 , the dimensional reduction of a bundle gerbe  on M specified by the cocycle M → B 2 U(1) will be a certain higher geometric structure on M 0 . More in detail, the dimensional reduction will be a map Crucially, a particular example of this reduction is well-known in DFT since the work by [53] and by [10]. In fact, if we consider the particular example of a T n -equivariant gerbe on a T n -bundle, we automatically have that it dimensionally reduce as follows: ( Indeed, a String(T n ×T n )-bundle on M 0 is a particular principal ∞-bundle whose curvature forms are the following (see [1] for details):

2.5)
which are nothing but the usual equations of doubled torus bundles we find in [53]. Notice that, indeed, if we forget the higher Fortschr. Phys. 2021, 69, 2000102 www.advancedsciencenews.com www.fp-journal.org form field, we are left with a principal T 2n -bundle on M 0 with curvature  = d ( ) ∈ Ω 2 (M 0 , ℝ 2n ), which is exactly the correspondence space of a topological T-duality. In other words, such a T 2nbundle on M 0 will be the fiber product K := M × M 0M and we will have the diagram of the form Therefore, the equivariant case recovers the correspondence space of Topological T-duality. Now we can ask ourselves: what is the most general doubled geometry that we can obtain by dimensional reduction of bundle gerbes? We will deal with this question in the rest of this subsection. Topology and Non-Geometry: The Generalized Correspondence Space: As firstly noticed by [10], a bundle gerbe  ↠ M which satisfies the T-dualizability condition  i H = 0 gives rise to a well-defined T n -bundle K ↠ M. Thus we obtain the following diagram (3.2.8) Now K is called generalized correspondence space and it can be interpreted as the correspondence space for a non-geometric T-duality, i.e. the case where the T-dual background is nongeometric and hence a T-fold. We can then write (3.2.9) where the dotted arrows, since T-folds are not smooth manifolds, are not well-defined smooth maps between manifolds. For a proper formalization of the concept of T-fold as a noncommutative T n -bundle on M 0 see [14] and more recently [2] .
Notice that, on any patch U of the base manifold M 0 , the generalized correspondence space is locally T 2n -bundle K| U ≅ U × T 2n . The difference of the non-geometric case with the geometric case is in how these patches are globalized. Therefore non-geometry is a global property of the topology of K or, equivalently, of the topology of .
The Generalized Correspondence Space of Poisson-Lie T-Duality: In [1] we derived that something very similar happens for nonabelian T-duality and more generally for Poisson-Lie T-duality. A bundle gerbe  ↠ M on a G-bundle which satisfies the Poisson-Lie T-dualizability condition we get a generalized correspondence space of the form

The Atlas of Higher Kaluza-Klein Theory
The aim of this section is finding an answer to the following question: if the doubled space  is not a smooth manifold, but a bundle gerbe, then how can we define local coordinates?
We will also show that the natural notion of local coordinates for the bundle gerbe coincides with the notion of local coordinates for DFT.
We will need first to investigate what gluing charts means in theoretical physics. The notions we are going to use were introduced in String Theory by [38,40] and [41] 1 . For a brief glossary of the fundamental notions in higher geometry, see Appendix A.

Review: Atlases in Higher Geometry
In ordinary differential geometry, given a smooth manifold M, we can define an atlas. This is given by an : We physicists, in fact, usually work not directly on a manifold M, but in coordinates on the atlas ⨆ ∈I ℝ d . Here, by using the pullback * of the projection (4.1.1), we are allowed to write a 1 The author thanks Urs Schreiber for explaining the notion of atlas of a stacks and, in particular, how it was firstly applied in the context of Super-Exceptional Geometry by [38] and [40]. We will try here to apply the definition to a simpler bundle gerbe and we will focus more on its global aspects. The objective of this subsection is to find a generalisation of this atlas coordinate description for bundle gerbes.
Atlas for A Stack: For a geometric stack , the notion of atlas is generalized as it follows: an atlas for is a smooth manifold  equipped with a morphism of stacks which is, in particular, an effective epimorphism, i.e. whose 0truncation 0 :  ←↠ 0 is a projection of smooth manifolds 2 (see [44] and [65] for details). This formalizes the idea that any geometric stack can be described by using an atlas that is an ordinary manifold  .

Example: Atlas for A Smooth Manifold:
If our geometric stack is an ordinary smooth manifold := M, we can choose an atlas given by  := ⨆ ∈I ℝ d and by a surjective map : This corresponds to the well-known idea in differential geometry that the total space P locally looks like a Cartesian space ℝ d+1 .
Any such map uniquely factorizes as : where the first map is just the surjection which is the identity on ℝ 1,d and the quotient map ℝ ↠ U(1) = ℝ∕2 ℤ. Crucially, the map F is an atlas of ordinary Lie groups.
The surjective map → P is an atlas for the total space P, in the stacky sense of the word. This corresponds to the intuitive idea that the total space of a circle bundle locally looks like the Lie group ℝ d × U (1).
TheČech Nerve of An Atlas: When we defined the atlas :  ←↠ for the stack , we said that it must be an effective epimorphism. An effective epimorphism is defined as the colimit of a 2 Let us mention that there exists a well-defined notion of a functor 0 which is called 0-truncation. This sends a higher stack to its restricted sheaf 0 at the 0-degree. For a geometric stack, its 0-truncation is just a smooth manifold. See [73] for details.
certain simplicial object which is calledČech nerve. In other words we have TheČech nerve of an atlas can be interpreted as a ∞-groupoid, which we will callČech groupoid. This groupoid encodes the global geometry of the stack in terms of the smooth manifold  , which makes it easier to deal with. Besides the original stack can always be recovered by the colimit of the nerve. How do we construct such a simplicial object? Let us firstly consider the kernel pair of the map , which is defined as the pullback (in the category theory meaning) of two copies of the map . The coequalizer diagram of this kernel pair will thus be of the following form: In this case the kernel pair, defined in (4.1.5), is the following: We notice that the kernel pair of the atlas encodes nothing but the information about how the charts are glued together over the manifold M. We intuitively have that the global geometry of a smooth manifold M is entirely encoded in itsČech groupoid ( ). We physicists are actually very familiar with this perspective: in fact we usually describe our fields as functions on the local charts ℝ d of a manifold M and, if we want to describe how they behave globally, we simply write how they transform on the overlaps ℝ d ∩ ℝ d of these charts. In the next paragraph we will formalize exactly this perspective on fields.

