Geometric T-duality: Buscher rules in general topology

The classical Buscher rules describe T-duality for metrics and B-fields in a topologically trivial setting. On the other hand, topological T-duality addresses aspects of non-trivial topology while neglecting metrics and B-fields. In this article we develop a new unifying framework for both aspects.


Introduction
Mathematical models for string theory are based on geometric backgrounds consisting of • a smooth manifold E (spacetime), • a Riemannian metric g on E (gravity field), and • a bundle gerbe G with connection over E (Kalb-Ramond field).
A special class of Kalb-Ramond fields is given by B-fields, i.e., 2-forms B ∈ Ω 2 (E); these are precisely the connections on the trivial bundle gerbe. Geometric backgrounds (are supposed to) determine 2dimensional quantum field theories, and an important question is when two geometric backgrounds determine the same theory.
In the context of T-duality, one assumes that spacetimes E have a toroidal symmetry: an action of the n-dimensional torus T n on E, such that g is invariant and E is a principal T n -bundle over the quotient X := E/T n . We will use the terminology geometric T-background for geometric backgrounds with toroidal symmetry. When are two geometric T-backgrounds (E, g, G) and (Ê,ĝ,Ĝ) T-dual, i.e., when do they determine the same quantum field theory? To the best of my knowledge, no general conditions are known -unless the data of a geometric T-background are simplified in one way or another. The purpose of the present paper is to propose such general conditions, implying those of all simplified situations.
Buscher provided conditions for T-duality [Bus87] in a topologically trivial situation, where E = X × T is the trivial circle bundle (i.e., n = 1) over an open subset X ⊆ R s , and the bundle gerbe G is just a B-field B ∈ Ω 2 (E). These conditions are the by now classical Buscher rules: Here, the indices label the coordinates of the direct product E = X × T, with α, β = 1, ..., s coordinates of X and θ the single fibre coordinate. The Buscher rules can be generalized to arbitrary torus dimension n, see [GPR94].
A groundbreaking observation of Giveon et al. [Giv,GK94] and Alvarez et al. [AAGBL94] was that (even in the case n = 1) the Buscher rules require a topology change as soon as X ⊆ R s is replaced by a topologically non-trivial manifold. The example studied in [GK94] is when E = S 3 is the Hopf fibration over X = S 2 , g is the round metric on S 3 , and B = 0. One can then cover S 2 by open subsets U i ⊆ S 2 over which E trivializes, and apply the Buscher rules on each patch to obtain locally defined dual metricsĝ i and a dual B-fieldsB i . The observation is then that these locally defined data do not glue to a new metric and B-field on the Hopf fibration, but rather to a new metric and B-field on the trivial bundleÊ := S 2 × T. In other words, spacetime changes its topology under Tduality! A second important development, due to Hori [Hor99], was a "Fourier-Mukai" transformation for Ramond-Ramond charges on D-branes accompanying T-duality, involving topological K-theory of spacetimes and the Poincaré bundle over T n × T n .
The topology change and the relation to K-theory sparked the interest of mathematicians in T-duality, and the question emerged for a formulation of Buscher rules in (more) general topology. Basically at the same time, string theorists and mathematicians explored topological aspects of B-fields. The first account in this direction was Gawȩdzki's work on topological effects in 2-dimensional sigma models using Deligne cohomology [Gaw88], and Alvarez' work on topological quantization [Alv85]. A major step was the invention of bundle gerbes by Murray [Mur96] that unleashed a number of advances, e.g. a complete classification of WZW models on compact simple Lie groups [GR03], a corresponding classification of D-branes in these models [GR02,Gaw05], a discussion of D-branes in terms of twisted K-theory [BCM + 02] or a classification of WZW orientifolds [SSW07,GSW07]. Bundle gerbes with connection have an underlying topological part, measured by their Dixmier-Douady class in H 3 (E, Z), and a curvature, a 3-form H ∈ Ω 3 (E) called H-flux . If the Dixmier-Douady class vanishes, then they reduce -up to isomorphism -to a trivial bundle gerbe I B carrying the former B-field B, such that H = dB. Despite these advances, the quite complicated interplay between metric and B-field, which is characteristic for the Buscher rules, did not have a straightforward generalization from B-fields to bundle gerbes with connection.
Bouwknegt-Evslin-Mathai observed in [BEM04b,BEM04a] that the topology change can also be observed by only looking at the H-flux, while discarding metrics and the remaining data of the bundle gerbe and its connection. An important result of the work of Bouwknegt-Evslin-Mathai was to establish the Fourier-Mukai transformation in twisted de Rham cohomology, an isomorphism Another important observation in this context was the eventual non-existence of T-duals in case of torus dimension n > 1. Mathai-Rosenberg explored these missing T-duals by invoking non-commutative geometry [MR06a,MR05,MR06b].
As the curvature H of a bundle gerbe with connection represents the Dixmier-Douady class only in real cohomology, it neglects torsion. Bunke-Rumpf-Schick invented a framework of topological T-duality that captures the full information of the two Dixmier-Douady classes, but now completely neglects connections and metrics [BS05,BRS06]. Their framework introduced a new and very enlightening aspect to T-duality. So far, T-duality was understood as a transformation, a map, taking one Tbackground to another, T-dual one. However, as mentioned above, some T-backgrounds do not have any T-duals. Even worse, if n > 1, T-backgrounds have many different T-duals. Thus, T-duality is by no means a map. Bunke-Rumpf-Schick implemented this insight by describing T-duality as a relation on the space of topological T-backgrounds (the latter consisting of a principal T n -bundle E and a bundle gerbe G over E without connection). It might be good to remark that this relation is not an equivalence relation; it is only symmetric, but neither reflexive nor transitive. The relation is established by the existence of an isomorphism pr * G ∼ =pr * Ĝ (1.1) between the pullbacks of the two bundle gerbes to the so-called correspondence space, the fibre product Moreover, the isomorphism (1.1) has to satisfy a certain Poincaré condition, relating it to the Poincaré bundle over T n ×T n . Bunke-Rumpf-Schick then started to explore the space of topological T-duality correspondences, consisting of two topological T-backgrounds (E, G) and (Ê,Ĝ), and an isomorphism (1.1), in its dependence on X.
The proof of (1) consists of some computations with differential forms, metrics, and connections performed in Section 3; the statement is Proposition 4.2.1 in the main text.
(2) and (3) follow directly from the definitions, see Propositions 4.3.3 and 4.4.3. The proof of (4) is rather involved due to the very different settings. In order to prove (4), we introduce in Section 6 another formalism that we call "differential T-duality"; we then show in Proposition 6.1.3 that it is a consequence of geometric T-duality, and prove in Proposition 6.3.3 that it is equivalent to Kahle-Valentino's setting.
We remark that our terminology "geometric" does not refer to the question whether or not dual T-backgrounds can be modelled on ordinary torus bundles, as opposed to the non-commutative ones of Mathai-Rosenberg. Instead, it will be used here in order to distinguish our setting from "topological" T-duality and "differential" T-duality.  Theorem 1.1 says that geometric T-duality reduces to several known forms of T-duality. We also consider the opposite question: can these other formulations of T-duality be upgraded to full geometric T-duality? Theorem 1.2.
(a) Locally, geometric T-duality is equivalent to the Buscher rules. More precisely, suppose (g, B) and (ĝ,B) satisfy the Buscher rules. Then, the geometric T-backgrounds (X × T n , g, I B ) and (X × T n ,ĝ, IB) are in geometric T-duality correspondence.
(b) Every topological T-duality correspondence can be lifted to a geometric T-duality correspondence.
More precisely, suppose (E, G) and (Ê,Ĝ) are topological T-backgrounds, and suppose D is a topological T-duality correspondence. Then, there exist T n -equivariant metrics g andĝ on E and E, and connections on G andĜ such that D is a geometric T-duality correspondence between (E, g, G) and (Ê,ĝ,Ĝ).
(c) Every differential T-duality pair can be lifted to a geometric T-duality correspondence. The precise statement is in Proposition 6.1.4.
(d) Every topological T-duality correspondence can be lifted to a differential T-duality pair. The precise statement is in Proposition 6.1.5.
The proof of (a) is rather straightforward, see Proposition 4.2.2. (b) follows from (c) and (d), see Proposition 4.3.5. (c) is a direct consequence of close relationship between geometric and differential T-duality. The proof of (d) is the hardest part, see Proposition 6.1.5. In order to prove it, we introduce in Section 5 a local formalism for geometric T-duality, i.e., we introduce a complete description in terms of functions and differential forms w.r.t. an open cover. Locally, on an open set U i this formalism gives precisely the Buscher rules. Additionally, it contains data and conditions on double, triple, and quadruple overlaps -higher order Buscher rules. To the best of my knowledge, these higher order Buscher rules have not been described before. Figure 1.2 summarizes our local description. For a more detailed explanation of these data and conditions we refer to Section 5.2.
winding numberŝ a ik =m ijk +âij +â jk gauge transformationŝ c ijk such that Local data for geometric T-backgrounds and geometric Tduality correspondences. The first line is the well-known local (topologically trivial) situation. The columns "background" and "dual background" each lists separately the local data from which one can glue a principal T n -bundle, an invariant metric, and a bundle gerbe with connection. The transition functions a ij andâ ij are taken to be R n -valued, revealing winding numbers m ijk andm ijk , respectively. The middle column shows how the (higher) Buscher rules mix these local data from both sides.
We summarize the local data described in Figure 1.2 (up to a certain notion of equivalence, and in the direct limit over refinements of open covers) in a set Loc geo (X). We also look at slightly smaller versions: • Loc diff (X), where the metrics are replaced by their Kaluza-Klein connections.
• Loc top (X), where all metrics and differential forms, and all conditions involving them, are removed.
These slightly smaller versions are very illuminating and important, not only for our proofs, but also because they can be related to another interesting quantity, namely the non-abelian differential cohomology with values in the T-duality 2-group TD,Ĥ 1 (X, TD). More precisely, it is its adjusted version H 1 (X, TD κ ) in the sense of Kim-Saemann [KS20,KS] that becomes relevant here. The 2-group TD has been introduced in [NW20], where we proved that the (non-differential) non-abelian cohomology H 1 (X, TD) classifies topological T-duality correspondences. The following result, in particular, extends this classification to differential and geometric T-duality correspondences. We denote by T-Corr geo (X), T-Corr diff (X), and T-Corr top (X) the sets of equivalence classes of geometric, differential, and topological T-duality correspondences, respectively.
in which all vertical arrows are bijections, and all horizontal arrows are surjections.
The surjectivity of the map (a) follows from Theorem 1.2 (c). The most laborious part in Theorem 1.3 is the construction of the map (b), establishing the relation between the global geometric formalism and the local formalism, and the proof that (b) is a bijection. This is undertaken in Sections 5.3 to 5.5, culminating in Proposition 5.5.1. That the map (c) is a bijection can then easily be deduced from the bijectivity of (b), see Proposition 6.2.1. Construction and a proof of bijectivity of the maps (e) and (f) are rather tedious calculations with local data and TD-cocycles, and are performed in Lemmas 5.6.2 and 6.2.4. The bijectivity of (f) together with above-mentioned classification result of [NW20] imply the bijectivity of (d), see Proposition 5.6.3. The final statement, the surjectivity of the forgetful map (g) from differential to non-differential non-abelian cohomology, is then a rather short -though important -calculation, performed in Proposition 6.2.5. Via the bijections (c) to (f), we obtain then the proof of Theorem 1.2 (d).
Apart from the results described above, we consider an interesting action of the (abelian) differential cohomologyĤ 3 (X) on the set T-Corr geo (X) of all geometric T-duality correspondences. This action has counterparts in the setting of differential and topological T-duality correspondences, and has also been studied by Bunke-Rumpf-Schick [BRS06], see Propositions 6.2.6 and 4.1.12.
Finally, we remark that Theorem 1.2 (d) guarantees the existence of many examples of geometric T-duality correspondences. In Section 7 we describe explicitly two full examples of geometric T-duality correspondences. The first concerns a geometric T-background of the form (E, g, I 0 ), i.e., an arbitrary principal T n -bundle E with an arbitrary metric g and trivial B-field. Reducing this to the case in which E = S 3 → S 2 is the Hopf fibration, and g is the round metric on S 3 , we reproduce the example of Alvarez et al. [AAGBL94] and the observation of a topology change, now in the full setting of geometric T-duality. The second example is again the Hopf fibration and the round metric, but now equipped with the "basic" gerbe of S 3 ∼ = SU(2). It was known in the setting of T-duality with H-flux that this T-background is self-dual. We confirm that self-duality persists in the full setting of geometric T-duality, see Proposition 7.3.1. In particular, it follows from Theorem 1.1 that self-duality holds in pure topological T-duality, and that the Buscher rules are satisfied locally.
Acknowledgements. I would like to thank Ines Kath, Christian Saemann, and Tilmann Wurzbacher for helpful discussions and comments. The beginning of this work was the PhD project of Malte Kunath, whose thesis [Kun21] treats the case n = 1 and -for this case -derives parts 1,2, and 3 of Theorem 1.1.

