Geometric T-Duality: Buscher Rules in General Topology

Abstract

The classical Buscher rules d escribe 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.

1 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 $\mathcal {G}$ with connection over E (Kalb–Ramond field).

A special class of Kalb–Ramond fields is given by B-fields, i.e., 2-forms $B \in \Omega ^2(E)$; these are precisely the connections on the trivial bundle gerbe. Geometric backgrounds (are supposed to) determine 2-dimensional 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 ${\mathbb {T}}^{n}$ on E, such that g is invariant and E is a principal ${\mathbb {T}}^{n}$-bundle over the quotient $X:=E/{\mathbb {T}}^{n}$. We will use the terminology geometric T-background for geometric backgrounds with toroidal symmetry. When are two geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$ 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 [14] in a topologically trivial situation, where $E = X \times {\mathbb {T}}$ is the trivial circle bundle (i.e., $n=1$) over an open subset $X \subseteq {\mathbb {R}}^{s}$, and the bundle gerbe $\mathcal {G}$ is just a B-field $B \in \Omega ^2(E)$. These conditions are the by now classical Buscher rules:

$$\begin{aligned}&\hat{g}_{\theta \theta }=\frac{1}{g_{\theta \theta }},\qquad \hat{g}_{\alpha \theta }= \frac{B_{\alpha \theta }}{g_{\theta \theta }},\qquad \hat{g}_{\alpha \beta }=g_{\alpha \beta }-\frac{1}{g_{\theta \theta }}(g_{\alpha \theta }g_{\beta \theta }-B_{\alpha \theta }B_{\beta \theta }) \\&\hat{B}_{\alpha \theta }=\frac{g_{\alpha \theta }}{g_{\theta \theta }},\qquad \hat{B}_{\alpha \beta }=B_{\alpha \beta }-\frac{1}{g_{\theta \theta }}(B_{\alpha \theta }g_{\beta \theta }-g_{\alpha \theta }B_{\beta \theta }) \end{aligned}$$

Here, the indices label the coordinates of the direct product $E=X \times {\mathbb {T}}$, with $\alpha ,\beta =1,\ldots ,s$ coordinates of X and $\theta$ the single fiber coordinate. The Buscher rules can be generalized to arbitrary torus dimension n, see [25].

A groundbreaking observation of Giveon et al. [22, 23] and Alvarez et al. [1] was that (even in the case $n=1$) the Buscher rules require a topology change as soon as $X \subseteq {\mathbb {R}}^{s}$ is replaced by a topologically non-trivial manifold. The example studied in [23] 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 $U_i \subseteq S^2$ over which E trivializes, and apply the Buscher rules on each patch to obtain locally defined dual metrics $\hat{g}_i$ and a dual B-fields $\hat{B}_i$. 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 bundle $\hat{E}:= S^2 \times {\mathbb {T}}$. In other words, spacetime changes its topology under T-duality! A second important development, due to Hori [30], 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 ${\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$.

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 [20], and Alvarez’ work on topological quantization [2]. A major step was the invention of bundle gerbes by Murray [41] that unleashed a number of advances, e.g., a complete classification of WZW models on compact simple Lie groups [27], a corresponding classification of D-branes in these models [21, 26], a discussion of D-branes in terms of twisted K-theory [3] or a classification of WZW orientifolds [29, 44]. Bundle gerbes with connection have an underlying topological part, measured by their Dixmier-Douady class in ${\textrm{H}}^3(E,{\mathbb {Z}})$, and a curvature, a 3-form $H\in \Omega ^3(E)$ called H-flux. If the Dixmier-Douady class vanishes, then they reduce—up to isomorphism—to a trivial bundle gerbe $\mathcal {I}_B$ carrying the former B-field B, such that $H={\textrm{d}}B$. 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 [4, 5] 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

$$\begin{aligned} {\textrm{H}}^{\bullet }_{{\textrm{dR}}}(E,H) \cong {\textrm{H}}^{\bullet +1}_{{\textrm{dR}}}(\hat{E},\hat{H}). \end{aligned}$$

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 [37,38,39].

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 [11, 13]. 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 T-background 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 ${\mathbb {T}}^{n}$-bundle E and a bundle gerbe $\mathcal {G}$ 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

$$\begin{aligned} {\textrm{pr}}^{*}\mathcal {G} \cong \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \end{aligned}$$

(1.1)

between the pullbacks of the two bundle gerbes to the so-called correspondence space, the fiber product

Moreover, the isomorphism (1.1) has to satisfy a certain Poincaré condition, relating it to the Poincaré bundle over ${\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. Bunke–Rumpf–Schick then started to explore the space of topological T-duality correspondences, consisting of two topological T-backgrounds $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$, and an isomorphism (1.1), in its dependence on X.

Bunke–Rumpf–Schick’s new perspective on topological T-duality was accompanied by a number of important results [11]: a precise criterion when a topological T-background $(E,\mathcal {G})$ admits a T-dual (the Dixmier-Douady class of $\mathcal {G}$ has to lie in the second step of the standard filtration of ${\textrm{H}}^3(E,{\mathbb {Z}})$) and a parameterization of all possible T-duals (the group $\mathfrak {so}(n,{\mathbb {Z}})$ of skew-symmetric integral $n\times n$-matrices acts freely and transitively on them). Moreover, Bunke–Rumpf–Schick obtained a version of the Fourier–Mukai transformation in topological twisted K-theory,

$$\begin{aligned} K^{\bullet }(E,\xi ) \cong K^{\bullet +1}(\hat{E},\hat{\xi })\text {,} \end{aligned}$$

where the twists $\xi ,\hat{\xi }$ are the Dixmier-Douady classes of the bundle gerbes $\mathcal {G}$ and $\hat{\mathcal {G}}$.

A further approach toward a unification of the topological data of a bundle gerbe and the differential data of its connection was proposed by Kahle–Valentino [35]. It can be understood as reducing the full information of a geometric T-background $(E,g,\mathcal {G})$ to the information of E, the full bundle gerbe with connection $\mathcal {G}$, and of the connection $\omega$ on E obtained from the metric g under Kaluza–Klein reduction. This approach remained rather unrelated to the previous approaches, in particular, to the Buscher rules, and also is formulated in a rather uncommon language of “differential cohomology groupoids”. Nonetheless, we will show here that the approach of Kahle–Valentino is very close to the formalism we will present below as “geometric T-duality”. Kahle–Valentino also propose a very interesting generalization of Bunke–Rumpf–Schick’s Fourier–Mukai transformation from twisted K-theory to twisted differential K-theory,

$$\begin{aligned} \hat{K}^{\bullet }(E,\mathcal {G}) \cong \hat{K}^{\bullet +1}(\hat{E},\mathcal {G}). \end{aligned}$$

Unfortunately, at that time, no general theory for twisted differential K-theory was available, and so Kahle–Valentino proposed an axiomatic description under which the isomorphism was deduced. However, it seems to be unclear if these axioms are met by existing models, e.g., [19], or if they can be proved in a modern framework of twisted differential cohomology, e.g., [9, 28].

In this article, we propose a new formalism which we call geometric T-duality. It is based on the full information of geometric T-backgrounds: a principal ${\mathbb {T}}^{n}$-bundle E over an arbitrary smooth manifold X, an invariant Riemannian metric g on E, and a bundle gerbe with connection $\mathcal {G}$ on E. The main point of our new formalism is a notion of geometric T-duality correspondence as a relation on the set of all such geometric T-backgrounds (Definition 4.9). The main ingredient is that the isomorphism (1.1) on the correspondence space is now a connection-preserving isomorphism

$$\begin{aligned} {\textrm{pr}}^{*}\mathcal {G} \cong \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \otimes \mathcal {I}_{\rho _{g,\hat{g}}}\text {,} \end{aligned}$$

(1.2)

where $\rho _{g,\hat{g}}$ is a certain 2-form produced from the metrics g and $\hat{g}$. The isomorphism (1.2) is then required to satisfy a differential version of Bunke–Rumpf–Schick’s Poincaré condition. The most important result about our new geometric T-duality is that it indeed unifies all aspects investigated before separately.

Theorem 1.1

Suppose two string backgrounds $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$ are in geometric T-duality correspondence. Then, the following statements are true:

• (1) Locally, the Buscher rules are satisfied. More precisely, there exist local trivializations $\varphi : U \times {\mathbb {T}}^{n} \rightarrow E$ and $\hat{\varphi }: U \times {\mathbb {T}}^{n} \rightarrow \hat{E}$ and bundle gerbe trivializations $\varphi ^{*}\mathcal {G} \cong \mathcal {I}_B$ and $\hat{\varphi }^{*}\hat{\mathcal {G}} \cong \mathcal {I}_{\hat{B}}$ such that (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules.

• (2) Discarding metrics and bundle gerbe connections, $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$ are in topological T-duality correspondence in the sense of Bunke–Rumpf–Schick.

• (3) Discarding metrics, and replacing the bundle gerbes by their curvature 3-forms, (E, H) and $(\hat{E},\hat{H})$ are T-dual as backgrounds with H-flux in the sense of Bouwknegt–Evslin–Mathai.

• (4) Replacing the metrics g and $\hat{g}$ by their Kaluza–Klein connections $\omega$ and $\hat{\omega }$, respectively, $(E,\omega )$ and $(\hat{E},\hat{\omega })$ form a differential T-duality pair in the sense of Kahle–Valentino.