Gluing a Field on A Stack
be an atlas for the stack and let ℱ be another stack, which we will interpret as the moduli-stack of some physical field. Now let A : → ℱ be a morphism of stacks (i.e. a physical field on ). We obtain an induced morphism A• :  → ℱ together with an isomorphism between the two induced morphisms which satisfies the cocycle condition on  ×  ×  . ifold induces a local 1-form A ( ) := A• ∈ Ω 1 (ℝ d , ) on each chart of the atlas. Notice that these 1-forms A ( ) (x) depends on local coordinates x ∈ ℝ d , like we physicists are used. On overlaps of charts we must also have an isomorphism between A ( ) and A ( ) given by a gauge transformation G). Again, these h ( ) (x) are not G-valued functions directly on the manifold, but on the atlas. Finally, these isomorphisms must satisfy the cocycle condition h ( ) h ( ) h ( ) = 1.

Example: Gluing A Gauge Field on A Smooth Manifold: Let
In this subsection we explained in an almost pedantic way how geometric structures on smooth manifolds become the familiar and more treatable objects on local ℝ d coordinates we physicists use. We will see in the next subsection that these intermediate steps become much less trivial if we want to glue local charts for DFT.

The Atlas of A Bundle Gerbe
The aim of this section will be finding an atlas for a bundle gerbe , seen as a stack.
An Atlas for The 2-Algebra: Let us call ℝ d ⊕ b (1) the 2algebra of the abelian 2-group ℝ d × BU(1). It is well-known that an L ∞ -algebra is equivalently described in terms of its Chevalley-Eilenberg differential graded algebra (see Appendix A). In our particular case this differential graded algebra will be where the {e a } with a = 0, … , d − 1 are elements in degree 1 and B is an element in degree 2. Since the 2-algebra is abelian, the differentials of the generators of its Chevalley-Eilenberg algebra are trivial. Now recall that an atlas is an effective epimorphism from our manifold to our stack. In this case it will be an effective epimorphism from an ordinary Lie algebra to our 2-algebra of the form Dually this can be given as an effective monomorphism between the respective Chevalley-Eilenberg differential graded algebras In other words we want to identify an ordinary Lie algebra such that its Chevalley-Eilenberg algebra contains an element := f * (B) ∈ CE( ) in degree 2 which is the image of the degree 2 generator of CE(ℝ d ⊕ b (1)) and which must satisfy the same equation since a homomorphism of differential graded algebras maps f * (0) = 0. Since must be an ordinary Lie algebra, its Chevalley-Eilenberg algebra CE( ) will have only degree 1 generators. Thus its generators must consist not only in the e a := f * (e a ), but also in an extra setẽ a for a = 0, … , d − 1 which satisfies

2.5)
Now the equation d = 0, combined with the equation de a = 0, implies that the differential of the new generator is zero, i.e. dẽ a = 0. Thus we found the differential graded algebra CE( ) = ℝ[e a ,ẽ a ]∕⟨de a = 0, dẽ a = 0⟩, (4.2.6) which must come from the ordinary Lie algebra Let us now call ℝ d,d := ℝ d ⊕ (ℝ d ) * and notice that the underlying vector space is 2d-dimensional. The differential graded algebra can be thought as where the notation Ω • li (G) means the complex of the left invariant differential forms on a Lie group G. But  In conclusion we constructed a homomorphism of 2-algebras which is a well-defined atlas for our 2-algebra. Now let us discuss the kernel pair of the atlas (4.2.10). As we have seen, this is defined as the pullback (in the category theory sense) of two copies of the map f of the atlas (4.2.10). The coequalizer diagram of these maps is (4.2.11) To deal with it, we can consider the Chevalley-Eilenberg algebras of all the involved L ∞ -algebras and look at the equalizer diagram of the cokernel pair which is dual to the starting kernel pair (4.2.11). This will be given by the following maps of differential graded algebras: Let us describe this in more detail. If composed with f * , the two maps send the generators e a to e a and the generator B to a couple =ẽ a ∧ e a and ′ =ẽ ′ a ∧ e a , whereẽ a andẽ ′ a are such that they both satisfy the same equation dẽ ′ a = dẽ a . This implies that they are related by a gauge transformationẽ ′ a =ẽ a + d a . This fact can be seen as a consequence of the gauge transformations B ′ = B + d with parameter := a e a .
is not an atlas, since its source is itself a stack and not an ordinary manifold. This map is still interesting, because it formalizes the idea that the total space of the gerbe  locally looks like the Lie 2-group ℝ d × BU(1). Now we know that each ℝ d × BU(1) has a natural atlas (4.2.13).
Thus by composition we can construct maps : . By combining them we can construct an atlas for the bundle gerbe: (4.2.14) Cech Nerve of The Atlas of The Bundle Gerbe: Let us assume that our bundle gerbe is specified by theČech cocycle (B ( ) , Λ ( ) , G ( ) ). We can now use the map (4.2.14) to explicitly construct theČech nerve of the atlas. We obtain the following simplicial object: Let us describe this diagram in more detail in terms of its dual diagram of Chevalley-Eilenberg algebras. The two maps of the kernel pair send the local degree 1 generator to dx and the local degree 2 generator to a couple of local 2-forms triv ( ) = dx ( ) ∧ dx and triv ( ) = dx ( ) ∧ dx on the fiber product of the -th andth charts. Now the local 1-forms dx ( ) and dx ( ) are required to be related by a gauge transformation dx ( ) = dx ( ) + dΛ ( ) where the gauge parameters Λ ( ) are given by the cocycle of the bundle gerbe. Equivalently the two 2-forms must be related by a gauge transformation triv ( ) = triv ( ) + dΛ ( ) with gauge parameter Λ ( ) = Λ ( ) dx . The gauge parameters, as expected, are required to satisfy the cocycle condition Λ ( ) + Λ ( ) + Λ ( ) = dG ( ) on three-fold fiber products of charts.

The Atlas of DFT
In the previous subsection we constructed the atlas of a bundle gerbe and we showed that it is made up of ℝ d,d -charts, which we interpret as the coordinate charts of Double Field Theory. In this perspective, the main problem of the traditional approaches to geometry of DFT is trying to glue the left-hand-side ⨆ ∈I ℝ d,d of the atlas (4.2.14) to form a global 2d-dimensional smooth manifold, not recognizing that it is the atlas of a bundle gerbe.
Natural Interpretation for The Extra Dimensions: The 2ddimensional atlas of the bundle gerbe is the natural candidate for being an atlas for the doubled space of DFT. This means that we can avoid the conceptual issue of requiring a 2d-dimensional spacetime (or even a much higher-dimensional one for Exceptional Field Theory), because the extra d coordinates of the charts locally describe the remaining degree of freedom of the bundle gerbe. In this sense, DFT on a chart ℝ d,d is a local description for a field theory on the bundle gerbe.
Principal Connection of The Gerbe: On the atlas of a bundle gerbe we can define its principal connection ∈ Ω 2 ( ⨆ ∈I ℝ d,d ) by the difference ( ) := triv ( ) − B ( ) of the local 2-form triv ( ) we obtained in the previous subsection and the pullback of the local connection 2-form B ( ) of the bundle gerbe living on the base manifold. This definition assures that ( ) = ( ) on overlaps of the ℝ d,d -charts. Thus in local coordinates we can write Notice that the form is invariant under gauge transformations of the bundle gerbe, i.e. of the Kalb-Ramond field. In general it is also possible to express the principal connection =ẽ a ∧ e a in terms of the globally defined 1-formsẽ a = dx ( )a + B ( )a dx and e a = dx a on the atlas. We can also pack both the left invariant differential forms in a single 1-form E A with index A = 1, … , 2d which is defined by E a := e a and E a :=ẽ a . In this notation we have that the connection can be expressed by