Preliminaries
In this section we recall structures, terminology, and conventions that will be used throughout this article. To start with, we recall that a connection on a principal H-bundle E over a smooth manifold M is a 1-form ω ∈ Ω 1 (E, g) such that where R denotes the principal action, p the projection to E, h the projection to H, and θ is the left-invariant Maurer-Cartan form on H. If H is abelian, we identify the curvature of ω with the unique 2-form F ∈ Ω 2 (M, h) such that π * F = dω, where π : E → M denotes the bundle projection.
We denote by I := M × H the trivial principal H-bundle over a smooth manifold M . We may identify connections ω on I with h-valued 1-forms A ∈ Ω 1 (M, h) in the usual way, i.e., where p : M × H → M and h : M × H → H are the projections. We write I A for the trivial bundle equipped with the connection (2.1). If H is abelian, and A 1 , · a is a connection-preserving bundle isomorphism. On an overlap U i ∩ U j , we consider the transition function g ij :

Bundle gerbes with connection
We use the definitions and conventions of [Wal07]. The reader familiar with bundle gerbes can safely skip this subsection. We write T := U(1) = R/Z. Definition 2.1.1. A bundle gerbe G with connection over a smooth manifold M consists of the following structure: 1. A surjective submersion π : Y → M , and a 2-form B ∈ Ω 2 (Y ) called "curving".
If B ∈ Ω 2 (M ) is a 2-form, then there is a "trivial" bundle gerbe with connection I B , with surjective submersion π = id M , the trivial T-bundle with connection P = I 0 , and the trivial bundle isomorphism If we further assume that the non-empty double intersections U i ∩ U j are contractible, we may choose sections s ij : This implies an equality A ik = A ij + A jk + c * ijk θ. Finally, the associativity condition for µ implies ǎ Cech cocycle condition c jkl · c ijl = c ijk · c ikl . The "local data" (B i , A ij , c ijk ) yield a degree-2-cocycle in Deligne cohomology, and thus represent a class in degree three differential cohomologyĤ 3 (M ) of M .
It will be important to consider the full bicategorical structure of bundle gerbes with connection.
Definition 2.1.2. Suppose G and G ′ are bundle gerbes with connection. A connection-preserving isomorphism A : G → G ′ consists of the following structure: and ξ [2] and ξ ′[2] denote the induced maps on double fibre products.
It is required that the following compatibility condition holds for all (z 1 , z 2 , z 3 ) ∈ Z [3] , for which we set ζ(z i ) =: (y i , y ′ i ): We remark that the curvature of G and G ′ coincide if there exists a connection-preserving isomorphism. The set of isomorphism classes of bundle gerbes with connections over M is denoted by Grb ∇ (M ). This set is actually a group, whose multiplication is given by the tensor product of bundle gerbes, see [Wal07].
Suppose we have chosen sections s i and s ij for G as above, and similar sections s ′ i and s ′ ij for G ′ , with corresponding local data ( . After a further refinement, we may assume that (s . We may then assume that t * i Q admits a local section u i , with corresponding 1-forms C i . Note that

This implies an equality
Finally, the compatibility condition yields an equality The data (C i , d ij ) constitute an equivalence between the Deligne 2-cocycles (B i , A ij , c ijk ) and . This establishes an isomorphism Grb ∇ (M ) ∼ =Ĥ 3 (M ) between the set of isomorphism classes of bundle gerbes with connection and degree three differential cohomology [MS00,Ste00].
Concerning local data, we may assume that the sections t 1,i : U i → Z 1 and t 2,i : U i → Z 2 lift to W , i.e., that there are sections v i : U i → W such that ω • v i = (t 1,i , t 2,i ). Then, v * i η : t * 1,i Q 1 → t * 2,i Q 2 is a connection-preserving bundle isomorphism, and there exists a unique smooth map z i : . This yields an equality C 2,i = C 1,i + z * i θ. The diagram leads to d 1,ij · z i = z j · d 2,ij .
The (vertical) composition of connection-preserving 2-isomorphisms is obtained by going to a common refinement and composing the bundle isomorphisms there. This way, we obtain a category Hom(G, G ′ ). There is a (horizontal) composition functor which turns bundle gerbes with connection into a bicategory. The following statement about the morphism category between trivial bundle gerbes will be very important later.
where the right hand side denotes the category of principal T-bundles with connection of fixed curvature F = B 2 − B 1 . Under this equivalence, the composition of connection-preserving isomorphisms corresponds to the tensor product of bundles with connection.

Poincaré bundles and equivariance
We summarize some required facts about the Poincaré bundle, also see [NW20, Appendix B]. We write T n := R n /Z n additively, and identify its Lie algebra with R n , and again T = T 1 = R/Z. The n-fold Poincaré bundle is the following principal T-bundle P over T 2n = T n × T n . Its total space is with (a,â, t) ∼ (a + m,â +m, mâ + t) for all m,m ∈ Z n and t ∈ T, and mâ is the standard inner product. The bundle projection is (a,â, t) → (a,â), and the T-action is (a,â, t) · s := (a,â, t + s).
(a) The following maps are well-defined bundle isomorphisms over T 3n : They express that the Poincaré bundle is "bilinear" in the two factors T n × T n . Using the given formulas, one can check that ϕ l satisfies the following associativity condition: P 1,4 ⊗ P 2,4 ⊗ P 3,4 id⊗ϕ l / / ϕ l ⊗id P 1,4 ⊗ P 2+3,4 ϕ l P 1+2,4 ⊗ P 3,4 ϕ l / / P 1+2+3,4 . (2.2.1) is a well-defined section along T n × R n → T n × T n . These restrict further to sections along the inclusion T n → T n × T n into one of the factors. The transition functions w.r.t. these sections are the following. Suppose a, a ′ ∈ R n such that m := a ′ − a ∈ Z n andâ ∈ T n . Then, (2.2.4) If a ∈ T n andâ,â ′ ∈ R n withm :=â ′ −â ∈ Z n , then we have χ r (a,â) = χ r (a,â ′ ) · am.
Over R n × R n the sections χ l and χ r do not coincide, but differ by the T-valued function m : R n × R n → T, m(a,â) := aâ, i.e., χ r = χ l · m. They do coincide when pulled back to Z n × Z n .
(c) We recall that the dual P ∨ of a principal T-bundle P has the same total space but T acting through inverses. The map λ : P → P ∨ 2,1 : (a,â, t) → (â, a, aâ − t) is a well-defined bundle isomorphism over the identity of T n × T n . It expresses that the Poincaré bundle is "skew-symmetric". We have λ 2 = id. Moreover, the isomorphism λ exchanges χ l with χ r .
where θ ∈ Ω 1 (T n , R n ) is the Maurer-Cartan form, and∧ denotes the wedge product of R n -valued forms using the standard inner product of R n on the values.
(iii) An identity expressing the 2-form Ω in terms of the Maurer-Cartan form on T is Since H * (T 2n , Z) is torsion free, this shows that the first Chern class of P is where pr i : T 2n → T is regarded as a representative for [T 2n , T] = H 1 (T 2n , Z).
The following discussion concerns the quite difficult equivariance properties of the Poincaré bundle and its connection. They will be used only in Section 5, so that the reader may also continue with Sections 3 and 4 first.
Remark 2.2.2. Restricting to Z n ⊆ R n , we have In particular, we obtain from (2.2.9) where e ∈ E sits in the fibre over x ∈ X, and the matrix on the right hand side refers to the decomposition T e E ∼ = T x M ⊕ h induced by the connection ω.
The connection ω that appears on the right hand side will be called the Kaluza-Klein connection associated to the metric g.
Remark 2.3.2. Theorem 2.3.1 is compatible with bundle isomorphisms: for a bundle isomorphism ϕ : E 1 → E 2 it is equivalent to be isometric for the metrics on E 1 and E 2 or to be connectionpreserving w.r.t. the Kaluza-Klein connections ω 1 and ω 2 .
In the literature, this is sometimes written as where, unfortunately, h x is suppressed or assumed to be constant.

Buscher rules revisited
The Buscher rules formulate the local behaviour of metrics and B-fields under T-duality. A priori, they only apply to trivial torus bundles over Euclidean space, for instance, over a coordinate patch. We first review the Buscher rules for higher tori, give then a reformulation when the metrics are replaced by their Kaluza-Klein connections, and finally produce a completely coordinate-free reformulation. It is this latter formulation that we generalize in Section 4.