The proof of (1) consists of some computations with differential forms, metrics, and connections performed in Sect. 3; the statement is Proposition 4.14 in the main text. (2) and (3) follow directly from the definitions, see Proposition 4.18 and 4.25. The proof of (4) is rather involved due to the very different settings. In order to prove (4), we introduce in Sect. 6 another formalism that we call “differential T-duality”; we then show in Proposition 6.3 that it is a consequence of geometric T-duality, and prove in Proposition 6.14 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 modeled 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. Figure 1 expresses the implications between the various notions of T-duality.

Fig. 1

A schematic overview about the various versions of T-duality considered here, and how they imply each other

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?

Fig. 2

Local data for geometric T-backgrounds and geometric T-duality 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 ${\mathbb {T}}^{n}$-bundle, an invariant metric, and a bundle gerbe with connection. The transition functions $a_{ij}$ and $\hat{a}_{ij}$ are taken to be ${\mathbb {R}}^{n}$-valued, revealing winding numbers $m_{ijk}$ and $\hat{m}_{ijk}$, respectively. The middle column shows how the (higher) Buscher rules mix these local data from both sides

Theorem 1.2

• (a) Locally, geometric T-duality is equivalent to the Buscher rules. More precisely, suppose (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules. Then, the geometric T-backgrounds $(X \times {\mathbb {T}}^{n},g,\mathcal {I}_B)$ and $(X \times {\mathbb {T}}^{n},\hat{g},\mathcal {I}_{\hat{B}})$ are in geometric T-duality correspondence.

• (b) Every topological T-duality correspondence can be lifted to a geometric T-duality correspondence. More precisely, suppose $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$ are topological T-backgrounds, and suppose $\mathcal {D}$ is a topological T-duality correspondence. Then, there exist ${\mathbb {T}}^{n}$-equivariant metrics g and $\hat{g}$ on E and $\hat{E}$, and connections on $\mathcal {G}$ and $\hat{\mathcal {G}}$ such that $\mathcal {D}$ is a geometric T-duality correspondence between $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$.

• (c) Every differential T-duality pair can be lifted to a geometric T-duality correspondence. The precise statement is in Proposition 6.4.

• (d) Every topological T-duality correspondence can be lifted to a differential T-duality pair. The precise statement is in Proposition 6.5.

The proof of (a) is rather straightforward, see Proposition 4.15. (b) follows from (c) and (d), see Proposition 4.20. (c) is a direct consequence of close relationship between geometric and differential T-duality. The proof of (d) is the hardest part, see Proposition 6.5. In order to prove it, we introduce in Sect. 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 2 summarizes our local description. For a more detailed explanation of these data and conditions, we refer to Sect. 5.2.

We summarize the local data described in Fig. 2 (up to a certain notion of equivalence, and in the direct limit over refinements of open covers) in a set ${\textrm{Loc}}^{{\textrm{geo}}}(X)$. We also look at slightly smaller versions:

• ${\textrm{Loc}}^{{\textrm{diff}}}(X)$, where the metrics are replaced by their Kaluza–Klein connections.

• ${\textrm{Loc}}^{{\textrm{top}}}(X)$, 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 $\mathbb{T}\mathbb{D}$, $\hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D})$. More precisely, it is its adjusted version $\hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa })$ in the sense of Kim-Saemann [32, 33] that becomes relevant here. The 2-group $\mathbb{T}\mathbb{D}$ has been introduced in [42], where we proved that the (non-differential) non-abelian cohomology ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D})$ classifies topological T-duality correspondences. The following result, in particular, extends this classification to differential and geometric T-duality correspondences. We denote by ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$, ${\text {T{-}Corr}}^{{\textrm{diff}}}(X)$, and ${\text {T{-}Corr}}^{{\textrm{top}}}(X)$ the sets of equivalence classes of geometric, differential, and topological T-duality correspondences, respectively.

Theorem 1.3

There is a commutative diagram

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 Sects. 5.3 to 5.5, culminating in Proposition 5.17. That the map (c) is a bijection can then easily be deduced from the bijectivity of (b), see Proposition 6.6. Construction and a proof of bijectivity of the maps (e) and (f) are rather tedious calculations with local data and $\mathbb{T}\mathbb{D}$-cocycles, and are performed in Lemma 5.19 and 6.9. The bijectivity of (f) together with above-mentioned classification result of [42] imply the bijectivity of (d), see Proposition 5.20. 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 6.10. Via the bijections (c) to (f), we obtain then the proof of Theorem 1.2(b).

Apart from the results described above, we consider an interesting action of the (abelian) differential cohomology $\hat{{\textrm{H}}}^3(X)$ on the set ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$ 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 [11], see Proposition 4.12 and 6.11.

Finally, we remark that Theorem 1.2(b) guarantees the existence of many examples of geometric T-duality correspondences. In Sect. 7, we describe explicitly two full examples of geometric T-duality correspondences. The first concerns a geometric T-background of the form $(E,g,\mathcal {I}_0)$, i.e., an arbitrary principal ${\mathbb {T}}^{n}$-bundle E with an arbitrary metric g and trivial B-field. Reducing this to the case in which $E=S^3 \rightarrow S^2$ is the Hopf fibration, and g is the round metric on $S^3$, we reproduce the example of Alvarez et al. [1] 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 . 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 7.4. 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.

2 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 $\omega \in \Omega ^1(E,\mathfrak {g})$ such that

$$\begin{aligned} R^{*}\omega = {\textrm{Ad}}_{h}^{-1}(p^{*}\omega ) + h^{*}\theta \end{aligned}$$

holds over $E \times H$, where R denotes the principal action, p the projection to E, h the projection to H, and $\theta$ is the left-invariant Maurer–Cartan form on H. If H is abelian, we identify the curvature of $\omega$ with the unique 2-form $F \in \Omega ^2(M,\mathfrak {h})$ such that $\pi ^{*}F={\textrm{d}}\omega$, where $\pi :E \rightarrow M$ denotes the bundle projection.

We denote by $\textbf{I}:= M \times H$ the trivial principal H-bundle over a smooth manifold M. We may identify connections $\omega$ on $\textbf{I}$ with $\mathfrak {h}$-valued 1-forms $A\in \Omega ^1(M,\mathfrak {h})$ in the usual way, i.e.,

$$\begin{aligned} \omega = {\textrm{Ad}}_h^{-1}(p^{*}A) + h^{*}\theta \end{aligned}$$

(2.1)

where $p:M \times H \rightarrow M$ and $h:M \times H \rightarrow H$ are the projections. We write $\textbf{I}_A$ for the trivial bundle equipped with the connection (2.1). If H is abelian, and $A_1,A_2\in \Omega ^1(M,\mathfrak {h})$, there is a bijection

(2.2)

under which $\varphi (x,h) = (x,h+f(x))$.

If E is a principal H-bundle over M with connection $\omega$, and $s_i: U_i \rightarrow E$ are local sections, then $\tau _i:\textbf{I}_{s_i^{*}\omega } \rightarrow E|_{U_i}: (x,a) \mapsto s_i(x)\cdot a$ is a connection-preserving bundle isomorphism.

On an overlap $U_i \cap U_j$, we consider the transition function $g_{ij}:U_i \cap U_j \rightarrow H$ defined by $s_i(x) = s_j(x)\cdot g_{ij}(x)$.

In particular, we have $s_i^{*}\omega = s_j^{*}\omega + g_{ij}^{*}\theta$.

2.1 Bundle Gerbes with Connection

We use the definitions and conventions of [48]. The reader familiar with bundle gerbes can safely skip this subsection. We write ${\mathbb {T}}:={\textrm{U}}(1)={\mathbb {R}}/{\mathbb {Z}}$.

Definition 2.1

A bundle gerbe $\mathcal {G}$ with connection over a smooth manifold M consists of the following structure:

• 1. A surjective submersion $\pi :Y \rightarrow M$, and a 2-form $B \in \Omega ^2(Y)$ called “curving”.

• 2. A principal ${\mathbb {T}}$-bundle P with connection over the double fiber product $Y^{[2]}=Y \times _M Y$, whose curvature is $F_P = {\textrm{pr}}_2^{*}B-{\textrm{pr}}_1^{*}B$.

• 3. A connection-preserving bundle isomorphism $\mu : {\textrm{pr}}_{12}^{*}P \otimes {\textrm{pr}}_{23}^{*}P \rightarrow {\textrm{pr}}_{13}^{*}P$ over $Y^{[3]}$, called “bundle gerbe product”.

It is required that over a point $(y_1,y_2,y_3,y_4) \in Y^{[4]}$ the following associativity condition holds:

The curvature of $\mathcal {G}$ is the unique 3-form $H\in \Omega ^3(M)$ such that $\pi ^{*}H={\textrm{d}}B$.

If $B \in \Omega ^2(M)$ is a 2-form, then there is a “trivial” bundle gerbe with connection $\mathcal {I}_B$, with surjective submersion $\pi ={\textrm{id}}_M$, the trivial ${\mathbb {T}}$-bundle with connection $P=\textbf{I}_0$, and the trivial bundle isomorphism $\textbf{I}_0 \otimes \textbf{I}_0 \cong \textbf{I}_0$. Its curvature is $H=\textrm{d}B$.

If $\{U_i\}_{i\in I}$ is an open cover of M admitting smooth local sections $s_i: U_i \rightarrow Y$, then we may define the 2-forms $B_i:= s_i^{*}B\in \Omega ^2(U_i)$. If we further assume that the non-empty double intersections $U_i \cap U_j$ are contractible, we may choose sections $s_{ij}:U_i \cap U_j \rightarrow (s_i,s_j)^{*}P$, inducing 1-forms $A_{ij}\in \Omega ^1(U_i\cap U_j)$ satisfying $B_j = B_i + {\textrm{d}}A_{ij}$. Next, there exists a unique smooth map $c_{ijk}: U_i \cap U_j \cap U_k \rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} \mu (s_{ij}(x) \otimes s_{jk}(x)) \cdot c_{ijk}(x)=s_{ik}(x). \end{aligned}$$

This implies an equality $A_{ik}=A_{ij}+A_{jk}+c_{ijk}^{*}\theta$.