3.2)
where AB is the 2d-dimensional standard symplectic matrix. Notice that we recover the curvature of the bundle gerbe by This is completely analogous to the curvature of a circle bundle P ↠ M being the differential of its connection ∈ Ω 1 (P), i.e. it is F = d ∈ Ω 2 cl (M). Global Generalized Metric on The Gerbe: A global generalized metric can now be defined just as an orthogonal structure  ←→ GL(2d)∕∕O(2d), (4.3.4) on the bundle gerbe itself, just like a Riemannian metric on a manifold. As explained in [1], if we require the generalized metric structure to be invariant under the principal action of the bundle gerbe, this will have to be of the form www.advancedsciencenews.com www.fp-journal.org (4.3.5) and in in terms of local coordinates we find the usual expression Para-Complex Geometry: Our local chart is canonically split by ℝ d,d = ℝ d ⊕ (ℝ d ) * , where the restriction ℝ d can be seen as a chart for the d-dimensional base manifold M of the gerbe. This immediately implies that the tangent bundle of the local chart splits by Then on each chart the gerbe connection becomes a projector to the vertical bundle Recall that an Ehresmann connection for an ordinary principal bundle defines a projection : TP ↠ VP onto the vertical subbundle: for the gerbe it is not so different. If we consider a vector X = X +X̃on ℝ d,d , this will be mapped by the connection to X V := (X + B ( ) X )̃. Thus, if we call {E A } a basis of leftinvariant vectors on ℝ d,d dual to the 1-forms {E A }, we obtain vectors of the form Notice that, if we restrict ourselves to strong constrained vectors, these are immediately globalized to sections of a Courant algebroid twisted by the gerbe with connection B ( ) . See [1] for more details about the tangent stack of the gerbe. The para-complex structure can be defined by using the gerbe connection by J := id Tℝ d,d − 2 , in analogy with a principal connection. If we split a vector in horizontal and vertical projection X = X H + X V , this will be mapped to J(X) = X H − X V .
Thus every chart (ℝ d,d , J, ) is a para-Hermitian vector space. In this specific sense a bundle gerbe is naturally equipped with an atlas of para-Hermitian charts, even if its total space is not a a globally well-defined smooth manifold.
Local Doubled-Yet-Gauged Geometry: The principal action of the bundle gerbe will be given on a local chart of the atlas by a shift (x ,x )  → (x ,x + ) in the unphysical coordinates, identified with a gauge transformation B  → B + d( dx ) of the Kalb-Ramond field. This matches with the coordinate gauge symmetry discovered by [78], upon application of the section condition. Moreover the global bundle gerbe property ∕∕BU(1) ≅ M, when written on a local chart of the atlas, can be identified with the property that physical points corresponds to gauge orbits of the doubled space: this gives a global geometric interpretation of the strong constraint. Thus the local charts of the bundle gerbe match with the doubled-yet-gauged patches by [78]. In this sense the Higher Kaluza-Klein formalism can also be seen as a globalization of the local geometry underlying the doubled-yet-gauged space proposal which we briefly reviewed in Section 2.

The T-Dual Spacetime is A Submanifold of The Gerbe
If we accept the identification of the global doubled space with the total space of the bundle gerbe, then how can we obtain the T-dual spacetime to the starting one? Let us briefly explain it with a concrete example.
Abelian T-Duality: Let our gerbe  ↠ M have a base manifold which is itself the total space of a T n -bundle M ↠ M 0 . Moreover we will assume that the gerbe bundle satisfies the T-duality condition  i H = 0. Now let (x ( ) , i ( ) ) be the local coordinates of a chart ℝ d = ℝ d−n × ℝ n of M adapted to the torus fibration and let i = d i ( ) + A i ( ) ∈ Ω 1 (M) be the global connection 1-form of the torus bundle. This means that the gerbe connection will be where we called B (2) ( ) the horizontal part of the 2-form B ( ) respect to the torus fibration and where (x ( )i ,̃( )i ) are the local coordinates of (ℝ d ) * . Thus we obtain the following forms Thanks to the T-dualizability condition satisfied by the bundle gerbe, something special happens: the 1-formẽ i = d̃( )i − i B ( ) becomes the global connection of a well-defined T n -bundle K ↠ M. See [1] for more details about abelian T-duality in the bundle gerbe picture.
Therefore, in the special case of a T-dualizable bundle gerbe, the (ℝ n ) * ⊂ (ℝ d ) * part of the charts are glued together to form an extra manifold: an extraT n -bundle. This manifold can be seen as the Tdual spacetime and this gives rise to T-duality. On the other hand the remaining (ℝ d−n ) * ⊂ (ℝ d ) * part of the charts still cannot be glued to form a manifold. Let us remark that in the general case the local charts (ℝ d ) * of the bundle gerbe cannot be glued to form a manifold at all. Whenever the gerbe contains such a fiber bundle K ↠ M, which we will call generalized correspondence space, there is T-duality.
See Section 5 for a deeper discussion of more general cases of T-duality and generalized correspondence space, including the ones whose fibers are Drinfel'd doubles.
Para-Hermitian Geometry of The Internal Space: The atlas also explains the effectiveness of Born Geometry when dealing with internal spaces, such as tori or Drinfel'd doubles. In fact, the restriction of the gerbe connection to the fiber of the generalized correspondence space K is the fundamental 2-form of an almost symplectic structure. For example, for the previous example abelian T-duality, the connection restricts to | T 2n = (d̃( )i + ( )ij := i j B ( ) the moduli field of the Kalb-Ramond field. It is not hard to see that the fibers of the generalized correspondence space K are almost para-Hermitian manifolds, equipped with Born Geometry. In Section 5 we will deal with more general examples, whose internal space is more generally a Drinfel'd double and this principle can www.advancedsciencenews.com www.fp-journal.org be generalized to them too. In such cases, the connection of the bundle gerbe, restricted to the Drinfel'd double D-fiber, | D will be the fundamental 2-form of para-Hermitian geometry.
Relaxation of The Strong Constraint: In principle, if we want to relax the strong constraint, we can consider fields on our bundle gerbe which are not necessarily invariant under its principal action. We will have to glue these new fields according to the rules that we exposed in the last subsection.