Buscher rules for toroidal symmetries
We first recall the classical Buscher rules for a T n -symmetry. The case n = 1 has been treated by Buscher in [Bus87]. Rules for higher n are described, e.g., in [GPR94], also see [Bou10] for a review.
The Buscher rules apply to a manifold E = R s ×T n . On E we consider a Riemannian T n -invariant metric g and a T n -invariant 2-form B, the "B-field". With respect to the standard coordinates, we may identify g with a block matrix g = g bas g mix g tr mix g f ib with symmetric matrices g bas ∈ C ∞ (R s ) s×s and g f ib ∈ C ∞ (R s ) n×n , and an arbitrary matrix g mix ∈ C ∞ (R s ) s×n . We also identity B with a block matrix We make the assumption that B f ib = 0, so that the B-field must not have components purely in fibre direction. Note that for n = 1 this is automatic. Pairs (g, B) with the condition B f ib = 0 will be called Buscher pairs.
To proceed, we form the "background" matrix The dualization process requires to form the "dual" matrix Dual metric and B-field are now obtained by taking the symmetric and anti-symmetric parts ofQ, respectively, i.e.ĝ := 1 2 (Q +Q tr ) andB := 1 2 (Q −Q tr ).
A standard calculation shows that (ĝ,B) is again a Buscher pair, and that the relation between the Buscher pairs (g, B) and (ĝ,B) is described by the following equations: It is straightforward to see that these rules reduce in the case of n = 1 to the usual Buscher rules. It is also straightforward to see thatQ = Q, implying that the Buscher rules are symmetric in the data. For completeness, let us fix the following definition.

Buscher rules in terms of Kaluza-Klein connections
We consider again a metric g on E = R s × T n , and consider E as a principal T n -bundle over R s . We apply Theorem 2.3.1 and Remark 2.3.3, to obtain a triple (A, g ′ , h) consisting of a Riemannian metric g ′ on R s , a 1-form A ∈ Ω 1 (R s , R n ), and a family h of inner products on R n parameterized by R s . Now we consider Buscher quadruples (A, g ′ , h, B) instead of Buscher pairs (g, B). By Theorem 2.3.1 there is a bijection between Buscher quadruples and Buscher pairs.
The expression for the metric g given in Remark 2.3.4 now reads In other words, we have We employ the same procedure on the dual side, gettinĝ The Buscher rules now attain the following simple form: Again for completeness, we fix the following definition and result.

Buscher rules in terms of Poincaré forms
Next we want to give a coordinate-independent description of the Buscher rules of Definition 3.2.1, which will again make them simpler. Let ω,ω ∈ Ω 1 (R s × T n , R n ) be the Kaluza-Klein connections on E = R s × T n corresponding to A andÂ, respectively, i.e., ω := A 1 + θ 2 andω :=Â 1 + θ 2 . Here, the indices refer to the pullback along the projections to the two factors, as explained in Section 2.2. We introduce the 2-form where the symbol∧ means that the standard scalar product on R n is used in the values the forms.
Proof. (a) and (b) are (3.2.1) and (3.2.3). We have We change to coordinates w.r.t. R s × T n × T n , which we label by i, µ, andμ. Then, we obtain and similarly and all other components vanish. Further, we have with again all other components vanishing. Thus, (c) is equivalent to the following set of equations: The second and third equation are (3.2.2) and (3.2.5). The first equation, using second and third, is equivalent toB and this is precisely (3.2.4).
A straightforward computation using Lemma 3.3.1 (c) shows the following.

Geometric T-duality
In this section we give the central definitions of this article: we introduce geometric T-backgrounds (Definition 4.1.1) and geometric T-duality correspondences between them (Definition 4.1.9). We deduce a number of first consequences; in particular, we relate geometric T-duality to T-duality with H-flux and to topological T-duality.

Basic definitions
Definition 4.1.1. A geometric T-background over a smooth manifold X is a triple (E, g, G) consisting of a principal T n -bundle E over X, a T n -invariant Riemannian metric g on E, and a bundle gerbe G over E with connection. Two geometric T-backgrounds (E 1 , g 1 , G 1 ) and (E 2 , g 2 , G 2 ) over X are equivalent, if there exists a bundle isomorphism f : E 1 → E 2 that is isometric with respect to the metrics g 1 and g 2 , and a connection-preserving bundle gerbe isomorphism G 1 ∼ = f * G 2 . The set of equivalence classes of geometric T-backgrounds over X is denoted by T-BG geo (X).
As every bundle gerbe with connection has a curvature 3-form, every geometric T-background carries a 3-form H ∈ Ω 3 (E), the H-flux . Note that H is closed, but in general not exact. The H-fluxes of equivalent geometric T-backgrounds satisfy H 1 = f * H 2 .
If (E, g, G) and (Ê,ĝ,Ĝ) are geometric T-backgrounds over the same manifold X, then the principal T 2n -bundle E × XÊ is called the correspondence space. It fits into an important commutative diagram: Let ω ∈ Ω 1 (E, R n ) andω ∈ Ω 1 (Ê, R n ) be the Kaluza-Klein connections of the metrics g andĝ, respectively, under Theorem 2.3.1. Then, we consider the 2-form where∧ denotes the wedge product of R n -valued forms w.r.t. the standard inner product. Since ω andω are T n -invariant (they are connections on a principal bundle with abelian structure group), the 2-form ρ g,ĝ is T 2n -invariant. We remark that the 2-form ρ g,ĝ also appeared in [Hor99,BEM04b].
Definition 4.1.2. A geometric correspondence over X consists of two geometric T-backgrounds (E, g, G) and (Ê,ĝ,Ĝ) over X, and a connection-preserving bundle gerbe isomorphism Remark 4.1.3. We shall explore some consequences of the isomorphism D in a geometric correspondence. For this, we will denote by F,F ∈ Ω 2 (X) the curvatures of the connections ω andω, respectively.
(a) Since the curvatures of isomorphic bundle gerbes with connection coincide, we have which is a condition in the context of T-duality with H-flux, see [BEM04b, Eq. 1.12] and Definition 4.4.2. From (4.1.2) and the definition of ρ g,ĝ one can deduce the equivariance rule where R is the principal action, e the projection to E, and h the projection to T n . Similarly, on the dual side we obtain In particular, these formulas show that H andĤ are T n -invariant.
(b) We consider the 3-forms Using (4.1.3) and (4.1.4) one can show that R * K = e * K and R * K =ê * K , so that these forms descend to X. In fact,K andK both descend to the same 3-form K ∈ Ω 3 (X), i.e., p * K =K and p * K =K. To see this, it suffices to note that the pullbacks ofK andK to the correspondence space coincide, which again can be checked using (4.1.1) and (4.1.2). Summarizing, every geometric correspondence determines a 3-form K ∈ Ω 3 (X) such that Note that dK = F∧F .
Remark 4.1.4. Geometric correspondence is a symmetric relation on the set T-BG geo (X). If D is a correspondence from (E, g, G) to (Ê,ĝ,Ĝ), then we construct a correspondence from (Ê,ĝ,Ĝ) to (E, g, G) as follows. Let s :Ê × X E → E × XÊ denote the swap map. Then, we consider Since −s * ρ g,ĝ = ρĝ ,g , this is again a geometric correspondence.
Definition 4.1.5. Two geometric correspondences over X,

connection-preserving bundle gerbe isomorphisms
A : G → f * G ′ andÂ :Ĝ →f * Ĝ′ , and a connection-preserving 2-isomorphism The set of equivalence classes of geometric correspondences over X is denoted by Corr geo (X).
Remark 4.1.6. In above definition we have implicitly used that F * ρ g ′ ,ĝ ′ = ρ g1,ĝ1 , which follows from the fact that f andf are connection-preserving, which in turn follows from the assumption that f and f are isometric (Remark 2.3.2).
Remark 4.1.7. Let H be a bundle gerbe with connection over X. Then, we may send This gives a well-defined action of the group of isomorphism classes of bundle gerbes with connection on the set of equivalence classes of geometric correspondences, Remark 4.1.8. It is straightforward to see that equivalent geometric correspondences determine the same 3-form K. The action of Remark 4.1.7 shifts this 3-form by curv(H).
Definition 4.1.9. A geometric correspondence D between two geometric T-backgrounds (E, g, G) and (Ê,ĝ,Ĝ) over X is called geometric T-duality correspondence if the following conditions hold: (T1) The Riemannian metrics g ′ andĝ ′ on X determined by the metrics g andĝ, respectively, under Theorem 2.3.1 coincide, i.e., g ′ =ĝ ′ .
(T2) The families of inner products h andĥ on R n determined by the metrics g andĝ, respectively, under Theorem 2.3.1, satisfy h −1 =ĥ under their identification with (n × n)-matrices.
(T3) Every point x ∈ X has an open neighborhood U ⊆ M such that the following structures exist: (c) Consider U × T 2n with projection maps pr,pr to U × T n . Further, consider the map Φ : Let P denote the principal T-bundle with connection over U × T 2n that corresponds to the isomorphism y y r r r r r r r r r r r r r r under the equivalence of Proposition 2.1.4. We require a connection-preserving isomorphism where P is the n-fold Poincaré bundle with its canonical connection.
The set of equivalence classes of geometric T-duality correspondences over X (with the equivalence relation just as in Definition 4.1.5) is denoted by T-Corr geo (X). Remark 4.1.11. It is straightforward to see that the action of Remark 4.1.7 restricts to an action of H 3 (X) on T-Corr geo (X). The properties of this action are best studied in the context of differential Tduality and carried out in differential cohomology, see Proposition 6.2.6. The result of Proposition 6.2.6 is the following.
Proposition 4.1.12. Let be the projection to the isomorphism classes of the principal T n -bundles E andÊ and their Kaluza-Klein connections ω andω induced by the metric g andĝ, respectively. We denote by (F,F ) ∈ Ω 2 (X) × Ω 2 (X) the well-defined pair of curvature forms. Consider the subgroup Then, the quotient Grb ∇ (X)/F F,F acts free and transitively in the fibre of (4.1.5) over an element with curvature pair (F,F ).
Remark 4.1.13. The assignments X → T-BG geo (X) and X → T-Corr geo (X) are presheaves on the category of smooth manifolds. In fact, it is straightforward and only omitted for brevity to enhance the sets T-BG geo (X) and T-Corr geo (X) to bicategories, which then form sheaves of bicategories on the site of smooth manifolds.