Finally, the associativity condition for $\mu$ implies a Čech cocycle condition $c_{jkl}\cdot c_{ijl} = c_{ijk}\cdot c_{ikl}$. 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 cohomology $\hat{{\textrm{H}}}^3(M)$ of M.

It will be important to consider the full bicategorical structure of bundle gerbes with connection.

Definition 2.2

Suppose $\mathcal {G}$ and $\mathcal {G}'$ are bundle gerbes with connection. A connection-preserving isomorphism $\mathcal {A}:\mathcal {G} \rightarrow \mathcal {G}'$ consists of the following structure:

• 1. A surjective submersion $\zeta : Z \rightarrow Y \times _M Y'$, and a principal ${\mathbb {T}}$-bundle Q with connection over Z whose curvature is $F_Q=\zeta ^{*}{\textrm{pr}}_{Y'}^{*}B' - \zeta ^{*}{\textrm{pr}}_Y^{*}B$.

• 2. A connection-preserving bundle isomorphism $\chi : (\xi ^{[2]})^{*}P \otimes {\textrm{pr}}_2^{*}Q \rightarrow {\textrm{pr}}_1^{*}Q \otimes (\xi '^{[2]})^{*}P'$ over the double fiber product $Z^{[2]}=Z \times _M Z$, where $\xi := {\textrm{pr}}_Y\circ \zeta : Z \rightarrow Y$ and $\xi ':= {\textrm{pr}}_{Y'} \circ \zeta : Z \rightarrow Y'$, and $\xi ^{[2]}$ and $\xi '^{[2]}$ denote the induced maps on double fiber products.

It is required that the following compatibility condition holds for all $(z_1,z_2,z_3)\in Z^{[3]}$, for which we set $\zeta (z_i)=: (y_i,y_i')$:

We remark that the curvature of $\mathcal {G}$ and $\mathcal {G}'$ coincide if there exists a connection-preserving isomorphism. The set of isomorphism classes of bundle gerbes with connections over M is denoted by $\textrm{Grb}^{\nabla }(M)$. This set is actually a group, whose multiplication is given by the tensor product of bundle gerbes, see [48].

Suppose we have chosen sections $s_{i}$ and $s_{ij}$ for $\mathcal {G}$ as above, and similar sections $s_i'$ and $s'_{ij}$ for $\mathcal {G}'$, with corresponding local data $(B_i,A_{ij},c_{ijk})$ and $(B_i',A'_{ij},c'_{ijk})$. After a further refinement, we may assume that $(s_{i},s_i'):U_i \rightarrow Y \times _M Y'$ lifts to Z, i.e., we may choose $t_i: U_i \rightarrow Z$ such that $\zeta \circ t_i=(s_i,s_i')$. We may then assume that $t_i^{*}Q$ admits a local section $u_i$, with corresponding 1-forms $C_i$. Note that $B_i'=B_i+{\textrm{d}}C_i$. There exists a unique smooth map $d_{ij}:U_i \cap U_j \rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} \chi (s_{ij}(x)\otimes u_j(x))\cdot d_{ij}(x) = u_i(x) \otimes s'_{ij}(x). \end{aligned}$$

This implies an equality

$$\begin{aligned} A'_{ij} = A_{ij}+C_j-C_i+d_{ij}^{*}\theta . \end{aligned}$$

Finally, the compatibility condition yields an equality

$$\begin{aligned} d_{ik}\cdot c_{ijk}=d_{ij}\cdot d_{jk}\cdot c_{ijk}'. \end{aligned}$$

The data $(C_i,d_{ij})$ constitute an equivalence between the Deligne 2-cocycles $(B_i,A_{ij},c_{ijk})$ and $(B'_i,A'_{ij},c'_{ijk})$.

This establishes an isomorphism $\textrm{Grb}^{\nabla }(M) \cong \hat{{\textrm{H}}}^3(M)$ between the set of isomorphism classes of bundle gerbes with connection and degree three differential cohomology [40, 45].

Definition 2.3

Suppose $\mathcal {G}$ and $\mathcal {G}'$ are bundle gerbes with connection, and suppose that $\mathcal {A}_1,\mathcal {A}_2: \mathcal {G} \rightarrow \mathcal {G}'$ are connection-preserving isomorphisms. A connection-preserving 2-isomorphism $\eta : \mathcal {A}_1 \Rightarrow \mathcal {A}_2$ is an equivalence class of triples $(W,\omega ,\eta )$, where is a surjective submersion, and $\eta : \omega ^{*}{\textrm{pr}}_{Z_1}^{*}Q_1 \rightarrow \omega ^{*}{\textrm{pr}}_{Z_2}^{*}Q_2$ is a connection-preserving bundle isomorphism. It is required that for all $(w,w')\in W \times _M W$ the following diagram is commutative:

where $\omega (w)=:(z_1,z_2)$ and $\omega (w')=:(z_1',z_2')$, as well as $\zeta _i(z_i)=:(y_1,y_1')$ and $\zeta _i(z_i')=:(y_2,y_2')$. Two triples are equivalent if their bundle isomorphisms coincide when pulled back to a common refinement.

Concerning local data, we may assume that the sections $t_{1,i}:U_i \rightarrow Z_1$ and $t_{2,i}: U_i \rightarrow Z_2$ lift to W, i.e., that there are sections $v_i: U_i \rightarrow W$ such that $\omega \circ v_i=(t_{1,i},t_{2,i})$. Then, $v_i^{*}\eta : t_{1,i}^{*}Q_1 \rightarrow t_{2,i}^{*}Q_2$ is a connection-preserving bundle isomorphism, and there exists a unique smooth map $z_i: U_i \rightarrow {\mathbb {T}}$ such that $v_i^{*}\eta (u_{1,i}(x))\cdot z_i(x)=u_{2,i}(x)$. This yields an equality $C_{2,i} = C_{1,i} + z_i^{*}\theta$. The diagram leads to $d_{1,ij} \cdot z_i = z_j \cdot d_{2,ij}$.

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 $\mathcal {H}\!om(\mathcal {G},\mathcal {G}')$. There is a (horizontal) composition functor

$$\begin{aligned} \mathcal {H}\!om(\mathcal {G}'',\mathcal {G}') \times \mathcal {H}\!om(\mathcal {G}',\mathcal {G}) \rightarrow \mathcal {H}\!om(\mathcal {G},\mathcal {G}'') \end{aligned}$$

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.

Proposition 2.4

[48, Prop. 4]. There is a canonical equivalence of categories

where the right hand side denotes the category of principal ${\mathbb {T}}$-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.

2.2 Poincaré Bundles and Equivariance

We summarize some required facts about the Poincaré bundle, also see [42, Appendix B]. We write ${\mathbb {T}}^{n}:= {\mathbb {R}}^{n}/{\mathbb {Z}}^{n}$ additively, and identify its Lie algebra with ${\mathbb {R}}^{n}$, and again ${\mathbb {T}}={\mathbb {T}}^1={\mathbb {R}}/{\mathbb {Z}}$. The n-fold Poincaré bundle is the following principal ${\mathbb {T}}$-bundle $\mathcal {P}$ over ${\mathbb {T}}^{2n}={\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. Its total space is

$$\begin{aligned} \mathcal {P}:= ({\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \times {\mathbb {T}}) \;/\; \sim \end{aligned}$$

with $(a,\hat{a},t)\sim (a+m,\hat{a}+\hat{m},m\hat{a}+t)$ for all $m,\hat{m}\in {\mathbb {Z}}^{n}$ and $t\in {\mathbb {T}}$, and $m\hat{a}$ is the standard inner product. The bundle projection is $(a,\hat{a},t) \mapsto (a,\hat{a})$, and the ${\mathbb {T}}$-action is $(a,\hat{a},t)\cdot s:= (a,\hat{a},t+s)$.

For maps ${\mathbb {T}}^{p} \rightarrow {\mathbb {T}}^q$ between different tori we use a notation

$$\begin{aligned} m_{1+2,-3,5}: {\mathbb {T}}^{5} \rightarrow {\mathbb {T}}^{3},\quad (a_1,a_2,a_3,a_4,a_5) \mapsto (a_1+a_2,-a_3,a_5). \end{aligned}$$

For pullbacks, we then write $(\ldots )_{1+2,-3,5}$ instead of $m_{1+2,-3,5}^{*}(\ldots )$. The following structures and properties are straightforward to check.