Global Tensor Hierarchy in Higher Kaluza-Klein Theory
In this section we propose a definition of global tensor hierarchy which is more general than a higher gauge theory. We will argue that strong constrained tensor hierarchies should be globalized and geometrized by dimensional reductions of the differential data of a bundle gerbe. This global definition allow us to identify the field content of a T-fold (which is not a higher gauge theory) with a global tensor hierarchy. As particular examples of this perspective we will deal with two example of T-folds. For a brief glossary of higher geometry, see Appendix A.

Global Tensor Hierarchies, Topology and Non-Geometry
Let us consider the idea of tensor hierarchies as higher gauge theories from Section 2. By using the global higher geometric machinery from Section 3, we will discuss this definition and propose a slight generalisation for it.
Motivations for A Slightly More General Definition: • In Section 2, we reviewed the definition of tensor hierarchy as a higher gauge theory. This definition can be immediately globalized, since higher gauge theories are globally well-defined. Moreover, the fields of a higher gauge theories can be identified with the connection of some (non-abelian) bundle gerbe. This would suggest that any global tensor hierarchy is geometrized by a (non-abelian) bundle gerbe structure. However, in [50], tensor hierarchies are introduced as the result of a general dimensional reduction (i.e. split of the coordinates) of a doubled space. As we have seen in Section 3, the dimensional reduction of a geometric structure does not give, in general, something as nice and regular as a globally well-defined bundle, but the split coordinates will be generally glued by monodromies. Therefore, generally, the dimensional reduction of the doubled space will not lead to a globally well-defined higher gauge theory. • As seen by [1], the global doubled space encoding a T-fold is obtained by dimensionally reducing the global doubled space, i.e. the bundle gerbe (see Section 3). Therefore, by following the idea by [50], the doubled space underlying the Tfold should be considered a global tensor hierarchy. However, such doubled spaces cannot be obtained by gauging the local tensor hierarchy algebra of Section 2, i.e. the local prestack Ω(U, (ℝ 2n )), because the fields are patched also by cocycles of monodromies. Therefore, they are not globally given by a higher gauge theory.
These two points suggest that, even if the picture of tensor hierarchy as higher gauge theory holds locally, it could be not the most useful global picture. In this section we will propose a definition of global strong constrained tensor hierarchies which is slightly more general than the higher gauge theory definition of Section 2. Then we will show that both abelian and Lie-Poisson T-folds match this enlarged definition.
Let us recall that tensor hierarchies require the strong constraint to be well-defined. We can thus replace the C-bracket with the anti-symmetrized Roytenberg bracket of Generalized Geometry. We can then solve the strong constraint and obtain locally the curvature as sc (ℝ 2n )-valued differential forms where now the bracket [−, −] Roy is the anti-symmetrized Roytenberg bracket. In coordinates this corresponds to setting̃i = 0 on any field so we will also have ( Analogously for all the others sc (ℝ 2n )valued differential forms. This reads as follows: any gerbe on the total space of a T nbundle M ↠ M 0 which is equivariant under the principal T naction is equivalently a String(T n ×T n )-bundle on the base manifold M 0 . If we forget the higher form fields, we remain with a T n ×T n -bundle, which is nothing but the correspondence space K = M × M 0M of a topological T-duality. For a review of nonabelian bundle gerbes whose structure group is a String-group, see [74]. Thus, unsurprisingly, our definition of global tensor hierarchy includes doubled torus bundles.

T-Folds as Global Tensor Hierarchies
In this subsection we will briefly explain the global geometry of a T-fold, which is obtained by dimensionally reducing a bundle gerbe on a torus bundle spacetime. Then we will explain how the geometric structure underlying the T-fold can be naturally interpreted as a particular case of the global tensor hierarchies we defined. Crucially, these T-fold geometries cannot be obtained by gauging the algebra of local tensor hierarchies from Section 2. This will give practical motivation to the definition of the previous subsection.
The Generalized Correspondence Space of T-Duality: Let us start from the T n -bundle M ↠ M 0 , whose total space M is equipped with a Riemannian metric g and gerbe structure with curvature H ∈ Ω 3 cl (M). In the following we will use the underlined notation for the fields living on the total space M. We can now use the principal connection ∈ Ω 1 (M, ℝ n ) of the torus bundle to expand metric g and curvature H in horizontal and vertical components respect to the fibration. We will obtain i , H ij , H ijk as globally defined differential forms which are pullbacks from base manifold M 0 , so that they do not depend on the torus coordinates. Recall that the differential data of a bundle gerbe on M with curvature H ∈ Ω 3 cl (M) is embodied by aČech cocycle (B ( ) , Λ ( ) , g ( ) ) satisfying the following patching conditions: Now, on patches and two-fold overlaps of patches of a good cover of M we can use the connection of the torus bundle to split the differential local data of the connection of the gerbe in horizontal and vertical part too. We obtain The Bianchi identity of the gerbe on the total space M reduces to the base M 0 as it follows: where d and d are respectively the exterior derivative on the total space M and on the base manifold M 0 . Analogously, the expression of the curvature of bundle gerbe on local patches becomes

2.6)
where i = ∕ i is the derivative respect to the i-th coordinate of the torus fiber. The patching conditions of the connection 2-form on two-fold overlaps of patches are as following: And the patching conditions of the 1-forms on three-fold overlaps of patches become Fortschr. Phys. 2021, 69, 2000102 www.advancedsciencenews.com www.fp-journal.org