Relation to Buscher rules
We will now make a deeper analysis of condition (T3) (c), and in particular show that the Buscher rules are satisfied over U .
Conversely, geometric T-duality locally does not pose any more conditions than the Buscher rules. To see this, we observe that any Buscher pair (g, B) extends to a geometric T-duality background, with E s,n := R s × T n , the given metric g, and the trivial bundle gerbe I B . If Buscher pairs (g, B) and (ĝ,B) satisfy the Buscher rules, then we havepr * B − pr * B + ρ g,ĝ = pr * T 2n Ω by Lemma 3.3.1. Thus, pr * T 2n P corresponds under Proposition 2.1.4 to a connection-preserving isomorphism D : pr * I B →pr * IB ⊗ I ρ g,ĝ (4.2.1) over the correspondence space E s,n × R s E s,n .
Proposition 4.2.2. Suppose (g, B) and (ĝ,B) are Buscher pairs and satisfy the Buscher rules. Then, the connection-preserving isomorphism (4.2.1) establishes a geometric T-duality correspondence between (E s,n , g, I B ) and (E s,n ,ĝ, IB).
Proof. Conditions (T1) and (T2) of Definition 4.1.9 are Lemma 3.3.1 (a) and (b). That condition (T3) is satisfied can be seen using the identity trivializations ϕ,φ and T ,T .

Relation to topological T-duality
We shall first recall the definition of topological T-duality following [BS05,BRS06,MR06a].
Every geometric T-background (E, g, G) induces a topological T-background (E, G) by forgetting the metric and forgetting the gerbe connection. Conversely, if (E, G) is a topological T-background, one can choose any T n -invariant metric on E and use the fact that every bundle gerbe admits a connection [Mur96], to upgrade it to a geometric T-background. Thus, we have a surjective map has an open neighborhood U ⊆ X such that the following structures exist: corresponds under the equivalence of Proposition 2.1.4 to pr * T 2n P.
of twisted K-theory groups is an isomorphism.
There is also an interesting converse question. Suppose two topological T-backgrounds are in topological T-duality correspondence. Can one lift them to geometric T-backgrounds that are in geometric T-duality correspondence? Proposition 4.3.5. Every topological T-duality correspondence can be lifted to a geometric T-duality correspondence. In more detail, suppose (E, G) and (Ê,Ĝ) are topological T-backgrounds, and suppose D is a topological T-duality correspondence. Then, there exist T n -equivariant metrics g andĝ on E and E, connections on G andĜ and a connection on D such that D is a geometric T-duality correspondence between (E, g, G) and (Ê,ĝ,Ĝ).
Proof. Combines Propositions 6.1.4 and 6.1.5, to be proved later using the local formalism.

Relation to T-duality with H-flux
In this section we show that geometric T-duality implies T-duality with H-flux in the sense developed by Bouwknegt-Evslin-Mathai in [BEM04b] and Bouwknegt-Hannabuss-Mathai in [BHM04]. In these papers, T-duality is not considered as a relation between T-backgrounds, but rather as a transformation that takes a T-background to another. A description of T-duality with H-flux as a relation on a class of suitable backgrounds has been given by Gualtieri-Cavalcanti in [CG10] based on [BEM04b,BHM04], and we will use this here.
Every geometric T-background (E, g, G) induces one with H-flux where the metric g is forgotten and H is the curvature of G. Conversely, every T-background with H-flux (E, H) can be upgraded to a geometric T-background by choosing some T n -invariant metric and some bundle gerbe with connection of curvature H.
Remark 4.4.4. For a general base manifold X, one cannot expect that every given T-duality correspondence with H-flux can be upgraded to a geometric (or only topological) T-duality correspondence. Indeed, a topological T-duality correspondence implies the triviality of the class c 1 (E)∪c 1 (Ê) ∈ H 4 (X, Z), while a T-duality correspondence with H-flux only implies the triviality of that class in de Rham cohomology.

Local perspective to geometric T-duality
We may see condition (T3) of Definition 4.1.9 as enforcing a geometric T-duality correspondence to be locally trivial . Just as for locally trivial fibre bundles, one may then extract "local data", or "gluing data". It is instructive to first do this in an ad hoc manner, which is the content of Section 5.1. In Section 5.2 we organize local data in a more systematic way, establishing the table in Figure 1.2 of Section 1. Sections 5.3 to 5.5 are devoted to a full proof of a bijection between the set T-Corr geo (X) of equivalence classes of geometric T-duality correspondences and a set Loc geo (X) of equivalence classes of local data. In Section 5.6 we reduce the discussion of local data to topological T-duality, and show that this reduction becomes the non-abelian cohomology with values in the T-duality 2-group.

Extraction of local data
We suppose that we have a geometric T-duality correspondence D as in Definition 4.1.9, between geometric T-backgrounds (E, g, G) and (Ê,ĝ,Ĝ) over X. We assume then that X is covered by open sets U i over which condition (T3) holds, and that corresponding bundle trivializations ϕ i ,φ i , bundle gerbe trivializations T i ,T i and 2-isomorphisms ξ i are chosen for all U i , where ξ i are the connectionpreserving 2-isomorphisms Let a ij : U i ∩U j → T n be the transition functions of E, which are determined by the trivializations ϕ i and ϕ j , i.e., ϕ j (x, a) · a ij (x) = ϕ i (x, a). It will soon become necessary to choose and fix lifts of these transition functions along R n → T n , which is always possible after eventually passing to a refinement of the open cover. The former cocycle condition then reveals "winding numbers" m ijk ∈ Z n such that a ij + a jk + m ijk = a ik (5.1.2) and these integers m ijk themselves satisfy the usualČech cocycle condition. We will also denote by a ij the corresponding map that multiplies by a ij (x); note that this map satisfies ϕ i = ϕ j • a ij . Next, we consider the composite I Bi ij Bj of bundle gerbe isomorphisms over (U i ∩U j )×T n , which corresponds by Proposition 2.1.4 to a principal T-bundle L ij over (U i ∩ U j ) × T n with connection of curvature a * ij B j − B i . The same works on the dual side, resulting in transition functionsâ ij : U i ∩ U j → R n , winding numbersm ijk ∈ Z n satisfyingâ ij +â jk +m ijk =â ik , (5.1.3) and principal T-bundlesL ij over (U i ∩ U j ) × T n with connection of curvatureâ * ijB j −B i . Before we proceed, we remark that the local trivializations ϕ i ,φ i also define local T n -invariant metrics g i := ϕ * i g andĝ i :=φ * iĝ on U i × T n . Due to the T n -invariance of g andĝ, we have As seen in Proposition 4.2.1, the pairs (g i , B i ) and (ĝ i ,B i ) satisfy the Buscher rules.
Lemma 5.1.1. The principal T-bundles L ij andL ij are trivializable. Thus, there exist A ij ,Â ij ∈ Ω 1 ((U i ∩ U j ) × T n ) and connection-preserving isomorphisms Proof. We assume that all non-empty double intersections U i ∩ U j are contractible; this can again be achieved by passing to a refinement. Then, the first Chern classes of L ij andL ij must be pullbacks from T n . We have H 2 (T n , Z) ∼ = so(n, Z), the group of skew-symmetric integral (n × n)-matrices, and this isomorphism can be realized explicitly using the Poincaré bundle P over T 2 : we send a matrix D ∈ so(n, Z) to the principal T-bundle see [NW20,§B]. Thus, there exist unique matrices D ij ,D ij ∈ so(n, Z) and (non-unique) bundle isomorphisms L ij ∼ = pr * T n P Dij andL ij ∼ = pr * T n PD ij . Taking connections into account, there exist 1-forms A ij ,Â ij ∈ Ω 1 ((U i ∩ U j ) × T n ) and connection-preserving isomorphisms We show next that D ij =D ij = 0, implying the claim of the lemma. This will be a consequence of the geometric T-duality correspondence, and so we need to work over (U i ∩ U j ) × T 2n . We consider the following maps: and construct with them the following diagram of bundle gerbes with connections and connection-preserving isomorphisms over (U i ∩ U j ) × T 2n : The unlabelled double arrows are the canonical unit and counit 2-isomorphisms of the adjunction between a 1-isomorphism and its inverse. The rectangular subdiagram in the middle commutes on the nose. The outer shape of the diagram is, via Proposition 2.1.4, a connection-preserving bundle isomorphism We shall forget the connections (and thus all trivial bundles) for a moment. Due to the equivariance of the Poincaré bundle discussed in Section 2.2, the lifts a ij andâ ij determine an isomorphism a * ij pr T 2n P ∼ = pr * T 2n P. Using this in (5.1.7), we are in the situation that all bundles are pulled back along the projection (U i ∩ U j ) × T 2n → T 2n . Hence, these bundles must already have been isomorphic before pullback; and we conclude that there exists a bundle isomorphism P ⊗ pr * P Dij ∼ =pr * PD ij ⊗ P.
over T 2n . Hence, there also exists a bundle isomorphism pr * P Dij ∼ =pr * PD ij , and this shows that both bundles separately are trivializable. This implies D ij =D ij = 0.
Remark 5.1.2. The principal T-bundle L ij andL ij can be regarded as part of the gluing data for the bundle gerbes G andĜ, respectively. Their triviality in case of geometric (or only topological) T-duality shows that T-backgrounds that can be part of a T-duality correspondence are of a special kind. More precisely, it means exactly that the Dixmier-Douady classes of G andĜ are in the second step of the filtration of H 3 (E, Z) that comes from the Serre spectral sequence, see [BRS06] and [NW20, §2.1].
Next we will spend some time on finding trivializations λ ij andλ ij with particular covariant derivatives A ij andÂ ij . We start with arbitrary choices as they exist by Lemma 5.1.1 and will then perform three revisions of the isomorphisms λ andλ, and accordingly shift the 1-forms A ij andÂ ij , finally arriving at (5.1.18).
We will only discuss A ij , the treatment ofÂ ij is analogous. We remark that due to Lemma 3.3.2, (3.3.1), the 2-form a * ij B j − B i is T n -invariant; moreover, we have Here, we use the notation introduced in Section 2.2: an index (..) α means a pullback from the α-th T n -factor, and the index (..) 1+2 means a pullback along the addition of two T n -factors. (5.1.5) and (5.1.8) imply This shows that we have a closed 1-form Since the de Rham cohomology class of α ij can only have contributions from the torus, and these contributions must be linear combinations of the generators [θ] ∈ H 1 (S 1 , R), there exists a smooth map β ij : (U i ∩ U j ) × T 2n → R and vectors p ij , q ij ∈ R n such that Moreover, since the definition of α ij is skew-symmetric with respect to the exchange of the two T nfactors; we have q ij = −p ij . We may now shift the isomorphism λ ij by the smooth map Its derivative is −p ij θ; thus, A ij becomes replaced by A ij + p ij θ, and (5.1.10) is replaced by just α ij = dβ ij . (5.1.11) In particular, we have shown that λ ij can be chosen such that α ij is trivial in de Rham cohomology. The left hand side is still skew-symmetric, and so we have d(β ij + s * β ij ) = 0, where s is the map that swaps the T n factors. This means that c ij := β ij (x, a, b) + β ij (x, b, a) is a constant function. Shifting β ij by − 1 2 c ij , we can achieve that c ij = 0, i.e., achieve that β ij is skew-symmetric in a and b. Over (U i ∩ U j ) × T 3n one can deduce from (5.1.9) the cocycle condition (dβ ij ) 1,3 = (dβ ij ) 1,2 + (dβ ij ) 2,3 . This shows that there exists a constant c ij ∈ R such that Thus, we may defineβ ij : (U i ∩ U j ) × T n → R byβ ij (x, a) := β ij (x, a, 0) and obtain, using the skew-symmetry of β ij , We are now in position to make a second revision of the choice of the isomorphism λ ij , and shift it by the smooth map (U i ∩ U j ) × T n → T : (x, a) → −β ij (x, a). This shifts A ij by dβ ij . Then, (5.1.11) is replaced by α ij = 0, and (5.1.9) results in On the dual side, we obtain analogously Next we have to bring A ij andÂ ij together, and consider for this purpose the connectionpreserving isomorphism η ij of (5.1.7). By Lemma 5.1.1, it simplifies to a connection-preserving isomorphism η ij :ã * ij pr * T 2n P ⊗ I pr * Aij → Ip r * Â ij ⊗ pr * T 2n P. (5.1.14) As a result of the fixed lifts a ij andâ ij , we obtain canonically a connection-preserving isomorphism Under the isomorphism R ij of (5.1.15) we obtain from (5.1.14) a connection-preserving bundle isomorphism I ψij ⊗ I pr * Aij ∼ = Ip r * Â ij , which in turn corresponds via the bijection (2.2) to a smooth map h ij : Proof. Considering (5.1.17) over (U i ∩ U j ) × T 4n twice, taking their difference, and using (5.1.12), (5.1.13) and (5.1.16) yields This implies that a,â). This shows the claim.
We now make one last revision of the choice of the isomorphismλ ij , and shift it by h ij . This changesÂ ij by h * ij θ, and hence turns (5.1.17) intô Note that (5.1.12) and (5.1.13) continue to hold, as a change by a 1-form that does not depend on T n cancels itself on both sides.
The definition of the principal T-bundle L ij induces a canonical connection-preserving bundle isomorphism Under the trivialization λ ij , it corresponds to a smooth map Further, by going to a quadruple intersection, it is straightforward to see that we obtain a cocycle condition a * ij c jkl · c ijl = c ijk · c ikl . (5.1.20) The same holds on the dual side, leading to a smooth mapĉ ijk satisfyinĝ where f ijk is defined by the expression Proof. We put the diagrams (5.1.6) for ij and jk, respectively, next to each other. In the middle, two occurrences ofã * jk ξ j cancel, and we obtain the following equality of connection-preserving 2isomorphisms: Our choice of isomorphisms L ij ∼ = I Aij andL ij ∼ = IÂ ij is such that we have an equality q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qã * ij pr * T 2n P r r pr * T 2n P⊗I ψ ij I Substituting this in (5.1.23) we collect on the left hand side an isomorphism R jk •ã * jk R ij and on the right hand side an isomorphism R ik . We compute the relation between these two isomorphisms: with f ijk as defined above.
We will see in the following sections that the differential forms and functions collected so far, and the conditions derived for them, are sufficient.