• (a) The following maps are well-defined bundle isomorphisms over ${\mathbb {T}}^{3n}$:

  $$\begin{aligned} \varphi _l:\mathcal {P}_{1,3} \otimes \mathcal {P}_{2,3} \rightarrow \mathcal {P}_{1+2,3}&: (a,c,t) \otimes (b,c',s) \mapsto (a+b,c,s+t) \\ \varphi _r:\mathcal {P}_{1,2} \otimes \mathcal {P}_{1,3} \rightarrow \mathcal {P}_{1,2+3}&: (a,b,t) \otimes (a',c,s) \mapsto (a,b+c,(a-a')c+s+t) \end{aligned}$$

  They express that the Poincaré bundle is “bilinear” in the two factors ${\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. Using the given formulas, one can check that $\varphi _l$ satisfies the following associativity condition:

  

  (2.3)

  An analogous condition holds for $\varphi _r$. Another compatibility condition that one can easily check is the commutativity of the following pentagon diagram:

  

  (2.4)

• (b) The map

  $$\begin{aligned} \chi _l: {\mathbb {R}}^{n} \times {\mathbb {T}}^{n} \rightarrow \mathcal {P}: (a,\hat{a}) \mapsto (a,\hat{a},0) \end{aligned}$$

  (2.5)

  is a well-defined section along ${\mathbb {R}}^{n} \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$, and the map

  $$\begin{aligned} \chi _r: {\mathbb {T}}^{n} \times {\mathbb {R}}^{n} \rightarrow \mathcal {P}: (a,\hat{a}) \mapsto (a,\hat{a},a\hat{a}) \end{aligned}$$

  is a well-defined section along ${\mathbb {T}}^{n} \times {\mathbb {R}}^{n} \rightarrow {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. These restrict further to sections along the inclusion ${\mathbb {T}}^{n} \rightarrow {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$ into one of the factors. The transition functions w.r.t. these sections are the following. Suppose $a,a'\in {\mathbb {R}}^{n}$ such that $m:=a'-a\in {\mathbb {Z}}^{n}$ and $\hat{a} \in {\mathbb {T}}^{n}$. Then,

  $$\begin{aligned} \chi _l(a,\hat{a}) = \chi _l(a',\hat{a})\cdot m\hat{a}. \end{aligned}$$

  (2.6)

  If $a\in {\mathbb {T}}^{n}$ and $\hat{a}, \hat{a}'\in {\mathbb {R}}^{n}$ with $\hat{m}:=\hat{a}'-\hat{a}\in {\mathbb {Z}}^{n}$, then we have

  $$\begin{aligned} \chi _r(a,\hat{a})=\chi _r(a,\hat{a}')\cdot a\hat{m}. \end{aligned}$$

  Over ${\mathbb {R}}^{n} \times {\mathbb {R}}^{n}$ the sections $\chi _l$ and $\chi _r$ do not coincide, but differ by the ${\mathbb {T}}$-valued function $m:{\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \rightarrow {\mathbb {T}}$, $m(a,\hat{a}):= a\hat{a}$, i.e., $\chi _r = \chi _l \cdot m$. They do coincide when pulled back to ${\mathbb {Z}}^{n} \times {\mathbb {Z}}^{n}$.

• (c) We recall that the dual $\mathcal {P}^{\vee }$ of a principal ${\mathbb {T}}$-bundle $\mathcal {P}$ has the same total space but ${\mathbb {T}}$ acting through inverses. The map

  $$\begin{aligned} \lambda :\mathcal {P}\rightarrow \mathcal {P}_{2,1}^{\vee }:(a,\hat{a},t) \mapsto (\hat{a},a,a\hat{a}-t) \end{aligned}$$

  is a well-defined bundle isomorphism over the identity of ${\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. It expresses that the Poincaré bundle is “skew-symmetric”. We have $\lambda ^2={\textrm{id}}$. Moreover, the isomorphism $\lambda$ exchanges $\chi _l$ with $\chi _r$.

• (d) The Poincaré bundle $\mathcal {P}$ carries a canonical connection $\omega$, which descends from the 1-form $\tilde{\omega }\in \Omega ^1({\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \times {\mathbb {T}})$ defined by $\tilde{\omega }:= -a {\textrm{d}}\hat{a}+{\textrm{d}}t$. It is straightforward to see that the isomorphisms $\varphi _l$ and $\varphi _r$ are connection-preserving. Moreover, the sections $\chi _l$ and $\chi _r$ have covariant derivatives

  $$\begin{aligned} \chi _l^{*}\omega =-a{\textrm{d}}\hat{a} \quad \text { and }\quad \chi _r^{*}\omega = -a{\textrm{d}}\hat{a}+{\textrm{d}}(a\hat{a})=\hat{a}{\textrm{d}}a. \end{aligned}$$

  (2.7)

  In other words, they establish trivializations $\chi _l^{*}\mathcal {P}\cong \textbf{I}_{-a{\textrm{d}}\hat{a}}$ over ${\mathbb {R}}^{n} \times {\mathbb {T}}^{n}$ and $\chi _r^{*}\mathcal {P}\cong \textbf{I}_{\hat{a}{\textrm{d}}a}$ over ${\mathbb {T}}^{n} \times {\mathbb {R}}^{n}$.

Remark 2.5

• (i) The curvature of $\omega$ is

  $$\begin{aligned} \Omega :=\theta _2 \dot{\wedge }\theta _1 \in \Omega ^2({\mathbb {T}}^{2n}), \end{aligned}$$

  where $\theta \in \Omega ^1({\mathbb {T}}^{n},{\mathbb {R}}^{n})$ is the Maurer–Cartan form, and $\dot{\wedge }$ denotes the wedge product of ${\mathbb {R}}^{n}$-valued forms using the standard inner product of ${\mathbb {R}}^{n}$ on the values.

• (ii) Note that $\Omega _{1,3}+\Omega _{2,3}=\Omega _{1+2,3}$ and $\Omega _{1,2}+\Omega _{1,3}=\Omega _{1,2+3}$, as well as $\Omega _{1,-2}=-\Omega _{1,2}=\Omega _{-1,2}$.

• (iii) An identity expressing the 2-form $\Omega$ in terms of the Maurer–Cartan form on ${\mathbb {T}}$ is

  $$\begin{aligned} \Omega = \sum _{i=1}^{n} {\textrm{pr}}_{i+n}^{*}\theta \wedge {\textrm{pr}}_{i}^{*}\theta \in \Omega ^2({\mathbb {T}}^{2n}). \end{aligned}$$

  Since ${\textrm{H}}^{*}({\mathbb {T}}^{2n},{\mathbb {Z}})$ is torsion free, this shows that the first Chern class of $\mathcal {P}$ is

  $$\begin{aligned} \sum _{i=1}^{n}{\textrm{pr}}_{i+n} \cup {\textrm{pr}}_i \in {\textrm{H}}^2({\mathbb {T}}^{2n},{\mathbb {Z}}) \end{aligned}$$

  where ${\textrm{pr}}_i: {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}$ is regarded as a representative for $[{\mathbb {T}}^{2n},{\mathbb {T}}]={\textrm{H}}^1({\mathbb {T}}^{2n},{\mathbb {Z}})$.

The following discussion concerns the quite difficult equivariance properties of the Poincaré bundle and its connection. They will be used only in Sect. 5, so that the reader may also continue with Sects. 3 and 4 first.

We remark that the curvature form $\Omega$ is ${\mathbb {T}}^{2n}$-invariant. However, the Poincaré bundle itself is not equivariant with respect to left or right multiplication of ${\mathbb {T}}^{2n}$ on itself. We construct a connection-preserving isomorphism

$$\begin{aligned} R^{l}: \mathcal {P}_{1+3,2+4} \rightarrow \mathcal {P}_{3,4} \otimes \textbf{I}_{\psi ^{l}} \end{aligned}$$

(2.8)

over ${\mathbb {R}}^{2n} \times {\mathbb {T}}^{2n}={\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \times {\mathbb {T}}^{n} \times {\mathbb {T}}^n$ in the following way, where $\psi ^{l}\in \Omega ^1({\mathbb {R}}^{2n}\times {\mathbb {T}}^{2n})$ is given at a point $((x,\hat{x}),(a,\hat{a}))$ by

$$\begin{aligned} \psi ^{l} := -x{\textrm{d}}\hat{x} - x{\textrm{d}}\hat{a}+\hat{x}{\textrm{d}}a. \end{aligned}$$

(2.9)

Indeed, we have, using $\varphi _l$ and $\varphi _r$, an isomorphism

$$\begin{aligned} \mathcal {P}_{1+3,2+4} \cong \mathcal {P}_{1,2} \otimes \mathcal {P}_{1,4} \otimes \mathcal {P}_{3,2} \otimes \mathcal {P}_{3,4} \end{aligned}$$

and due to (2.4) it does not matter how in which order these are used. In the next step we use the sections $\chi _l$ and $\chi _r$, and now we see that for the tensor factor $\mathcal {P}_{1,2}$ over ${\mathbb {R}}^{2n}$ it does matter how to trivialize it. Using $\chi _l$, we obtain

$$\begin{aligned} \mathcal {P}_{x,\hat{x}} \otimes \mathcal {P}_{x,\hat{a}} \otimes \mathcal {P}_{a,\hat{x}} \otimes \mathcal {P}_{a,\hat{a}}\cong \textbf{I}_{-x{\textrm{d}}\hat{x} - x{\textrm{d}}\hat{a}+\hat{x}{\textrm{d}}a} \otimes \mathcal {P}_{a,\hat{a}}= \textbf{I}_{\psi ^{l}}|_{(x,\hat{x}),(a,\hat{a})} \otimes \mathcal {P}_{a,\hat{a}}, \end{aligned}$$

all together resulting in the isomorphism (2.8). Explicitly,

$$\begin{aligned} R^{l}_{(x,\hat{x}),(a,\hat{a})}(x+a,\hat{x}+\hat{a},t)= (a,\hat{a},t-a\hat{x}). \end{aligned}$$

(2.10)

One might be tempted to assume that $R^{l}$ establishes an action of ${\mathbb {R}}^{2n}$ on $\mathcal {P}$ covering the action of ${\mathbb {R}}^{2n}$ on ${\mathbb {T}}^{2n}$, but this is not true. Instead, “acting twice” gives the formula

$$\begin{aligned} R^{l}_{(x_1,\hat{x}_1),(a,\hat{a})} \circ R^{l}_{(x_2,\hat{x}_2),(x_1+a,\hat{x}_1+\hat{a})} =R^{l}_{(x_2+x_1,\hat{x}_2+\hat{x}_1),(a,\hat{a})} \cdot (x_1\hat{x}_2)^{-1} \end{aligned}$$