The Tensor Hierarchy of A T-Fold:
To show that this geometric structure we obtained from the reduction of the bundle gerbe is a particular case of global tensor hierarchy, let us make the following redefinitions to match with the notation we used in Section 2: , where the local 1-forms A i ( ) and scalars i ( ) are respectively the local potential and the transition functions of the original torus bundle M ↠ M 0 . Since our fields are assumed to be strong constrained, we well have simply I = (0, i ) with i = ∕ i on horizontal forms. The Bianchi equation (5.2.5) of the gerbe curvature, together with the Bianchi equation dF = 0 of the curvature of the torus bundle can now be equivalently rewritten as which are a particular case of the Bianchi equations of a tensor hierarchy. We can now rewrite all the patching conditions in the following equivalent form: 2.11) which, at first look, appears a particular and strong constrained case of the global tensor hierarchy in (2.7.26). However we will see in the following that it is not completely the case. This will motivate more the identification of a T-fold with an element of ℋ T n sc (M 0 ).
It is well-known that, to be T-dualizable, the string background we started with must satisfy the T-duality condition  i H = 0 on the curvature of the bundle gerbe. From now on we will assume a simple solution for this equation: the invariance of Kalb-Ramond field under the torus action. In other words we will require  i B ( ) = 0, but the other differential data Λ ( ) , g ( ) of the gerbe are still allowed to depend on the torus coordinates. See [1] for the general solution. Notice that this immediately implies that  ( ) = d ( ) in equation (5.2.11).
Topology of The Tensor Hierarchy of A T-Fold: Now it is important to show that the curvature  I ( ) is not in general the curvature of a T 2n -bundle on the (d − n)-dimensional base manifold M 0 . To see this let us split  I ( ) = ( i ( ) , ( )i ) and consider [ * H] ∈ H 2 (M, ℤ n ). We can see that Notice that the inequality (5.2.12) becomes an equality if and only if H (1) is an exact form on the base manifold M 0 , as it is showed in [10] and [1]. In this case we would have H (0) = 0 and H (1) As explained in [1], this particular case corresponds to geometric T-duality, which is the case where the T-dual spacetime is a well-defined manifold and not a non-geometric T-fold. Thus geometric T-duality is exactly the special case where  I ( ) is the curvature of a T 2n -bundle on M 0 . But what is the geometric picture for a T-fold?
In the T-fold case, as seen in [1], we can think about [ * H] ∈ H 2 (M 0 , ℤ n ) as the curvature of a T n -bundle K ↠ M over the total spacetime M. The total space K is called generalized correspondence space.
The geometrical meaning of the curvature  ( ) ∈ Ω 2 cl (U × T 2n ) is being at every patch the curvature of the local torus bundle U × T 2n , even if these ones are not globally glued to be a T nbundle on M 0 . This corresponds indeed to the well-known fact that a non-geometry is a global property. In fact we can always perform geometric T-duality if we restrict ourselves on any local patch: the problem is that all these T-dualized patches will in general not glue together. www.advancedsciencenews.com www.fp-journal.org As derived by [10] and more recently by [75], T-folds are characterized by a monodromy matrix cocycle [n ( ) ], which is a collection of an anti-symmetric integer-valued matrix n ( ) at each two-fold overlap of patches, satisfying the cocycle condition n ( ) + n ( ) + n ( ) = 0 on each three-fold overlap of patches. The monodromy matrix cocycle is nothing but the gluing data for the local B In this sense, a T-fold is patched by a cocycle e n ( ) ∈ O(n, n; ℤ) valued in the T-duality group. If, instead, we want to look at the T-fold as a globally defined T n -bundle K ↠ M with first Chern class [ * H] ∈ H 2 (M, ℤ n ), we can easily construct its connection by noticing that the following 1-form is global on the total space K of the bundle: We can thus define the global 1-form whose first n components are just the pullback of connection Ξ i = i of spacetime M ↠ M 0 and whose last n components Ξ i are the wanted connection of the generalized correspondence space K ↠ M. As desired, the differential dΞ i on K gives the pullback on K of the globally-defined curvature * H ∈ Ω 2 cl (M, ℝ n ). Similarly to  ( ) , the moduli field  ( )IJ of the generalized metric is not a global O(n, n)-valued scalar on the base manifold M 0 , but it is glued on two-fold overlaps of patches U ∩ U ⊂ M 0 by the integer B-shifts encoded by the monodromy matrix n ( ) of the T-fold as Only the 3-form field  of the tensor hierarchy, as seen in [1], is a globally defined (but not closed) differential form on the (d − n)dimensional base manifold M 0 .

T-Duality on The Tensor Hierarchy of A T-Fold:
Given any element  ∈ O(n, n; ℤ) of the T-duality group, we can see that there is a natural action on the local 2n coordinates of the torus by