Geometric T-duality cocycles
In this section we organize the local data extracted in the previous section. For this purpose, we fix the following definition. A geometric T-duality cocycle with respect to an open cover {U i } of X consists of the following data: 5. m ijk ,m ijk ∈ Z n , and 6. smooth maps c ijk ,ĉ ijk : This local data is subject to the following conditions (LD1) to (LD9).
(LD1) The pair (a ij , m ijk ) is local data for a principal T n -bundle E over X, i.e., a ik = m ijk + a ij + a jk m jkl + m ijl = m ikl + m ijk .
We remark that the second line follows from the first; it is only listed for convenience.
(LD3) The metrics g i yield a metric on E, i.e., a * ij g j = g i .
(LD5) The triple (B i , A ij , c ijk ) is local data for a bundle gerbe with connection over E, i.e., is local data for a bundle gerbe with connection overÊ, i.e., The pairs (g i , B i ) and (ĝ i ,B i ) satisfy the Buscher rules.
(LD8) The second order Buscher rules are satisfied: (LD9) The third order Buscher rules are satisfied: The data of a geometric T-duality cocycle are highly redundant; some of these redundancies are described in the following. A minimized version will be obtain in the context of topological T-duality (Section 5.6) and differential T-duality (Section 6.2). Despite of its lavish data content, a geometric T-duality cocycle clearly reflects the situation of a geometric T-duality correspondence, with data from both sides separated from each other, subject to the Buscher rules (LD7) to (LD9) relating them.
Remark 5.2.1. Let ω i ,ω i ∈ Ω 1 (U i × T n , R n ) be the connections on the trivial bundle U i × T n that are induced by the metrics g i andĝ i , respectively, under Theorem 2.3.1. We remark that by (LD3) and (LD4) the bundle isomorphisms a ij andâ ij are isometries, and hence connection-preserving by Remark 2.3.2. Thus, by bijection (2.2), the connections transform under the transition functions as The connections in turn correspond to 1-forms A i ,Â i ∈ Ω 1 (U i , R n ), via ω i = (A i ) 1 + θ 2 and ω i = (Â i ) 1 + θ 2 , which then transform as By (LD7) and Lemma 3.3.1, the equivariance rules of Lemma 3.3.2 apply to B i andB i , i.e.
In particular,B i and B i are T n -invariant. We may further consider the 3-forms where F andF are the globally defined curvatures of the connections ω andω, respectively. Using (LD7) one can show that K i =K i and that they are the pullback of a globally defined 3-form K ∈ Ω 3 (X) along U i × T n → X.
Remark 5.2.2. Similarly as proved in Section 5.1, (LD8) implies In particular, A ij andÂ ij are T n -invariant.
Remark 5.2.3. We notice that in (LD9) the right hand side is independent ofâ, and the left hand side is independent of a. In other words, the right hand side is constant in a, and the left hand side is constant inâ, and these two constants are equal. Explicitly, if we define to be this constant, then we get for all a,â ∈ T n . We deduce from this the equivariance rules Remark 5.2.4. The Buscher rules (LD7) to (LD9) determineĝ i ,B i ,Â ij , andĉ ijk uniquely. Ifĝ i , B i ,Â ij , andĉ ijk exist and satisfy (LD7) to (LD9), one can in fact show thatĝ i is a T n -invariant Riemannian metric satisfying (LD4), and that (B i ,Â ij ,ĉ ijk ) satisfy (LD6). The same holds upon exchanging quantities with hats and without. In other words, either (LD3) and (LD5), or (LD4) and (LD6) can be omitted in the above list of conditions. Since there is no way to decide which ones should be omitted, we kept both.
We will next describe the conditions under which two geometric T-duality cocycles are considered to be equivalent. We suppose that we have two cocycles 2. smooth maps p i ,p i : U i → R n , 3. numbers z ij ,ẑ ij ∈ Z n , and 4. smooth maps d ij ,d ij : such that the following conditions (LD-E1) to (LD-E8) are satisfied. Abusing notation, we consider in the following the functions p i ,p i eventually as maps p i ,p i : U i × T n → U i × T n given by (x, a) → (x, a + p i (x)) and (x,â) → (x,â +p i (x)), respectively.
(LD-E1) The bundles E and E ′ corresponding to (a ij , m ijk ) and (a ′ ij , m ′ ijk ) are isomorphic: We remark that the second line follows from the first and (LD1); it is only listed for convenience.
(LD-E2) The bundlesÊ andÊ ′ corresponding to (â ij ,m ijk ) and (â ′ ij ,m ′ ijk ) are isomorphic: Under the bundle isomorphism of (LD-E1), the metrics g and g ′ corresponding to g i and g ′ i are identified: Under the bundle isomorphism of (LD-E2), the metricsĝ andĝ ′ corresponding toĝ andĝ ′ are identified:p * iĝ ′ i =ĝ i (LD-E5) The pair (C i , d ij ) is a connection-preserving 1-isomorphism between the bundle gerbes corresponding to (B i , A ij , c ijk ) and (B ′ i , A ′ ij , c ′ ijk ): is a connection-preserving 1-isomorphism between the bundle gerbes cor- The following equality of 1-forms on U i × T 2n holds: (LD-E8) The following equality holds for all (x, a,â) ∈ (U i ∩ U j ) × T 2n : Remark 5.2.5. Let ω i , ω ′ i ∈ Ω 1 (U i × T n , R n ) be the connections on the trivial bundle U i × T n that are induced by the metrics g i and g ′ i , respectively, under Theorem 2.3.1. We remark that the bundle isomorphism p i is an isometry, and hence connection-preserving by Remark 2.3.2. Thus, the connections transform under the functions p i as ω i = ω ′ i + p * i θ. The connections in turn correspond to 1-forms Analogous formulas hold on the dual side, i.e., Remark 5.2.6. From (LD-E7) one can derive the following equivariance rules over U i × T 2n : Remark 5.2.7. We notice that in (LD-E8) the left hand side is independent ofâ, and the right hand side is independent of a. In other words, the right hand side is constant inâ, and the left hand side is constant in a, and these two constants are equal. If we define to be this constant, then we get, for all a,â ∈ T n , the equality From this, we can deduce the following equivariance properties: With this precise definition of local data at hand, we will prove in the following two sections that Loc geo (X) indeed classifies geometric T-duality correspondences over X.