(2.11)

exhibiting an error term which should not be present when $R^{l}$ were an action. If one uses $\chi _r$ instead of $\chi _l$, one obtains an isomorphism

$$\begin{aligned} R^{r}: \mathcal {P}_{1+3,2+4} \rightarrow \textbf{I}_{\psi ^{r}} \otimes \mathcal {P}_{3,4} \end{aligned}$$

(2.12)

for

$$\begin{aligned} \psi ^{r} := \hat{x}{\textrm{d}} x - x{\textrm{d}}\hat{a}+\hat{x}{\textrm{d}}a= \psi ^{l} + {\textrm{d}}(x \hat{x}). \end{aligned}$$

Remark 2.6

Restricting to ${\mathbb {Z}}^{n}\subseteq {\mathbb {R}}^{n}$, we have

$$\begin{aligned} \psi ^{l}_{(m,\hat{m}),(a,\hat{a})} =\psi ^{r}_{(m,\hat{m}),(a,\hat{a})} \quad \text { and }\quad R^{l}_{(m,\hat{m}),(a,\hat{a})}=R^{r}_{(m,\hat{m}),(a,\hat{a})} \end{aligned}$$

the latter corresponding to the automorphism of $\mathcal {P}$ given by multiplication with the map

$$\begin{aligned} f_{m,\hat{m}}:{\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}: (a, \hat{a}) \mapsto m\hat{a}-\hat{m} a. \end{aligned}$$

In particular, we obtain from (2.11)

$$\begin{aligned} R^{l}_{(x+m,\hat{x}+\hat{m}),(a,\hat{a})}= R^{l}_{(x,\hat{x}),(m+a,\hat{m}+\hat{a})} \cdot f_{m,\hat{m}}\cdot (m\hat{x}). \end{aligned}$$

(2.13)

2.3 Invariant Metrics on Principal Bundles

We review the mathematical basis of Kaluza–Klein theory, summarized in the following Theorem 2.7. Its relevance for T-duality has been recognized already in Buscher’s first paper [14] on the subject. General discussions can be found in [7, §9.3], [17] or [16, §3.4]; the latter reference contains a complete proof.

Theorem 2.7

Suppose that E is a principal H-bundle over X. Then, there is a bijection

under which g corresponds to $(\omega ,g',h)$ if and only if

$$\begin{aligned} g_e = \begin{pmatrix} g'_{x} &{}\quad 0 \\ 0 &{}\quad h_x \\ \end{pmatrix}, \end{aligned}$$

(2.14)

where $e\in E$ sits in the fiber over $x\in X$, and the matrix on the right hand side refers to the decomposition $T_eE \cong T_xM \oplus \mathfrak {h}$ induced by the connection $\omega$.

The connection $\omega$ that appears on the right hand side will be called the Kaluza–Klein connection associated to the metric g.

Remark 2.8

Theorem 2.7 is compatible with bundle isomorphisms: for a bundle isomorphism $\varphi :E_1 \rightarrow E_2$ it is equivalent to be isometric for the metrics on $E_1$ and $E_2$ or to be connection-preserving w.r.t. the Kaluza–Klein connections $\omega _1$ and $\omega _2$.

Remark 2.9

If the principal H-bundle is trivial, $E=\textbf{I}= X \times H$, we identify connections $\omega$ on E with $\mathfrak {h}$-valued 1-forms $A\in \Omega ^1(X,\mathfrak {h})$ as in (2.1). Under this identification, a metric g corresponds to a triple $(A,g',h)$ under the bijection of Theorem 2.7 if and only if

$$\begin{aligned} g_{x,h} = \begin{pmatrix} g' + h_x(\textrm{Ad}_h^{-1}(A_x(-)),\textrm{Ad}_h^{-1}(A_x(-))) &{}\quad h_x(\textrm{Ad}_h^{-1}(A_x(-)),-) \\ h_x(-,\textrm{Ad}_h^{-1}(A_x(-))) &{}\quad h_x(-,-) \\ \end{pmatrix}, \end{aligned}$$

(2.15)

where now the decomposition on the right hand side refers to the equality $T_{(x,h)}E = T_xM \oplus \mathfrak {h}$ induced by the direct product structure of E and the identification $T_ hH \cong \mathfrak {h}$ via left multiplication, see [17, p. 101].

Remark 2.10

In later applications of Theorem 2.7, H will be a torus, $H={\mathbb {T}}^{n}$. In particular, H is abelian. In this case, the $\textrm{Ad}$-invariance in Theorem 2.7 is vacuous. Moreover, (2.15) reduces to

$$\begin{aligned} g_{x,h} = \begin{pmatrix} g'_x + h_x(A_x(-),A_x(-)) &{}\quad h_x(A_x(-),-) \\ h_x(-,A_x(-)) &{}\quad h_x(-,-) \\ \end{pmatrix}. \end{aligned}$$

In the literature, this is sometimes written as

$$\begin{aligned} g = g' + A \odot A, \end{aligned}$$

where, unfortunately, $h_x$ is suppressed or assumed to be constant.

3 Buscher Rules Revisited

The Buscher rules formulate the local behavior 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 Sect. 4.

3.1 Buscher Rules for Toroidal Symmetries

We first recall the classical Buscher rules for a ${\mathbb {T}}^n$-symmetry. The case $n=1$ has been treated by Buscher in [14]. Rules for higher n are described, e.g., in [25], also see [10] for a review.

The Buscher rules apply to a manifold $E={\mathbb {R}}^s \times {\mathbb {T}}^n$. On E we consider a Riemannian ${\mathbb {T}}^n$-invariant metric g and a ${\mathbb {T}}^n$-invariant 2-form B, the “B-field”. With respect to the standard coordinates, we may identify g with a block matrix

$$\begin{aligned} g= \begin{pmatrix} g_{bas} &{}\quad g_{mix} \\ g_{mix}^{\textrm{tr}} &{}\quad g_{fib} \\ \end{pmatrix} \end{aligned}$$

with symmetric matrices $g_{bas}\in C^{\infty }({\mathbb {R}}^{s})^{s \times s}$ and $g_{fib}\in C^{\infty }({\mathbb {R}}^{s})^{n \times n}$, and an arbitrary matrix $g_{mix}\in C^{\infty }({\mathbb {R}}^{s})^{s \times n}$. We also identity B with a block matrix

$$\begin{aligned} B=\begin{pmatrix} B_{bas} &{}\quad B_{mix} \\ -B_{mix}^{\textrm{tr}} &{}\quad B_{fib} \\ \end{pmatrix} \end{aligned}$$

where $B_{bas}\in C^{\infty }({\mathbb {R}}^s)^{s\times s}$ and $B_{fib}\in C^{\infty }({\mathbb {R}}^{s})^{n \times n}$ are skew-symmetric and $B_{mix}\in C^{\infty }({\mathbb {R}}^s)^{s \times n}$ is arbitrary. We make the assumption that $B_{fib}=0$, so that the B-field must not have components purely in fiber direction. Note that for $n=1$ this is automatic. Pairs (g, B) with the condition $B_{fib}=0$ will be called Buscher pairs.

To proceed, we form the “background” matrix

$$\begin{aligned} Q = \begin{pmatrix} Q_{bas} &{}\quad Q_{mix} \\ Q_{mix}' &{}\quad Q_{fib} \\ \end{pmatrix} := g+B. \end{aligned}$$

The dualization process requires to form the “dual” matrix

$$\begin{aligned} \hat{Q} := \begin{pmatrix} Q_{bas} - Q_{mix}Q_{fib}^{-1}Q'_{mix} &{}\quad Q_{mix}Q_{fib}^{-1} \\ -Q_{fib}^{-1}Q'_{mix} &{}\quad Q_{fib}^{-1} \\ \end{pmatrix}. \end{aligned}$$

Dual metric and B-field are now obtained by taking the symmetric and anti-symmetric parts of $\hat{Q}$, respectively, i.e.,

$$\begin{aligned} \hat{g} := \frac{1}{2}(\hat{Q} + \hat{Q}^{\textrm{tr}}) \quad \text { and }\quad \hat{B} := \frac{1}{2}(\hat{Q} - \hat{Q}^{\textrm{tr}}). \end{aligned}$$

A standard calculation shows that $(\hat{g},\hat{B})$ is again a Buscher pair and that the relation between the Buscher pairs (g, B) and $(\hat{g},\hat{B})$ is described by the following equations:

$$\begin{aligned} \hat{g}_{bas}&= g_{bas} -g_{mix} g_{fib}^{-1}g_{mix}^{\textrm{tr}} + B_{mix}g_{fib}^{-1}B_{mix}^{\textrm{tr}} \end{aligned}$$

(3.1)

$$\begin{aligned} \hat{g}_{mix}&= B_{mix}g_{fib}^{-1} \end{aligned}$$

(3.2)

$$\begin{aligned} \hat{g}_{fib}&= g_{fib}^{-1} \end{aligned}$$

(3.3)

$$\begin{aligned} \hat{B}_{bas}&= B_{bas} - B_{mix}g_{fib}^{-1}g_{mix}^{\textrm{tr}}+g_{mix} g_{fib}^{-1}B_{mix}^{\textrm{tr}} \end{aligned}$$

(3.4)

$$\begin{aligned} \hat{B}_{mix}&= g_{mix}g_{fib}^{-1} \end{aligned}$$

(3.5)

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 that $\hat{\hat{Q}} = Q$, implying that the Buscher rules are symmetric in the data. For completeness, let us fix the following definition.