Poisson-Lie T-folds as Non-Abelian Global Tensor Hierarchies
Non-abelian T-duality is a generalization of abelian T-duality for string backgrounds whose group of isometries is non-abelian.
Poisson-Lie T-duality is a further generalization of this concept where the string background is not even required to have isometries, but which relies on the existence of a more subtle rigid group structure. See [16] for discussion of Poisson-Lie T-duality of a -model in a group manifold and [28,42,46] for discussion of Poisson-Lie T-duality in DFT. For recent applications concerning the Drinfel'd double SL(2, ℂ) = SU(2) ⋈ SB(2, ℂ) see [12,91] and [15]. In this subsection we will introduce the notion of Poisson-Lie T-fold [1] and we will show that it can be formalized by our definition of global tensor hierarchy.
The Generalized Correspondence Space of Poisson-Lie T-Duality: As we have just seen, a bundle gerbe on a T n -bundle spacetime whereC jk i are the structure constants of the Lie algebrã:= Lie(G). Notice that the local 1-form − e i B ( ) is now the local potential of a non-abelian principal bundle. In analogy with abelian T-duality we call the total space K the generalized correspondence space of the Poisson-Lie T-duality. Therefore we have a diagram of the following form: Crucially the composition K ↠ M 0 is a fiber bundle on M 0 with fiber G ×G, but it is not a principal bundle. However, for any good cover {U } of the base manifold M 0 , the total space K will be still locally of the form K| U ≅ U × G ×G.
The Hidden Drinfel'd Double Fiber: The generalized correspondence space K, on any patch U of the base manifold M 0 , can be restricted to a local trivial bundle K| U ≅ U × G ×G. For the fiber G ×G we can introduced the parametrization defined by ( ) = exp( i ( ) e i ) and̃( ) = exp(̃( )iẽ i ), where ( ) and̃( ) are local coordinates on the group G ×G near the identity element. Now, on each trivial local G ×G-bundle K| U ≅ U × G × G we can construct the following local ⊕̃-valued differential 1-form: ( )i ⊗ẽ i . Here we used the vector notation for elements of ⊕̃. Notice that this is a local G ×G-connection 1-form for our local bundle U × G ×G. Now, the global connection data of the generalized correspondence space K is given by the connection of the G-bundle M ↠ M 0 and the one of theG-bundle K ↠ M. We can combine www.advancedsciencenews.com www.fp-journal.org them in a global ⊕̃-valued 1-form on the total space K as it follows ∈ Ω 1 (K, ⊕̃). (5.3.4) The relation between the global 1-form Ξ encoding the global connection data of the generalized correspondence space and the local G ×G-connections ( ) ,̃( ) defined in (5.3.3) is given by ) . (5.3.5) We can rewrite the relation by making the generators {e i ,ẽ i } i=1,…,n of the algebra ⊕̃explicit where the submatricesũ ( ) andb ( ) depend only on the local coordinates ofG andb ( ) is skew-symmetric. Similarly, the adjoint action of the subgroup G on the Lie algebra is given on generators by , (5.3.9) where this time the matrices u ( ) and ( ) depend only on the local coordinates of G and ( ) is skew-symmetric.
Recall that we are parametrizing the points of our local bundle by (x ( ) , Γ ( ) ) ∈ U × D. It was shown by [49] that on each D fiber the Maurer-Cartan 1-form is given by Now, on our local bundle K| U ≅ U × D, we can define the following local -valued 1-form by requiring the identity When the Drinfel'd double D is an abelian group we immediately recover the usual abelian 1-form potentials We can now combine equation (5.3.6) with equation (5.3.13) to find the relation between the global 1-form Ξ I , encoding the global connection data of the generalized correspondence space, and the local D-bundle connection in (5.3.15). The relation is thus given as it follows: where we defined the following matrix (5.3.17) which generally depends on both the local coordinates ( i ( ) ,̃( )i ) of the fibers. Now we must calculate its inverse matrix and find (5.3.18) Finally we can define aČech cocycle which is given on two-fold overlaps of patches by We can calculate this matrix and find Thus we can finally write the patching conditions for our local D-bundle connections by Therefore the cocycle N ( ) represents the obstruction of the generalized correspondence space K from being a global D-bundle on the base manifold M 0 . In physical terms this means that, whenever the cocycle N ( ) is non-trivial, the Poisson-Lie T-dual spacetime is not a geometric background, but a T-fold. This is directly analogous to how the abelian T-fold rises from the generalized cor-respondence space not being a global T 2n -bundle (see previous section). Moreover, if we include the higher form field, we have that the cocycle N ( ) is also the obstruction of the bundle gerbe  from being equivalent to a global String(G ⋈G)-bundle on M 0 . This observation is the key to understand how the tensor hierarchy of the Poisson-Lie T-fold is globalized on the base manifold.
In the next part of the subsection, like we did for the abelian T-fold, we will compare this structure with the one emerging by gauging the algebra of tensor hierarchies we defined in Section 2. We will briefly show that they do not perfectly match, like in the abelian case.
Tensor Hierarchy of A Poisson-Lie T-Fold: Let us define the 2algebra of doubled vectors (D) on the Drinfel'd double D = G ⋈ G by directly generalizing the 2-algebra (ℝ 2n ) we saw in (2.7.6). We can then consider the 2-algebra ) . (5.3.22) The manifestly strong constrained version of the 2-algebra (D) will be given by the following Now let us try to construct the stackification of the prestack Ω(U, sc (D)) of local tensor hierarchies and let us consider a cocycle ( ( ) ,  ( ) , ( ) , Ξ ( ) , g ( ) ) where the local differential forms are given as it follows: where we used the map log : D → and where D is the covariant derivative of the field A ( ) . Again, the globalization of tensor hierarchy that we obtain by gauging the local prestack of tensor hierarchies is not the most general globalization we can think of. This is because it does not take into account the obstruction N ( ) cocycle, appearing in equation (5.3.21), which we get by dimensional reduction of the bundle gerbe.
Conclusion: Poisson-Lie T-Fold as Global Tensor Hierarchy: Thus this discussion motivates again the definition of global strong constrained tensor hierarchy by the following dimensional reduction of a bundle gerbe:

Example: Semi-Abelian T-Fold
In this subsection we will consider (1) a spacetime which is a general As seen in the previous subsection the dimensional reduction of these gerbe contains a global bundle K, the generalized correspondence space, defined by the following diagram: This case is often called semi-abelian, because spacetime is a principal fibration whose algebra has non-zero structure constants [e i , e j ] (2) = k ij e k , but the dual onesC jk i = 0 vanish. From calculations which are analogous to the ones for the abelian T-fold we find that the moduli of the flux are related to the moduli of the Kalb-Ramond field by H Also notice that the moduli of the Kalb-Ramond field is patched on overlaps of patches by B ij . This will be useful very soon. We must now apply all the machinery from the previous subsection to this particular example.
• Now we can consider a local patch U ⊂ M 0 of the base manifold. The total space K restricted on this local patch will be isomorphic to K| U = U × SU(2) × T 3 . These local bundles can be equipped with local connections As derived in [ [1], pag. 61], these local connections are glued on two-fold overlaps of patches (U ∩ U ) × SU(2) × T 3 by a cocycle of B-shifts of the form where we defined the matrix n ( )ij := [i Λ (0) ( )j] . Notice that the T-dualizability condition we imposed on the gerbe implies that n ( )ij is a ∧ 2 ℤ 3 -valuedČech cocycle, similarly to the monodromy matrix cocycle appearing in the abelian T-fold.
As we will explain later, these are the local connection used in most of the non-abelian T-duality literature (before the introduction of Drinfel'd doubles). As noticed by [ [19], pag. 13] they look very similar to an abelian T-fold, but with the monodromy depending on the coordinates via the term k ij Λ (0) ( )k . However we will see that it is better to construct and use proper local D-connections to make the tensor hierarchy really manifest.
• Now we must use the fact that the adjoint action of T 3 on D = SU(2) ⋉ T 3 is specified by settingũ i ( )j = i j andb ( )ij = k ij̃( )k . Thus we can construct the local connection for each local Dbundle U × D byb ( ) -twisting the local connection (5.4.3), obtaining ∈ Ω 1 inv (U × G). Thus this gives a proper local connection for the internal manifold-fibration rising from the vertical part of the dimensional reduction of the bundle gerbe.
To verify that the 1-form (5.4.5) is a proper connection we need to verify that the local potential  ( ) ∈ Ω 1 (U , ) is actually the pullback of a 1-form from the base U . Since for D = SU(2) ⋉ T 3 we have ij = 0, the first component  i = A i is just the local potential of the SU(2)-bundle. To check the second component  ( )i , let us notice that the T-dualizability condition  e i B ( ) = 0 on the gerbe immediately implies  e i (B (1) ( )k ∧ k ) = 0. Now notice that, since the matrix u T ( ) encompasses the adjoint action of the inverse of ( ) = exp( i ( ) e i ), it must be equal to the exponential of the matrix i jk k ( ) . Therefore we can re-write the 1-form B (1) where the ( )i depend only on the base U . The field strengths of these principal connections is then given by their covariant derivative In components of the generators of the algebra , these assume the following form: What we need to find out now is how these local D-bundles are globally glued together. We can immediately see that the global connections of the generalized correspondence space encoded in the global 1-form Ξ I are related to the local SU(2) ⋉ T 3 -connections (5.4.5) by a local patch-wise B-shift U I ( )  of the following form: which, crucially, depends both on the physical and on the extra coordinates. The geometric flux is here just given by the structure constants k ij of (2). Thus from this expression we immediately get that the wanted patching condition on twofold overlaps of patches are