Reconstruction of a geometric T-duality correspondence
In the following we describe a procedure that constructs from a geometric T-duality cocycle a geometric T-duality correspondence in the sense of Definition 4.1.9. First of all, the maps a ij andâ ij become (after exponentiation) T n -valued transition functions, and we let E andÊ be the corresponding principal T n -bundles. Note that these come with canonical trivializations ϕ i andφ i over U i , which induce the given transition functions. Due to (LD3) and (LD4), the locally defined metrics g i andĝ i yield metrics on E andÊ, respectively, which are Riemannian and T n -invariant.
Next we construct the bundle gerbe G over E. We define the surjective submersion π : Y → E by putting Y := i∈I U i × T n and π| Ui×T n := ϕ i . Over Y we consider the 2-form B defined by B| Ui×T n := B i . The fibre products over E can be identified in the following way: where the projection maps pr j : Y [k] → Y become, under this identification, We remark that the more general projections pr j1,...,j l : can then be described using (5.3.2) in each component of the range separately.
On Y [2] we define the 1-form A by A| Yij := A ij ; then, the first line of (LD5) implies pr * 2 B − pr * 1 B = dA. We may interpret A as a connection on the trivial principal T-bundle L over Y [2] , so that dA is its curvature. Finally, we define an isomorphism µ : pr * 12 L ⊗ pr * 23 L → pr * 13 L over Y [3] as multiplication by the smooth map −c : Y → T, i.e., −c| Y ijk := −c ijk . The second line of (LD5) implies that µ is connection-preserving, and the third line implies that it satisfies the cocycle condition. This finishes the construction of the bundle gerbe G.
So far we have provided the structure of a geometric T-duality correspondence. It remains to prove the axioms. Conditions (T1) and (T2) of Definition 4.1.9 follow from (LD7) via Lemmas 3.3.1 and 3.2.2. For (T3), consider one of the open sets U i , over which we have the trivializations ϕ i and ϕ i , and the trivializations T i : ϕ * i G → I Bi andT i :φ * iĜ → IB i mentioned above.
Lemma 5.3.2. The principal T-bundle with connection over U i × T 2n that corresponds to the connection-preserving bundle gerbe isomorphism y y s s s s s s s s s s s s s s is given w.r.t. the covering Z i → U i × T 2n by the connection 1-form ω i ∈ Ω 1 (Z i ) and the transition function z ii : Z Proof. All bundle gerbes and bundle gerbe isomorphisms that appear in the composition above just involve trivial principal T-bundles. The composition has to be computed over a common refinement of all involved surjective submersions; here, Z i → U i × T 2n is sufficient. The trivializations contribute, since we work over a single open set U i , the trivial functions c iii = 1 andĉ iii = 1. It remains the contribution of Φ * i D, which is z ii . For the connections, it is similar: the trivializations contribute A ii = 0 andÂ ii = 0, and Φ * i D contributes ω i .
It remains to notice that z ii (x, a,â, m,m) =âm. This function, as well as the 1-form ω i , are obviously pulled back along the following map of coverings: Comparing with (2.2.4) and (2.2.5), we see that z ii and ω i are the local data of the Poincaré bundle and its connection, w.r.t. the section χ l : R 2n → T 2n . This shows that (T3) is satisfied.

Well-definedness of reconstruction under equivalence
In this section we show that the reconstruction of a geometric T-duality correspondence from a geometric T-duality cocycle described in Section 5.3 is compatible with equivalences between correspondences (Definition 4.1.5) and cocycles (Section 5.2). For this purpose, we consider two geometric T-duality cocycles and an equivalence between them provided by a tuple (C i ,Ĉ i , p i ,p i , z ij ,ẑ ij , d ij ,d ij ). Moreover, we let ((E, g, G), (Ê,ĝ,Ĝ), D) and ((E ′ , g ′ , G ′ ), (Ê ′ ,ĝ ′ ,Ĝ ′ ), D ′ ) be the geometric T-duality correspondences reconstructed from the two cocycles.
The functions p i andp i define bundle isomorphisms p : E → E ′ andp :Ê →Ê ′ due to (LD-E1) and (LD-E2). It is straightforward to see using (LD-E3) and (LD-E4) that p andp are isometric. Concerning the bundle gerbe G and G ′ , we have a commutative diagram i.e., p ′ | Yi = p i . Thus, we may construct a bundle gerbe isomorphism A : G → p * G ′ using the common refinement of their surjective submersions. We define the 1-form C ∈ Ω 1 (Y ) by setting C| Ui×T n := C i , and consider the trivial bundle Q := I C over Y . Then, the first ingredient of the isomorphism A is the equation p ′ * B ′ = B + curv(Q), which follows immediately from the first equation in (LD-E5). The next part is to provide a connection-preserving bundle isomorphism . Since all bundles are trivial (L = I A and L ′ = I A ′ ), this is the same as a smooth map d : thus, we can take the given data d ij according to the second equation in (LD-E5). Finally, we have to show that the diagram is commutative. Restricting to Y ijk , this means that which is the third equation in (LD-E5). The dual side works precisely in an analogous way, using (LD-E6).
It remains to produce the connection-preserving 2-isomorphism ξ of Definition 4.1.5. We consider a commutative diagram where P := p × p ′ , and P ′ : Z → Z is defined by The 2-isomorphism ξ is given by a function w : Z → T satisfying: Here, ω, ω ′ are the 1-forms (5.3.6) from the reconstruction of D and D ′ , respectively, and z, z ′ are the corresponding T-valued functions (5.3.7). Proof. We set w i := w| Zi . Employing definitions, we find under which (5.4.1) becomes (LD-E7). In order to treat (5.4.2) we need to compute the induced map , resulting in (i, j, x, a,â, m 2 ,m 2 ) → (i, j, x, a + p i (x),â +p i (x), m 2 − z ij ,m 2 −ẑ ij ).
Using this, (5.4.2) becomes equivalent to Inserting the definitions of z ij and w i , and once using (LD-E2), one can see that the latter equation is equivalent to (LD-E8), hence satisfied. In the next section we show that it is a bijection.

Local-to-global equivalence
In this section we prove the following result.
One can then trivialize the Poincaré bundle using the section χ l : R 2n → P, see Section 2.2. This results into a 2-isomorphic 1-morphism D ′′ . As the covariant derivative of χ l is the 1-form ω := −adâ on R 2n , the principal T-bundle of D ′′ is I ω . Its isomorphism is the composite where the projections pr 1 , pr 2 : Z ′[2] → Z ′ are as in the proof of Lemma 5.3.1. Using the formulas (2.2.3) and (2.2.8) we can calculate this isomorphism explicitly: where z ij was defined in (5.3.7). This shows that the bundle isomorphism of D ′′ is multiplication with z ij . Hence, D ′′ is precisely the reconstructed isomorphism, proving the surjectivity in Proposition 5.5.1.
It remains to prove injectivity of reconstruction. For this purpose, we look at two geometric T-duality cocycles, consider the corresponding reconstructed geometric T-duality correspondences ((E, g, G), (Ê,ĝ,Ĝ), D) and ((E ′ , g ′ , G ′ ), (Ê ′ ,ĝ ′ ,Ĝ ′ ), D ′ ), and assume that these are equivalent in the sense of Definition 4.1.5. Thus, there exist isometric bundle isomorphisms p : E → E ′ andp :Ê →Ê ′ , connection-preserving bundle gerbe isomorphisms A : G → p * G ′ andÂ :Ĝ →p * Ĝ′ , and a connection-preserving 2isomorphism It is straightforward to see that the isomorphisms p andp induce smooth maps p i ,p i : U i → R n and z ij ,ẑ ij ∈ Z n satisfying (LD-E1) to (LD-E4). Note that the surjective submersions of all 4 bundle gerbes have the same domain Y = Y i , with Y i := U i × T n , and the bundle isomorphisms p andp lift to Y [k] as the component-wise defined maps and the analogousp i . We may thus assume that the isomorphisms A andÂ consist of principal T-bundles Q andQ with connections over Y . Their restrictions to Y i will be denoted by Q i and Q i , respectively. The curvatures are curv( and their connection-preserving bundle isomorphisms over Y [2] ∼ = Y ij are component-wise and an analogousχ ij . As explained in the proof of Lemma 5.1.1, there exist bundle isomorphisms Q i ∼ = pr * T n P Fi and Z). The isomorphism χ ij shows that F i = F j andF i =F j , so that we can omit the indices. The 2-isomorphism ξ induces over (U i ∩ U j ) × T 2n an isomorphism pr * T 2n P ⊗ pr * T 2npr * PF ∼ = pr * T 2n pr * P F ⊗ pr * T 2n P which then implies F =F = 0. Thus, there exist 1-forms C i ,Ĉ i ∈ Ω 1 (Y i ) and connection-preserving bundle isomorphisms κ i : Q i → I Ci andκ i :Q i → IĈ i . The isomorphisms χ ij andχ ij then induce functions d ij ,d ij : (U i ∩ U j ) × T n → T such that (LD-E5) and (LD-E6) are satisfied.
over U i ×T n ×T n . Now we proceed similar as in Section 5.1. We have a closed 1-form α i ∈ Ω 1 cl (U i ×T 2n ) defined by α i := (C i ) 1,3 − (C i ) 1,2 +p i θ 3−2 . Since the de Rham cohomology of U i × T 2n only has torus contributions, there exist a smooth map β i : U i × T 2n → R and vectors r i , s i ∈ R n such that Moreover, since the definition of α i is skew-symmetric with respect to the exchange of a with b; this implies that r i = −s i . We may now shift the isomorphism κ i by the smooth map U i × T n → T : (x, a) → r i a. This shifts C i by r i θ and shows that Again, the left hand side is skew-symmetric, so that where s swaps the two T n -factors. Thus, c i := β i + s * β i ∈ R is a constant. Shifting β i , we can achieve that this constant is zero, and that β i is skew-symmetric; moreover, definingβ i : x, a, 0), we obtain We may now shift κ i by the function (x, a) →β i (x, a), getting the formula (5.5.1) On the dual side, we obtain analogously We continue by looking at the local description of the 2-isomorphism ξ. We pullback to the space Z = U i × T 2n , where, as D and D ′ are obtained by reconstruction, they consist of the trivial bundles with connections ω i , ω ′ i and of the bundle morphisms z ij , z ′ ij defined in (5.3.6) and (5.3.7). Note that ω ′ i = ω i , whereas z ij and z ′ ij are different. Thus, the 2-isomorphism ξ consists of smooth maps w i : U i × R 2n → T such that also see (5.4.1) and (5.4.2). We study now the dependence of w i on the first and the second R n -factor. From (5.5.1) and (5.5.3) one can show that over (x, a, a ′ ,â) ∈ U i × R n × R n × R n holds. Similarly, (5.5.2) and (5.5.3) imply that In particular, definingw i : Putting a =â = 0 shows that z i = 0. We make a final revision of the isomorphism κ i by the functioñ w i . This changes C i to C i +w * i θ, and changes w i to just