Definition 3.1

Two Buscher pairs (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules if (3.1) to (3.5) are satisfied.

3.2 Buscher Rules in Terms of Kaluza–Klein Connections

We consider again a metric g on $E={\mathbb {R}}^{s} \times {\mathbb {T}}^{n}$, and consider E as a principal ${\mathbb {T}}^{n}$-bundle over ${\mathbb {R}}^{s}$. We apply Theorem 2.7 and Remark 2.9, to obtain a triple $(A,g',h)$ consisting of a Riemannian metric $g'$ on ${\mathbb {R}}^{s}$, a 1-form $A\in \Omega ^1({\mathbb {R}}^{s},{\mathbb {R}}^{n})$, and a family h of inner products on ${\mathbb {R}}^{n}$ parameterized by ${\mathbb {R}}^{s}$. Now we consider Buscher quadruples $(A,g',h,B)$ instead of Buscher pairs (g, B). By Theorem 2.7 there is a bijection between Buscher quadruples and Buscher pairs.

The expression for the metric g given in Remark 2.10 now reads

$$\begin{aligned} g = \begin{pmatrix} g'+ A^{\textrm{tr}}hA &{}\quad A^{\textrm{tr}}h \\ hA &{}\quad h \\ \end{pmatrix}. \end{aligned}$$

In other words, we have

$$\begin{aligned} g_{bas} = g' + A^{\textrm{tr}}hA \quad \text {, }\quad g_{mix} = A^{\textrm{tr}}h \quad \text { and }\quad g_{fib} = h. \end{aligned}$$

We employ the same procedure on the dual side, getting

$$\begin{aligned} \hat{g}_{bas} = \hat{g}' + \hat{A}^{\textrm{tr}}\hat{h}\hat{A} \quad \text {, }\quad \hat{g}_{mix} = \hat{A}^{\textrm{tr}}\hat{h} \quad \text { and }\quad \hat{g}_{fib} = \hat{h}. \end{aligned}$$

The Buscher rules now attain the following simple form:

$$\begin{aligned} \hat{g}'&= g' \end{aligned}$$

(3.6)

$$\begin{aligned} \hat{A}^{\textrm{tr}}&= B_{mix} \end{aligned}$$

(3.7)

$$\begin{aligned} \hat{h}&= h^{-1} \end{aligned}$$

(3.8)

$$\begin{aligned} \hat{B}_{bas}&= B_{bas} - B_{mix}A+(B_{mix}A)^{\textrm{tr}} \end{aligned}$$

(3.9)

$$\begin{aligned} \hat{B}_{mix}&= A^{\textrm{tr}} \end{aligned}$$

(3.10)

Again for completeness, we fix the following definition and result.

Definition 3.2

Two Buscher quadruples $(A,g',h,B)$ and $(\hat{A},\hat{g}',\hat{h},\hat{B})$ satisfy the Buscher rules, if the (3.6) to (3.10) are satisfied.

Lemma 3.3

Under the bijection between Buscher pairs and Buscher quadruples, the Buscher rules of Definitions 3.1 and 3.2 are equivalent.

3.3 Buscher Rules in Terms of Poincaré Forms

Next we want to give a coordinate-independent description of the Buscher rules of Definition 3.2, which will again make them simpler. Let $\omega ,\hat{\omega } \in \Omega ^1({\mathbb {R}}^{s} \times {\mathbb {T}}^{n},{\mathbb {R}}^{n})$ be the Kaluza–Klein connections on $E={\mathbb {R}}^{s}\times {\mathbb {T}}^{n}$ corresponding to A and $\hat{A}$, respectively, i.e., $\omega := A_1 + \theta _2$ and $\hat{\omega }:= \hat{A}_1 + \theta _2$. Here, the indices refer to the pullback along the projections to the two factors, as explained in Sect. 2.2. We introduce the 2-form

$$\begin{aligned} \rho := \hat{{\textrm{pr}}}^{*}\hat{\omega } \dot{\wedge }{\textrm{pr}}^{*}\omega \in \Omega ^2({\mathbb {R}}^{s} \times {\mathbb {T}}^{2n}), \end{aligned}$$

where the symbol $\dot{\wedge }$ means that the standard scalar product on ${\mathbb {R}}^{n}$ is used in the values the forms.

Lemma 3.4

Buscher quadruples $(g',A,h,B)$ and $(\hat{g}',\hat{A},\hat{h},\hat{B})$ satisfy the Buscher rules of Definition 3.2 if and only if the following conditions are satisfied:

• (a) $\hat{g}'=g'$

• (b) $\hat{h} = h^{-1}$

• (c) $\hat{{\textrm{pr}}}^{*}\hat{B} - {\textrm{pr}}^{*}B = {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\Omega -\rho$.

Proof

(a) and (b) are (3.6) and (3.8). We have

$$\begin{aligned} -\rho _{x,t,\hat{t}}= & {} \omega _{x,t} \dot{\wedge }\hat{\omega }_{x,\hat{t}} =( A_{x} + \theta _t^{{\mathbb {T}}^{n}})\dot{\wedge }( \hat{A}_{x} + \theta _{\hat{t}}^{{\mathbb {T}}^{n}})\\ {}= & {} A_x \dot{\wedge }\hat{A}_x + A_x\dot{\wedge }\theta _{\hat{t}}^{{\mathbb {T}}^{n}} + \theta _t^{{\mathbb {T}}^{n}}\dot{\wedge }\hat{A}_x +\theta _t^{{\mathbb {T}}^{n}} \dot{\wedge }\theta _{\hat{t}}^{{\mathbb {T}}^{n}}. \end{aligned}$$

We change to coordinates w.r.t. ${\mathbb {R}}^{s} \times {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$, which we label by i, $\mu$, and $\hat{\mu }$. Then, we obtain

$$\begin{aligned}{} & {} -\rho (e_i,e_j)=A_i\hat{A}_j - A_j\hat{A}_i, -\rho (e_i,e_{\mu }) = -\hat{A}_{\mu i}, \\{} & {} \quad -\rho (e_i,e_{\hat{\mu }}) =A_{\hat{\mu } i} \quad \text { and }\quad -\rho (e_{\mu },e_{\hat{\nu }})=\delta _{\mu \hat{\nu }} \end{aligned}$$

Note that

$$\begin{aligned}{} & {} (B_{bas})_{ij} = ({\textrm{pr}}^{*}B)(e_i,e_j), (B_{mix})_{i\mu } = ({\textrm{pr}}^{*}B)(e_i,e_{\mu }) \quad \text { and }\quad \\{} & {} \quad (B_{fib})_{\mu \nu } = ({\textrm{pr}}^{*}B)(e_{\mu },e_{\nu }) \end{aligned}$$

and similarly

$$\begin{aligned}{} & {} (\hat{B}_{bas})_{ij} = (\hat{{\textrm{pr}}}^{*}\hat{B})(e_i,e_j), (\hat{B}_{mix})_{i\hat{\mu }} = (\hat{{\textrm{pr}}}^{*}\hat{B})(e_i,e_{\hat{\mu }}) \quad \text { and }\quad \\{} & {} \quad (\hat{B}_{fib})_{\hat{\mu }\hat{\nu }}= (\hat{{\textrm{pr}}}^{*}\hat{B})(e_{\hat{\mu }},e_{\hat{\nu }}), \end{aligned}$$

and all other components vanish. Further, we have

$$\begin{aligned} ({\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\Omega )(e_{\mu },e_{\hat{\nu }}) = -\delta _{\mu \hat{\nu }}, \end{aligned}$$

with again all other components vanishing. Thus, (c) is equivalent to the following set of equations:

$$\begin{aligned} (\hat{B}_{bas})_{ij} - (B_{bas})_{ij}&= A_i\hat{A}_j - A_j\hat{A}_i \\ (B_{mix})_{i\mu }&=\hat{A}_{\mu i} \\ (\hat{B}_{mix})_{i\hat{\mu }}&= A_{\hat{\mu } i} \end{aligned}$$

The second and third equation are (3.7) and (3.10). The first equation, using second and third, is equivalent to

$$\begin{aligned} \hat{B}_{bas} - B_{bas} = (B_{mix}A)^{\textrm{tr}} - B_{mix}A \end{aligned}$$

and this is precisely (3.9).

A straightforward computation using Lemma 3.3.1 (c) shows the following.

Lemma 3.5

Suppose $(A,g',h,B)$ and $(\hat{A},\hat{g}',\hat{h},\hat{B})$ are Buscher quadruples satisfying the Buscher rules of Definition 3.2. Then, we have

$$\begin{aligned} B_{1,2+3}&= B_{1,2}+ \hat{A}_1\wedge \theta _3 \end{aligned}$$

(3.11)

$$\begin{aligned} \hat{B}_{1,2+3}&= \hat{B}_{1,2}+ A_1 \wedge \theta _3 \end{aligned}$$

(3.12)

over ${\mathbb {R}}^{s} \times {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. In particular, B and $\hat{B}$ are ${\mathbb {T}}^{n}$-invariant.

4 Geometric T-Duality

In this section, we give the central definitions of this article: we introduce geometric T-backgrounds (Definition 4.1) and geometric T-duality correspondences between them (Definition 4.9). 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.