Some Physical Insights from the Semi-Abelian T-Fold
Let us conclude with some remarks about applications to String Theory.
The Puzzle of The Compactness of The T-Dual Fiber: For simplicity, let us consider a SU(2)-equivariant bundle gerbe ↠ M on a spacetime which is an SU(2)-bundle SU(2) → M ↠ M 0 . As seen in Subsection 5.4, this induces a generalized correspondence space K that is a principal D-bundle on M 0 , with D = SU(2) ⋉ T 3 . We can now verify that the extended fiber is compact and, in particular, a 3-torus T 3 .
Let us start from the patching equation B ( ) − B ( ) = dΛ ( ) on M. Similarly to Subsection 5.2, we can expand both the differential forms in the connection ∈ Ω 1 (M, (2)) of the SU (2) Analogously, the patching condition Λ ( ) + Λ ( ) + Λ ( ) = dg ( ) reduces to where g ( ) is theČech cocycle corresponding to the bundle gerbe ↠ M. Notice that, with this assumptions, the 1-form B ( )i is the connection of a principal T 3 -bundle. A similar, but more complicated, statement will hold for a general T-dualizable bundle gerbe on M. In the rest of this subsection we will attempt to clarify other geometric aspects of non-abelian T-duality.
Application to Relevant Holographic Backgrounds: Non-abelian T-duality has been used a fundamental tool in studying the structure of AdS/CFT correspondence and in generating new solutions. See seminal work by [58,62]. Moreover, T-duality in AdS/CFT correspondence is also closely related to the fundamental notion of integrability e.g. see [[ 29,52 ] [86]]. We redirect to [87] for a broad introduction to these topics.
Let us consider the spacetime M = AdS 3 × S 3 × T 4 , which underlies the geometry of a set of NS5-branes wrapped on a 4-torus T 4 and of fundamental strings smeared on the same T 4 such that they are all located at the same point in the transverse space. The S 3 -bundle AdS 3 × S 3 × T 4 ←↠ AdS 3 × T 4 is immediately topologically trivial. Therefore, to investigate its non-abelian T-dual, we need to apply our semi-abelian T-fold construction to a trivial S 3bundle. In particular, we can focus on the 3-sphere and consider a S 3 -bundle over the point S 3 ↠ * , i.e. where the base manifold M 0 = {0} is just a point. In this particular case, the generalised correspondence space will be just the Lie group D = SU(2) ⋉ T 3 . Now, let us momentarily forget of our geometric construction and follow the literature. This expedient will help us underlying some new insights. As discussed in equation (5.4.11), in the literature on non-abelian T-duality, one commonly starts from a metric g = g ij i ⊗ j and the Kalb-Ramond field B = B ij i ∧ j on S 3 , such that g ij and B ij are constant matrices and i are a basis of left SU(2)-invariant 1-forms. Then, one can T-dualize bỹ So that we obtain the T-dual tensorsg =g ij (̃)d̃i ⊗ d̃j andB = B ij (̃)d̃i ∧ d̃j. For simplicity, in the following discussion we can choose B ij = 0. Commonly, one defines a new set of coordinates 1 = r sin , 2 = r cos sin , 3 = r cos cos , (5.5.4) so that the T-dual metric and Kalb-Ramond field take the following simple form: This fact would lead one to think that the new fiber is a noncompact space ℝ 3 . However, as noticed by [19], there is no diffeomorphism relating our tensors at (r, , ) and at (r + Δr, , ) for any choice of radius Δr. Let us now naïvely combine these matricesg ij andB ij in a O(3, 3)-covariant matrix IJ . We immediately recognize that we can recast T-duality (5.5.3) as  ObjectsČech cocycles (A ( ) , f ( ) ) -valued 1-forms Morphisms Global gauge transformations Infinitesimal gauge transformations on the overlaps of patches of the base manifold W NS5 × ℝ + . Notice that, if the S 3 -fibration is trivial, e.g. the example M = AdS 3 × S 3 × T 4 we previously discussed, the cocycle n ( ) is immediately trivial.
In the context of holography, the global structure of the dualitycovariant fields on the general base manifold W NS5 × ℝ + , which is given in Equation 5.5.15, will be relevant for the understanding beyond the current known examples.
Finally, since the global geometry of the moduli space of the string compactifications is supposed to be related to nonperturbative effects in String Theory, the investigation of the global properties of such geometric and non-geometric compactifications is likely to have some relevance in the study of the String Landscape, or in the understanding the Swampland.

Outlook
We clarified some aspects of the Higher Kaluza-Klein approach to DFT. In particular we defined an atlas for the bundle gerbe which locally matches what we expect from the doubled space of DFT. Moreover we illustrated how (strong constrained) tensor hierarchies can be globalized by starting from the bundle gerbe.
Exceptional Field Theory as Geometrized M-Theory: One of the strengths of Higher Kaluza-Klein geometry is that it can, in principle, be generalized to any bundle n-gerbe and more generally to any non-abelian principal ∞-bundle. This can overcome the usual difficulty of DFT geometries in directly being generalized to M-theory. We are intrigued by the prospect that Exceptional Field Theory could be formalized as a Higher Kaluza-Klein Theory on the total space of the (twisted) M2/M5-brane bundle gerbe on the 11d super-spacetime, such as the one described by [38].
The cases of Heterotic DFT and Exceptional Field Theory will be explored in papers to come. Chevalley-Eilenberg dg-Algebras: For simplicity, let us denote, from now on, the underlying graded vector space of a L ∞ -algebra not anymore as V, but just as . Similarly, let us denote as the underlying graded vector bundle of an L ∞ -algebroid , instead that E.
An L ∞ -algebra structure on is equivalently a dg-algebra (i.e. a differential-graded algebra) structure on ∧ • * , which we will call Chevalley-Eilenberg dg-algebra CE( ) of . Given an L ∞ -algebra , let us define the following differential-graded algebra: where the underlying graded vector space is 1.8) and the +1-degree differential is defined by d : t a  ←→ dt a = − ∑ n∈ℕ + 1 n! [t a 1 , t a 2 , … , t a n ] a n t a 1 ∧ t a 2 ∧ ⋯ ∧ t a n , where {t a } is a basis of and {t a } is its dual basis of * . Thus, the differential encodes the n-ary brackets of the L ∞ algebra . Now, we will show that the differential condition d 2 = 0 on the dg-algebra CE( ) is equivalent to the condition Jac n = 0 for the Jacobiator for any n ∈ ℕ + on the L ∞ -algebra . Thus, we can directly calculate [t a 1 , … , t a n ] n , t b 2 , … , t b m ] a m × t a 1 ∧ ⋯ ∧ t a n ∧ t b 1 ∧ ⋯ ∧ t b m .
This can be produced by summing over all the unshuffles ∈ (n, m − 1) weighted by the Koszul-sign of the permutation, i.e. where a is the anchor map of the L ∞ -algebroid, in components. The differential of the generators in the higher degrees is given in analogy with the differential (A.1.9). NQ-Manifolds: An NQ-manifold, also known as differential graded manifold or dg-manifold (see [32] for a review), is a couple  = (||,  ∞ ) of a topological space || and a sheaf of differential graded algebras  ∞ on || such that, for any open set U ⊂ ||, where V is a graded vector space and Q is a differential. Crucially, an NQ-manifold is just an alternative description of a L ∞ -algebroid. The NQ-manifold  = (||,  ∞ ) corresponding to an L ∞ -algebroid is given by For example, the NQ-manifold corresponding to the tangent algebroid TM is the shifted tangent bundle T [1]M, whose coordinates {x , dx } are respectively of degree 0 and 1, and whose differential is where { i } are the 1-degree coordinates of [1] and C i jk are the structure constants of the Lie algebra. Clearly, we have CE( ) =  ∞ ( [1]).
Lie ∞-Groups and Lie ∞-Groupoids: Now, it is known that Lie algebras can be integrated to Lie groups. Similarly, there is a welldefined notion of integration of a L ∞ -algebra and such an object is called Lie ∞-group. This can be pictured as an object which satisfies the defining properties of a Lie group up to a potentially infinite tower of homotopies, i.e. gauge transformations. An useful example of this notion is the 2-group G = BU(1) we defined in Section 3.1. Finally, the Lie-integration of a L ∞ -algebroid is a Lie ∞-groupoid.