Local perspective to topological T-duality
In this section, we deduce from the local perspective to geometric T-duality obtained in Sections 5.1 to 5.5 a corresponding local perspective to topological T-duality, and relate that to the non-abelian cohomology with values in the T-duality 2-group.
We define a topological T-duality cocycle as a geometric T-duality cocycle with all metrics and differential forms stripped off. Thus, a topological T-duality cocycle is a tuple (a ij ,â ij , m ijk ,m ijk , c ijk ,ĉ ijk ) of data as in Section 5.2, subject to conditions (LD1) and (LD2), only the last equations of (LD5) and (LD6), and the third order Buscher rule (LD9). Two topological T-duality cocycles are considered to be equivalent if there exist equivalence data (z ij ,ẑ ij , p i ,p i , d ij ,d ij ) as in Section 5.2, satisfying (LD-E1) and (LD-E2), the last equations of (LD-E5) and (LD-E6), and (LD-E8). The direct limit of equivalence classes over refinement of open covers will be denoted by Loc top (X).
Applying the reconstruction procedure of Sections 5.3 and 5.4 to only the topological data establishes a map Loc top (X) → T-Corr top (X). (5.6.1) In principle it could be argued similarly as in Section 5.5 that this map is a bijection. However, we will prove this in a different way using the non-abelian differential cohomology H 1 (X, TD) of the T-duality 2-group TD, and a result of [NW20], see Proposition 5.6.3.
The T-duality 2-group TD has been introduced in [NW20, §3.2]. Its definition and a general definition of non-abelian cohomology can be found there. Here we only recall the resulting definition of the set H 1 (X, TD), see [NW20,Rem. 3.7]. An element in H 1 (X, TD) is represented with respect to an open cover {U i } by a TD-cocycle, a tuple (a ij ,â ij , m ijk ,m ijk , t ijk ), where the first four quantities are exactly as in geometric T-duality cocycles, and t ijk : U i ∩ U j ∩ U k → T are smooth functions. The cocycle conditions are (LD1) and (LD2), and (5.6.2) , with the first four quantities just as in the case of an equivalence between geometric T-duality cocycles, and smooth functionsẽ ij : Then, H 1 (X, TD) is a direct limit of equivalence classes of TD-cocycles over refinement of open covers. We recall the following result. We will now describe a map Loc top (X) → H 1 (X, TD) (5.6.4) and prove that it is a bijection, see Lemma 5.6.2. Let (a ij ,â ij , m ijk ,m ijk , c ijk ,ĉ ijk ) be a topological T-duality cocycle, representing an element in Loc top (X). In Remark 5.2.3 we have already defined the function t ijk (x) := −ĉ ijk (x, 0) − m ijkâik (x). (5.6.5) A straightforward calculation using (5.2.5) shows that t ijk indeed satisfies (5.6.2).
Given an equivalence between two topological T-duality cocycles established by a tuple (z ij ,ẑ ij , p i ,p i , d ij ,d ij ), we consider a slight modification of the function e ij defined in Remark 5.2.7, namely, we set

Differential T-duality
In this section we investigate the relation between geometric T-duality as discussed in Section 4 and a closed related notion of "differential T-duality". Differential T-duality can be seen as a reformulation of Kahle-Valentino's "differential T-duality pairs" [KV14]. It is an intermediate step between geometric and topological T-duality, in which just the metrics are replaced by their Kaluza-Klein connections. This intermediate step turns out to be useful for proving our main Theorem 1.2.

Differential T-duality correspondences
We first give a definition of differential T-duality that fits into the setting of geometric and topological T-duality. This definition is very natural, but has not appeared anywhere else, as far as I know. The relation to the work of Kahle-Valentino [KV14] will be described later in Section 6.3.
Definition 6.1.1. A differential T-background over X is a triple (E, ω, G) consisting of a principal T n -bundle E with connection ω over X and a bundle gerbe G with connection over E. Two differential T-backgrounds (E, ω, G) and (E ′ , ω ′ , G ′ ) over X are equivalent if there exists a connection-preserving bundle isomorphism p : E → E ′ and a connection-preserving bundle gerbe isomorphism G ∼ = p * G ′ . The set of equivalence classes of differential T-backgrounds is denoted by T-BG diff (X).
Obviously, every geometric T-background (E, g, G) induces a differential T-background (E, ω, G), where ω is the Kaluza-Klein connection of g. By Theorem 2.3.1, this establishes in fact a bijection where RieM(X) is the set of all Riemannian metrics on X, and PDS(R n ) is the manifold of all positivedefinite symmetric bilinear forms on R n . We see that differential T-backgrounds are almost as good as geometric T-backgrounds, up to independent global information.
Definition 6.1.2. A differential T-duality correspondence between two differential T-backgrounds (E, ω, G) and (Ê,ω,Ĝ) is a connection-preserving isomorphism D : pr * G →pr * Ĝ ⊗ I ρ ω,ω over E × XÊ , such that every point x ∈ X has an open neighborhood U ⊆ X over which condition (T3) of Definition 4.1.9 is satisfied.
Here, it is understood that the 2-form ρ g,ĝ that appears in (T3) is replaced by ρ ω,ω . We shall fix the following obvious observation.
We also have the following converse result.
Proof. We choose a Riemannian metric g ′ on X. Let h : R n × R n → R denote the standard inner product. We define g to be the T n -invariant metric on E corresponding to the triple (ω, g ′ , h) under Theorem 2.3.1, and we defineĝ to be the metric onÊ corresponding to (ω, g ′ , h). We have ρ ω,ω = ρ g,ĝ , so that D has the correct structure of a geometric T-duality correspondence. Finally, we observe that it satisfies all three conditions, (T1) to (T3).
The following result is more difficult to show, and its proof relies on the local formalism developed in Section 5 and extended to differential T-duality below in Section 6.2.
Proposition 6.1.5. Suppose (E, G) and (Ê,Ĝ) are topological T-backgrounds, and D is a topological T-duality correspondence between them. Suppose further that ω andω are connections on E andÊ, respectively. Then, there exist connections on G,Ĝ, and D, such that D becomes a differential T-duality correspondence between (E, ω, G) and (Ê,ω,Ĝ).
Proof. Proposition 6.2.5 in combination with Lemmas 5.6.2 and 6.2.4.
The obvious composition of Propositions 6.1.4 and 6.1.5, about lifting topological T-duality correspondences to geometric ones, is stated as Proposition 4.3.5 in Section 4.3. On the level of equivalence classes, it is clear that the map T-Corr geo (X) → T-Corr top (X) from Section 4.3 factors as where both maps are surjective.

Local perspective to differential T-duality
In this section we develop a local description of differential T-duality. We modify the geometric T-duality cocycles considered in Section 5.2 by replacing the metrics g i andĝ i by 1-forms A i ,Â i ∈ Ω 1 (U i , R n ), and replacing conditions (LD3) and (LD4) by the following new conditions: Concerning equivalences between cocycles, we keep the structure of an equivalence as it is, and replace conditions (LD-E3) and (LD-E4) by the new conditions: The corresponding set of equivalence classes, and its direct limit over refinements of open covers will be denoted by Loc diff (X). Enforced by Theorem 2.3.1, and using Remarks 5.2.1 and 5.2.5, there is a bijection Loc geo (X) ∼ = Loc diff (X) × RieM(X) × C ∞ (X, PDS(R n )), obtained by replacing the metrics g i andĝ i by the local connection 1-forms A i ,Â i of their Kaluza-Klein connections.
The reconstruction procedure described in Sections 5.3 and 5.4, together with the proof of Proposition 5.5.1, goes through with obvious small modifications, so that we infer the following result.
Proposition 6.2.1. Reconstruction is a bijection, Next we set differential T-duality in relation to the differential non-abelian cohomology of the T-duality 2-group TD, whose investigation was started recently by Kim-Saemann [KS]. Differential non-abelian cohomology in general has been studied by Breen-Messing [BM05] and further developed in [SW11,SW13,Sch11]. A common phenomenon in higher gauge theory is the appearance of several versions of connection-data, which, in my review in [Wal17, §2.2] are categorized into fake-flat , regular , and generalized , with increasing generality. Thus, there are (at least) 3 versions of non-abelian differential cohomology with values in some Lie 2-group Γ, related by mapŝ H 1 (X, Γ) f f →Ĥ 1 (X, Γ) reg →Ĥ 1 (X, Γ) gen that commute with the projections to the (non-differential) non-abelian cohomology H 1 (X, Γ).
In order to explain it on the basis of [Wal17, §2.2] and [KS], we need to express the Lie 2-group TD and its associated Lie 2-algebra as crossed modules (of Lie groups and Lie algebras, respectively). The crossed module of TD consists of the Lie group homomorphism The corresponding crossed module of Lie algebras is trivial: it consists of the induced Lie algebra homomorphism, τ * = 0, and the induced action of the Lie algebra g of G on the Lie algebra h of H, α * = 0. Of relevance is further the differential of the action of a fixed element of G, α g : H → H, which is here (α a,â ) * = id R , and the differential of the map which is here (α t,m,m ) * (a,â) = −âm.
With these expressions at hand, we can recall the definition ofĤ 1 (X, TD) gen on the basis of [Wal17, §2.2]. Thus, a generalized differential TD-cocycle consists of a TD-cocycle (a ij ,â ij , m ijk ,m ijk , t ijk ) as in Section 5.6, and additionally of 1-forms A i ,Â i ∈ Ω 1 (U i , R n ), a 2-form R i ∈ Ω 2 (U i ), and a 1-form ϕ ij ∈ Ω 1 (U i ∩ U j ) such that (LD3') and (LD4') and are satisfied. Indeed, for an equivalence between generalized differential TD-cocycles we require a tuple (φ i , p i ,p i , z ij ,ẑ ij ,ẽ ij ), where φ i ∈ Ω 1 (U i ), and (p i ,p i , z ij ,ẑ ij ,ẽ ij ) is, as in Section 5.6, an equivalence between the TD-cocycles (a ′ ij ,â ′ ij , m ′ ijk ,m ′ ijk , t ′ ijk ) and (a ij ,â ij , m ijk ,m ijk , t ijk ), i.e., it satisfies (LD-E1) and (LD-E2) and (5.6.3). Additionally, we require (LD-E3') and (LD-E4') and We remark that the 2-form R i does not appear in any of the above conditions. This will be fixed by considering an adjustment κ for TD. In general, an adjustment is a map κ : G × g → h, and in case of TD Saemann-Kim [KS] use κ((a,â), (b,b)) := ab.
Then, an adjusted differential TD-cocycle satisfies, in addition to the conditions listed above, the condition R j + dϕ ij = R i + a ijF , (6.2.3) whereF ∈ Ω 2 (X) is defined byF | Ui = dÂ i . Moreover, for an equivalence between adjusted differential TD-cocycles, we additionally require the condition Remark 6.2.2. The 3-curvature of an adjusted differential TD-cocycle is, by definition, Having recalled the definition of the κ-adjusted differential cohomology of TD, we are in position to construct a map Loc diff (X) →Ĥ 1 (X, TD κ ). (6.2.6) , we consider the underlying TD-cocycle (a ij ,â ij , m ijk ,m ijk , t ijk ), where t ijk was defined in Remark 5.2.3, namely, This coincides with the expression given in (5.6.5), using (LD9). We add the given 1-forms A i and A i , so that (LD3') and (LD4') are satisfied as before. Let σ : U i → U i × T n be the zero section, σ(x) := (x, 0). The 2-form R i is then defined by