4.1 Basic Definitions

Definition 4.1

A geometric T-background over a smooth manifold X is a triple $(E,g,\mathcal {G})$ consisting of a principal ${\mathbb {T}}^{n}$-bundle E over X, a ${\mathbb {T}}^{n}$-invariant Riemannian metric g on E, and a bundle gerbe $\mathcal {G}$ over E with connection. Two geometric T-backgrounds $(E_1,g_1,\mathcal {G}_1)$ and $(E_2,g_2,\mathcal {G}_2)$ over X are equivalent, if there exists a bundle isomorphism $f:E_1 \rightarrow E_2$ that is isometric with respect to the metrics $g_1$ and $g_2$, and a connection-preserving bundle gerbe isomorphism $\mathcal {G}_1 \cong f^{*}\mathcal {G}_2$. The set of equivalence classes of geometric T-backgrounds over X is denoted by ${{\mathrm {T-BG}}}^{{\textrm{geo}}}(X)$.

As every bundle gerbe with connection has a curvature 3-form, every geometric T-background carries a 3-form $H\in \Omega ^3(E)$, 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 $(E,g,\mathcal {G})$ and $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ are geometric T-backgrounds over the same manifold X, then the principal ${\mathbb {T}}^{2n}$-bundle $E \times _X \hat{E}$ is called the correspondence space. It fits into an important commutative diagram:

Let $\omega \in \Omega ^1(E,{\mathbb {R}}^{n})$ and $\hat{\omega }\in \Omega ^1(\hat{E},{\mathbb {R}}^{n})$ be the Kaluza–Klein connections of the metrics g and $\hat{g}$, respectively, under Theorem 2.7. Then, we consider the 2-form

$$\begin{aligned} \rho _{g,\hat{g}} := \hat{{\textrm{pr}}}^{*}\hat{\omega } \dot{\wedge }{\textrm{pr}}^{*}\omega \in \Omega ^2(E \times _X \hat{E}), \end{aligned}$$

(4.1)

where $\dot{\wedge }$ denotes the wedge product of ${\mathbb {R}}^{n}$-valued forms w.r.t. the standard inner product. Since $\omega$ and $\hat{\omega }$ are ${\mathbb {T}}^{n}$-invariant (they are connections on a principal bundle with abelian structure group), the 2-form $\rho _{g,\hat{g}}$ is ${\mathbb {T}}^{2n}$-invariant. We remark that the 2-form $\rho _{g,\hat{g}}$ also appeared in [5, 30].

Definition 4.2

A geometric correspondence over X consists of two geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ over X, and a connection-preserving bundle gerbe isomorphism

$$\begin{aligned} \mathcal {D} : {\textrm{pr}}^{*}\mathcal {G} \rightarrow \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \otimes \mathcal {I}_{\rho _{g,\hat{g}}}. \end{aligned}$$

Remark 4.3

We shall explore some consequences of the isomorphism $\mathcal {D}$ in a geometric correspondence. For this, we will denote by $F,\hat{F} \in \Omega ^2(X)$ the curvatures of the connections $\omega$ and $\hat{\omega }$, respectively.

• (a) Since the curvatures of isomorphic bundle gerbes with connection coincide, we have

  $$\begin{aligned} {\textrm{pr}}^{*}H - \hat{{\textrm{pr}}}^{*}\hat{H} = {\textrm{d}}\rho _{g,\hat{g}}, \end{aligned}$$

  (4.2)

  which is a condition in the context of T-duality with H-flux, see [5, Eq. 1.12] and Definition 4.24. From (4.2) and the definition of $\rho _{g,\hat{g}}$, one can deduce the equivariance rule

  $$\begin{aligned} R^{*}H=e^{*}H+e^{*}p^{*}\hat{F} \,\dot{\wedge }\, h^{*}\theta \end{aligned}$$

  (4.3)

  on $E \times {\mathbb {T}}^{n}$, where R is the principal action, e the projection to E, and h the projection to ${\mathbb {T}}^{n}$. Similarly, on the dual side we obtain

  $$\begin{aligned} R^{*}\hat{H}=\hat{e}^{*}\hat{H}+\hat{e}^{*}\hat{p}^{*}F \wedge h^{*}\theta \end{aligned}$$

  (4.4)

  on $\hat{E} \times {\mathbb {T}}^{n}$. In particular, these formulas show that H and $\hat{H}$ are ${\mathbb {T}}^{n}$-invariant.

• (b) We consider the 3-forms

  $$\begin{aligned} \tilde{K} := \omega \dot{\wedge }\, p^{*}\hat{F} - H \in \Omega ^3(E) \quad \text { and }\quad \hat{K}:=\hat{\omega }\dot{\wedge }\,\hat{p}^{*} F - \hat{H} \in \Omega ^3(\hat{E}) \end{aligned}$$

  Using (4.3) and (4.4), one can show that $R^{*}\tilde{K}=e^{*}\tilde{K}$ and $R^{*}\hat{K}=\hat{e}^{*}\hat{K}$, so that these forms descend to X. In fact, $\tilde{K}$ and $\hat{K}$ both descend to the same 3-form $K\in \Omega ^3(X)$, i.e., $p^{*}K=\tilde{K}$ and $\hat{p}^{*}K=\hat{K}$. To see this, it suffices to note that the pullbacks of $\tilde{K}$ and $\hat{K}$ to the correspondence space coincide, which again can be checked using (4.2) and (4.1). Summarizing, every geometric correspondence determines a 3-form $K\in \Omega ^3(X)$ such that

  $$\begin{aligned} p^{*}K=\omega \dot{\wedge }\, p^{*}\hat{F}-H \quad \text { and }\quad \hat{p}^{*}K=\hat{\omega }\dot{\wedge }\,\hat{p}^{*} F - \hat{H}. \end{aligned}$$

  Note that ${\textrm{d}}K = F \dot{\wedge }\hat{F}$.

Remark 4.4

Geometric correspondence is a symmetric relation on the set ${{\mathrm {T-BG}}}^{{\textrm{geo}}}(X)$. If $\mathcal {D}$ is a correspondence from $(E,g,\mathcal {G})$ to $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$, then we construct a correspondence from $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ to $(E,g,\mathcal {G})$ as follows. Let $s: \hat{E} \times _X E \rightarrow E \times _X \hat{E}$ denote the swap map. Then, we consider

Since $-s^{*}\rho _{g,\hat{g}}=\rho _{\hat{g},g}$, this is again a geometric correspondence.

Definition 4.5

Two geometric correspondences over X, $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$ and $((E',g',\mathcal {G}'),(\hat{E}',\hat{g}',\hat{\mathcal {G}'}),\mathcal {D}')$, are considered to be equivalent, if there exist isometric bundle isomorphisms $f:E \rightarrow E'$ and $\hat{f}: \hat{E} \rightarrow \hat{E}'$, connection-preserving bundle gerbe isomorphisms $\mathcal {A}:\mathcal {G} \rightarrow f^{*}\mathcal {G}'$ and $\hat{\mathcal {A}}:\hat{\mathcal {G}} \rightarrow \hat{f}^{*}\hat{\mathcal {G}}'$, and a connection-preserving 2-isomorphism

where $F:= f \times \hat{f}:E \times _X \hat{E} \rightarrow E' \times _X \hat{E}'$. The set of equivalence classes of geometric correspondences over X is denoted by ${\textrm{Corr}}^{{\textrm{geo}}}(X)$.

Remark 4.6

In above definition, we have implicitly used that $F^{*}\rho _{g',\hat{g}'}=\rho _{g_1,\hat{g}_1}$, which follows from the fact that f and $\hat{f}$ are connection-preserving, which in turn follows from the assumption that f and $\hat{f}$ are isometric (Remark 2.8).

Remark 4.7

Let $\mathcal {H}$ be a bundle gerbe with connection over X. Then, we may send

$$\begin{aligned} ((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D}) \mapsto ((E,g,\mathcal {G}\otimes p^{*}\mathcal {H}),(\hat{E},\hat{g},\hat{\mathcal {G}} \otimes \hat{p}^{*}\mathcal {H}),\mathcal {D} \otimes {\textrm{id}}_{\mathcal {H}}). \end{aligned}$$

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,

$$\begin{aligned} \textrm{Grb}^{\nabla }(X) \times {\textrm{Corr}}^{{\textrm{geo}}}(X) \rightarrow {\textrm{Corr}}^{{\textrm{geo}}}(X). \end{aligned}$$

Remark 4.8

It is straightforward to see that equivalent geometric correspondences determine the same 3-form K. The action of Remark 4.7 shifts this 3-form by $\textrm{curv}(\mathcal {H})$.

Definition 4.9

A geometric correspondence $\mathcal {D}$ between two geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ over X is called geometric T-duality correspondence if the following conditions hold:

• (T1) The Riemannian metrics $g'$ and $\hat{g}'$ on X determined by the metrics g and $\hat{g}$, respectively, under Theorem 2.7 coincide, i.e., $g'=\hat{g}'$.

• (T2) The families of inner products h and $\hat{h}$ on ${\mathbb {R}}^{n}$ determined by the metrics g and $\hat{g}$, respectively, under Theorem 2.7, satisfy $h^{-1}=\hat{h}$ under their identification with $(n\times n)$-matrices.

• (T3) Every point $x\in X$ has an open neighborhood $U \subseteq M$ such that the following structures exist:

  • (a) Trivializations $\varphi : U \times {\mathbb {T}}^{n} \rightarrow E|_U$ and $\hat{\varphi }: U \times {\mathbb {T}}^{n} \rightarrow \hat{E}|_U$ of principal ${\mathbb {T}}^{n}$-bundles over U.