A.2. Smooth Stacks
Higher geometry is based on the notion of stack. This can be intuitively though as a generalization of the notion of sheaf which takes value in Lie ∞-groupoids, instead of Lie groups. In other words, given an open cover {U } ∈I for a base manifold M, a stack (M) on such manifold will be given by Lie ∞-groupoids (U ) on each patch, which are glued by patching conditions on any n-fold intersection, i.e. on (U ∩ U ), (U ∩ U ∩ U ), etc… For an introduction to 1-stacks, i.e. stacks taking values in 1-groupoids, see [44]. For details on higher stacks, see [80].
Geometric Stacks: In this paper we will mostly be interested to a particular simple case of smooth stack, called geometric stack. These are, roughly, stacks which can be presented as a Lie ∞groupoid. More precisely, a geometric stack can be represented by a Lie ∞-groupoid G as follows: In geometric terms the objects are all the principal G-bundles over M and the morphisms are all the isomorphisms (i.e. gauge transformations) between them. Thus we will operatively define a principal G-bundle as just an object of groupoid H(M, BG).
The fundamental idea for defining principal ∞-bundles is letting this formalism work for ∞-groups too. Hence we can identify a principal ∞-bundle as an object of H(M, BG) where G is a Lie ∞-group. An example of such a principal ∞-bundle is bundle gerbe from Section 3, where G = BU(1). See [73] and [72] for details.

A.4. Principal ∞-Bundles with Connection
Here we briefly explain how a connection of a principal ∞-bundle is defined in higher geometry.
Given a Lie ∞-group G, the moduli stack of G-bundles with connection BG conn ∈ H is defined, on any smooth manifold M, by the equation where (M) is the path ∞-groupoid of the smooth manifold M, i.e. the ∞-groupoid whose objects are points x ∈ M, 1morphisms : x  → y are paths, 2-morphisms are homotopies of paths, 3-morphisms are homotopies of homotopies, etc… See [80] for more details.
Let us now provide an intuitive example for this definition: an ordinary G-bundle with connection. Given an ordinary Lie group G and a smooth manifold M, a functor tra A : (M) ←→ BG is called parallel transport and it is given by the map Thus, a map tra A is equivalently a cocycle (A ( ) , f ( ) ) ∈ H(M, BG conn ) which encodes the global differential data of a principal G-bundle with connection. The functorial nature of the parallel transport is clear from where • is the composition of paths.
Interestingly, there is a forgetful functor BG conn frgt ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → BG, which forgets the connection of the G-bundles. Thus, it is important to remark that a cocycle M → BG conn does not contain only local connection data, but it remembers the underlying bundle structure M → BG. For example, if G is an ordinary Lie group as in the example above, then cocycles are mapped as Fortschr. Phys. 2021, 69, 2000102 www.advancedsciencenews.com www.fp-journal.org so that the functor forgets the connection data, but retains the global geometric data. If we consider a principal ∞-bundle with a higher structure group G, e.g. a bundle gerbe, we have that the parallel transport is not only on paths, but also on surfaces, on volumes, etc… As a result, we obtain a connection which is made up not only of local 1-forms, but also of 2-forms, 3-forms, etc… See [92] for many details in the case of principal bundles with structure 2-groups.

A.5. Higher Gauge Fields, L ∞ -Algebras and Global Aspects
In this subsection we will discuss the relation between the definition of a higher gauge field as a cocycle M → BG conn and the one, more common, as a map T [1]M → [1] of NQ-manifolds. In particular we will discuss their global aspects.
In the previous subsection we learnt that aČech cocycle (A ( ) , f ( ) ) encoding the global data of a principal G-bundle with connection, where G is an ordinary Lie group, can be expressed as a map Notice that, equivalently, in NQ-manifold notation, this map can be rewritten as which is a definition commonly adopted of gauge field in literature. It is important to notice that the space of such maps is itself an algebroid, whose objects are A ∈ Ω 1 (M, ) and whose morphisms are infinitesimal gauge transformation where the gauge parameter is a global -valued function ∈  ∞ (M, ). For any fixed object A ∈ Ω 1 (M, ) we have a Lie algebra of gauge parameters, i.e. a gauge algebra: this Lie algebra structure is given by the Lie bracket 12 = [ 1 , 2 ] (A. 5.8) for any couple of gauge parameters 1 , 2 ∈  ∞ (M, ). It is intuitively clear that such an algebroid, which we may call Maps(T[1]M, [1]), is nothing but the infinitesimal version of the groupoid H( (M), BG).
Everything we said in this subsection can be immediately generalised to higher gauge fields by replacing the ordinary Lie group G with a Lie ∞-group.
Given a Lie ∞-group G, often in the literature a higher gauge field is defined as a map A : T [1]M → [1], where [1] is the NQ-manifold corresponding to b . Similarly to the case of the ordinary gauge field, by fixing any object in the L ∞ -algebroid Maps(T[1]M, [1]), we obtain the gauge L ∞ -algebra of the higher gauge field. However, as we explicitly worked out for the particular example of an ordinary Lie algebra, this definition does not capture at all the global geometry of the higher gauge field, which are encoded by H( (M), BG).