2.7)
and the 1-form ϕ ij is defined by It remains to check condition (6.2.1) for generalized differential cocycles and the additional condition (6.2.3) for adjusted differential cocycles. These are straightforward calculations; the first involving (LD4') and (LD5) and Remark 5.2.2, the second involving (LD5) and (3.3.1).
Let us now suppose that we have an equivalence between two differential T-duality cocycles, established by a tuple (C i ,Ĉ i , p i ,p i , z ij ,ẑ ij , d ij ,d ij ). We recall from Section 5.6 that the functions p i ,p i : U i → R 2n andẽ ij : U i ∩ U j → T, defined in (5.6.7) bỹ establish an equivalence between the underlying two TD-cocycles. Additionally, conditions (LD-E3') and (LD-E4') remain valid. It remains to provide 1-forms φ i ∈ Ω 1 (U i ) satisfying (6.2.2) and (6.2.4). We set This formula together with (LD-E1) and (LD4') proves (6.2.2). This completes the construction of the map (6.2.6).
Remark 6.2.3. We recall from Remark 5.2.1 that every geometric T-duality cocycle comes equipped with a globally defined 3-form K ∈ Ω 3 (X), which corresponds to the 3-form of a geometric T-duality correspondence, see Remarks 4.1.3 (b) and 5.3.3. Under the map Loc geo (X) → Loc diff (X), the same 3-form can be obtained from a differential T-duality cocycle, namely Under the map (6.2.6), Loc diff (X) →Ĥ 1 (X, TD κ ), the 3-form K is precisely the curvature of Remark 6.2.2.
Lemma 6.2.4. The map (6.2.6) is a bijection, Proof. We suppose that we have an adjusted differential TD-cocycle First, we reproduce, as in the proof of Lemma 5.6.2, the topological part of a differential T-duality cocycle, i.e., we define c ijk andĉ ijk as in (5.6.8) and (5.6.9). We further revert the assignments made in the definition of (6.2.6) using Lemma 3.3.2, and set Similarly, using Remark 5.2.2, we set One can then check using (6.2.1) and (6.2.3) that the first and second lines of (LD5) are satisfied (the third line is already checked in Lemma 5.6.2). Finally, we defineB i andÂ ij such that the Buscher rules (LD7) and (LD8) are satisfied. As mentioned in Remark 5.2.4, it then follows automatically that (LD6) is satisfied. This shows the surjectivity of our map.
For injectivity, we assume that two differential T-duality cocycles, become equivalent after passing toĤ 1 (X, TD κ ). That is, there exists a tuple (φ i , p i ,p i , z ij ,ẑ ij ,ẽ ij ) satisfying (LD-E3') and (LD-E4') and (6.2.2) and (6.2.4), as well as the usual (non-differential) cocycle conditions (LD-E1) and (LD-E2) and (5.6.3). We have seen in the proof of Lemma 5.6.2 how to obtain d ij andd ij such that the third lines of (LD-E5) and (LD-E6) and (LD-E8) are satisfied. It remains to provide 1-forms C i ,Ĉ i ∈ Ω 1 (U i ×R n ) such that the first two lines of (LD-E5) and (LD-E6), and (LD-E7) hold. We set The first line reverts (6.2.9), and the second is chosen such that (LD-E7) holds. The first line of (LD-E5) con now be verified using (5.2.1), (6.2.4) and (6.2.7), and the second line of (LD-E5) can be verified using Remark 5.2.2 and (6.2.2) and (6.2.8). The two first lines of (LD-E6) can be checked analogously. This shows that the given differential T-duality cocycles are equivalent.
The identification of differential T-duality correspondences with the adjusted differential cohomology of TD has the advantage that the presentation with differential TD-cocycles is less redundant than the one with differential T-duality cocycles: instead of two 2-forms B i andB i there is only a single 2-form R i , instead of A ij andÂ ij there is only ϕ ij , and instead of c ijk andĉ ijk there is only t ijk . Moreover, all data are defined on the open sets U i and intersections thereof, while the data of T-duality cocycles live on U i × T n and their intersections. The following two results show that (adjusted) differential cohomology is very efficient for calculations. The first, Proposition 6.2.5, delivers the core ingredient to the proofs of our main results Theorems 1.2 and 1.3. Proposition 6.2.5. Every TD-cocycle can be lifted to an adjusted differential TD-cocycle, i.e., the mapĤ 1 (X, TD κ ) → H 1 (X, TD) is surjective.
Proof. Given a TD-cocycle (a ij ,â ij , m ijk ,m ijk , t ijk ), by the well-known existence of connections on principal bundles we find 1-forms A i ,Â i ∈ Ω 1 (U i , R n ) satisfying (LD3') and (LD4'). We write (6.2.1) as (δϕ) ijk = t * ijk θ −Â k m ijk , where δ denotes theČech coboundary operator. It is easy to check using (5.6.2) that the right hand side is aČech 2-cocycle; then, by the exactness of theČech complex with values in the sheaf Ω 1 it follows that ϕ ij exist such that (6.2.1) is satisfied. Finally, we write (6.2.3) as (δR) ij = a ijF − dϕ ij where the left hand side denotes groupoid whose objects are all (p − 1)-forms on X, and which has only identity morphisms. A geometric trivialization of an object ξ ∈ H p (X) is a differential form K ∈ Ω p−1 (X) and an isomorphism τ : ξ → I K in H p (X). The setĤ p−1 (X) acts on the set of all geometric trivializations of ξ, where [η] ∈Ĥ 2 (X) sends τ to τ +η, and K gets shifted by the "curvature" of η. This action is free and transitive.
A concrete realization of these groupoids can be obtained using Deligne cocycles w.r.t. a fixed open cover with all finite non-empty intersections contractible, see [KV14, §A.2]. The objects of H p (X) are Deligne (p − 1)-cocycles ξ, and the morphisms ξ 1 → ξ 2 are equivalence classes [η] of (p − 2)-cochains η satisfying ξ 2 = ξ 1 + Dη, where D denotes the Deligne differential, and η 1 ∼ η 2 if there exists a (p − 3)-cochain β with η 2 = η 1 + Dβ. Composition of morphisms is just addition. The cup product on the level of objects is the usual cup product in Deligne cohomology, as recalled below. The functor I is the usual inclusion ϕ → (ϕ, 0, ., , , 0) of a globally defined differential form as a "topologically trivial" Deligne cocycle. For p = 2, the groupoid H 2 (X) is equivalent to the groupoid of principal T-bundles with connections, and connection-preserving bundle isomorphisms. Under this equivalence, a geometric trivialization is a (not necessarily flat) section. The free and transitive action byĤ 1 (X) = C ∞ (X, T) is the action of smooth T-valued functions on sections.
Kahle-Valentino claim in [KV14, §2.5] that differential T-duality pairs induce topological T-duality correspondences. We want to sharpen this relation and show that differential T-duality pairs are the same as our differential T-duality correspondences. Their relation to topological T-duality correspondences is then a consequence thereof. In order to proceed, it is necessary to consider an equivalence relation on the set of all differential T-duality pairs over X. Unfortunately, Kahle-Valentino do not introduce such relation. Apparently, the most natural definition is the following. Definition 6.3.2. Two differential T-duality pairs (ξ,ξ, K, τ ) and (ξ ′ ,ξ ′ , K ′ , τ ′ ) over X are equivalent if K ′ = K and there exist isomorphisms p : ξ → ξ ′ andp :ξ →ξ ′ in H 2 (X) such that the diagram in H 4 (X) is commutative. The set of equivalence classes of differential T-duality pairs is denoted by TDP(X).
Below, we will prove the following result. Proposition 6.3.3. There is a canonical bijection between equivalence classes of differential T-duality correspondences and equivalence classes of differential T-duality pairs, T-Corr diff (X) ∼ = TDP(X), such that the diagram is commutative.
Exploring the notion of a differential T-duality pair further, we spell out in the following what a geometric trivialization of ξ ∪ξ from (6.3.3) is. It consists of: (a) a 3-form K ∈ Ω 3 (X) (b) 2-forms R i ∈ Ω 2 (U i ), such that A i ∧F = K + dR i . (6.3.5) (c) 1-forms ϕ ij ∈ Ω 1 (U i ∩ U j ) such that a ijF = −R j + R i + dϕ ij . (6.3.6) (d) functions b ijk : U i ∩ U j ∩ U k → R such that m ijkÂk = ϕ ij + ϕ jk − ϕ ik + db ijk . (6.3.7) (e) numbers q ijkl ∈ Z satisfying and m ijkmklp = q ijkl − q ijkp + q ijlp − q iklp + q jklp . (6.3.9) At this point, it makes sense to discuss the action ofĤ 3 (X) on differential T-duality pairs, which is induced by the above-mentioned action ofĤ 3 (X) on all geometric trivializations of ξ ∪ξ. Here, this action takes the formĤ 3 (X) × TDP(X) → TDP(X) and is given, using the above description of geometric trivializations, by the formula Note that K is shifted by the globally defined 3-form H = dB i , the curvature. It is clear that this action restricts to the fibres of the map TDP(X) →Ĥ 2 (X) ×Ĥ 2 (X) and is transitive in each fibre.
Proof. We have to show that (K, R i , ϕ ij , b ijk , q ijkl ) and (K, R i +ŷF + yF , ϕ ij , b ijk , q ijkl ) define the same morphism. We consider the automorphism of ξ = (A i , a ij , m ijk ) given by (y, 0), this works as D(y, 0) = (0, 0, 0), and similarly, the automorphism ofξ given by (ŷ, 0). According to (6.3.4), we have This proves that F acts trivially. Conversely, if (K, R i , ϕ ij , b ijk , q ijkl ) and (K + dB i , R i + B i , ϕ ij + A ij , b ijk + c ijk , q ijkl + s ijkl ) are equivalent, we have to show that (B i , A ij , c ijk , s ijkl ) ∼ (ŷF − yF , 0, 0, 0). The proof of this is very similar to the one given in Proposition 6.2.6, and omitted for brevity.

A torus bundle with trivial B-field
We consider a geometric T-background (E, g, G) over a smooth manifold X, whose bundle gerbe is the trivial one, i.e., G = I 0 . In this section, we explicitly construct a geometric T-duality correspondence whose left leg is (E, g, I 0 ).
We let (ω, g ′ , h) be the triple corresponding to g under Theorem 2.3.1, and we let F ∈ Ω 2 (X) be the curvature of the connection ω. We consider the trivial bundleÊ := X × T n , and equip it with the trivial connection,ω := θ. We letĝ be the invariant metric onÊ that corresponds to the triple (ω, g ′ , h −1 ). Next we construct the bundle gerbeĜ overÊ.
Remark 7.1.1. If n = 1, thenĜ is precisely the cup product bundle gerbe pr * X E ∪ pr T , where pr X :Ê → X and pr T :Ê → T 1 are the projections; explicitly, pr * X E is a principal T-bundle overÊ with connection, and pr T is a T-valued function onÊ. A description of the cup product of such structures, resulting in a bundle gerbe with connection, has been given by Johnson in [Joh02]. Our construction above (for n = 1) reproduced exactly that description. Johnson also proved that the cup product of a principal T-bundle with connection and a T-valued function coincides with the cup product in Deligne cohomology [Joh02].