  • (b) Two 2-forms $B,\hat{B} \in \Omega ^2(U \times {\mathbb {T}}^{n})$ together with connection-preserving isomorphisms $\mathcal {T}: \varphi ^{*}\mathcal {G} \rightarrow \mathcal {I}_B$ and $\hat{\mathcal {T}}:\hat{\varphi }^{*}\hat{\mathcal {G}}\rightarrow \mathcal {I}_{\hat{B}}$ over $U \times {\mathbb {T}}^{n}$.

  • (c) Consider $U \times {\mathbb {T}}^{2n}$ with projection maps ${\textrm{pr}}, \hat{{\textrm{pr}}}$ to $U \times {\mathbb {T}}^{n}$. Further, consider the map $\Phi : U \times {\mathbb {T}}^{2n} \rightarrow E \times _X \hat{E}$ defined by $\Phi (x,a,\hat{a}):= (\varphi (x,a),\hat{\varphi }(x,\hat{a}))$. Let P denote the principal ${\mathbb {T}}$-bundle with connection over $U \times {\mathbb {T}}^{2n}$ that corresponds to the isomorphism

    

    under the equivalence of Proposition 2.4. We require a connection-preserving isomorphism

    $$\begin{aligned} P \cong {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}, \end{aligned}$$

    where $\mathcal {P}$ 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 4.5) is denoted by ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$.

Remark 4.10

If $\mathcal {D}$ is a geometric T-duality correspondence, then the inverse correspondence $s^{*}\mathcal {D}^{-1}$ of Remark 4.4 is also a geometric T-duality correspondence. Indeed, conditions (T1) and (T2) are obviously symmetric, and in (T3) it is straightforward to show that swapping s and inversion $\mathcal {D}^{-1}$ have both the effect of dualizing the bundle P. Thus, geometric T-duality is a symmetric relation on the set ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$.

Remark 4.11

It is straightforward to see that the action of Remark 4.7 restricts to an action of $\hat{{\textrm{H}}}^3(X)$ on ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$. The properties of this action are best studied in the context of differential T-duality and carried out in differential cohomology, see Proposition 6.11. The result of Proposition 6.11 is the following.

Proposition 4.12

Let

(4.5)

be the projection to the isomorphism classes of the principal ${\mathbb {T}}^{n}$-bundles E and $\hat{E}$ and their Kaluza–Klein connections $\omega$ and $\hat{\omega }$ induced by the metric g and $\hat{g}$, respectively. We denote by $(F,\hat{F}) \in \Omega ^2(X) \times \Omega ^2(X)$ the well-defined pair of curvature forms. Consider the subgroup

$$\begin{aligned} \mathcal {F}_{F,\hat{F}}:= \{ \mathcal {I}_{\hat{y} F+y\hat{F}} \;|\;y,\hat{y}\in {\mathbb {R}}\}\subseteq \textrm{Grb}^{\nabla }(X). \end{aligned}$$

Then, the quotient $\textrm{Grb}^{\nabla }(X)/\mathcal {F}_{F,\hat{F}}$ acts free and transitively in the fiber of (4.5) over an element with curvature pair $(F,\hat{F})$.

Remark 4.13

The assignments $X \mapsto {{\mathrm {T-BG}}}^{{\textrm{geo}}}(X)$ and $X \mapsto {\text {T{-}Corr}}^{{\textrm{geo}}}(X)$ are presheaves on the category of smooth manifolds. In fact, it is straightforward and only omitted for brevity to enhance the sets ${{\mathrm {T-BG}}}^{{\textrm{geo}}}(X)$ and ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$ to bicategories, which then form sheaves of bicategories on the site of smooth manifolds.

4.2 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.

Proposition 4.14

Let $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$ be a geometric T-duality correspondence. Consider an open set $U \subseteq X$ together with the structure listed in (T3) (b). Then, the Buscher pairs (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules.

Proof

Applying Proposition 2.4 to the bundle gerbe isomorphism in (T3) (c) yields

$$\begin{aligned} \hat{{\textrm{pr}}}^{*}\hat{B} - {\textrm{pr}}^{*}B = {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\Omega -\Phi ^{*}\rho _{g,\hat{g}}. \end{aligned}$$

In addition to (T1) and (T2), Lemmas 3.4 and 3.3 show that (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules.

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 $E^{s,n}:= {\mathbb {R}}^{s} \times {\mathbb {T}}^{n}$, the given metric g, and the trivial bundle gerbe $\mathcal {I}_B$. If Buscher pairs (g, B) and $(\hat{g},\hat{B})$ satisfy the Buscher rules, then we have $\hat{{\textrm{pr}}}^{*}\hat{B} -{\textrm{pr}}^{*}B+\rho _{g,\hat{g}}={\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\Omega$ by Lemma 3.4. Thus, ${\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}$ corresponds under Proposition 2.4 to a connection-preserving isomorphism

$$\begin{aligned} \mathcal {D}:{\textrm{pr}}^{*}\mathcal {I}_B \rightarrow \hat{{\textrm{pr}}}^{*}\mathcal {I}_{\hat{B}} \otimes \mathcal {I}_{\rho _{g,\hat{g}}} \end{aligned}$$

(4.6)

over the correspondence space $E^{s,n} \times _{{\mathbb {R}}^{s}} E^{s,n}$.

Proposition 4.15

Suppose (g, B) and $(\hat{g},\hat{B})$ are Buscher pairs and satisfy the Buscher rules. Then, the connection-preserving isomorphism (4.6) establishes a geometric T-duality correspondence between $(E^{s,n},g,\mathcal {I}_B)$ and $(E^{s,n},\hat{g},\mathcal {I}_{\hat{B}})$.

Proof

Conditions (T1) and (T2) of Definition 4.9 are Lemma 3.4 (a) and (b). That condition (T3) is satisfied can be seen using the identity trivializations $\varphi , \hat{\varphi }$ and $\mathcal {T},\hat{\mathcal {T}}$.

4.3 Relation to Topological T-Duality

We shall first recall the definition of topological T-duality following [11, 13, 38].

Definition 4.16

A topological T-background over X is a principal ${\mathbb {T}}^{n}$-bundle E together with a bundle gerbe $\mathcal {G}$ over E. Two topological T-backgrounds $(E,\mathcal {G})$ and $(E',\mathcal {G}')$ over X are equivalent if there exists a bundle isomorphism $\varphi : E \rightarrow E'$ and a bundle gerbe isomorphism $\mathcal {G} \cong \varphi ^{*}\mathcal {G}'$. Equivalence classes of topological T-backgrounds over X form a set ${{\mathrm {T-BG}}}^{{\textrm{top}}}(X)$.

Every geometric T-background $(E,g,\mathcal {G})$ induces a topological T-background $(E,\mathcal {G})$ by forgetting the metric and forgetting the gerbe connection. Conversely, if $(E,\mathcal {G})$ is a topological T-background, one can choose any ${\mathbb {T}}^{n}$-invariant metric on E and use the fact that every bundle gerbe admits a connection [41], to upgrade it to a geometric T-background. Thus, we have a surjective map

$$\begin{aligned} {{\mathrm {T-BG}}}^{{\textrm{geo}}}(X) \rightarrow {{\mathrm {T-BG}}}^{{\textrm{top}}}(X). \end{aligned}$$

Definition 4.17

A topological correspondence between T-backgrounds $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$ is an isomorphism $\mathcal {D}: {\textrm{pr}}^{*}\mathcal {G} \rightarrow \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}}$ over $E \times _X E$. A topological correspondence $\mathcal {D}$ is called topological T-duality correspondence, if each point $x\in X$ has an open neighborhood $U \subseteq X$ such that the following structures exist:

• (a) Trivializations $\varphi : U \times {\mathbb {T}}^{n} \rightarrow E|_U$ and $\hat{\varphi }: U \times {\mathbb {T}}^{n} \rightarrow \hat{E}|_U$ of principal ${\mathbb {T}}^{n}$-bundles over U.

• (b) Bundle gerbe isomorphisms $\mathcal {T}: \varphi ^{*}\mathcal {G} \rightarrow \mathcal {I}$ and $\hat{\mathcal {T}}:\hat{\varphi }^{*}\hat{\mathcal {G}}\rightarrow \mathcal {I}$ over $U \times {\mathbb {T}}^{n}$.

• (c) The bundle gerbe isomorphism

  

  over $U \times {\mathbb {T}}^{2n}$ corresponds under the equivalence of Proposition 2.4 to ${\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}$.

Proposition 4.18

If $\mathcal {D}$ is a geometric T-duality correspondence between two geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ over X, then $\mathcal {D}$ is a topological T-duality correspondence between the topological T-backgrounds $(E,\mathcal {G})$ and $(\hat{E}, \hat{\mathcal {G}})$.

Proof

Discarding all metrics and connections from Definitions 4.2 and 4.9 results precisely in Definition 4.16.

Corollary 4.19

If two geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E}, \hat{g}, \hat{\mathcal {G}})$ over X are in geometric T-duality correspondence, then the homomorphism

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 4.20

Every topological T-duality correspondence can be lifted to a geometric T-duality correspondence. In more detail, suppose $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$ are topological T-backgrounds, and suppose $\mathcal {D}$ is a topological T-duality correspondence. Then, there exist ${\mathbb {T}}^{n}$-equivariant metrics g and $\hat{g}$ on E and $\hat{E}$, connections on $\mathcal {G}$ and $\hat{\mathcal {G}}$ and a connection on $\mathcal {D}$ such that $\mathcal {D}$ is a geometric T-duality correspondence between $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$.

Proof

Combines Propositions 6.5 and 6.4, to be proved later using the local formalism.

Remark 4.21

It is straightforward to see that Proposition 4.18 induces a map

$$\begin{aligned} {\text {T{-}Corr}}^{{\textrm{geo}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X). \end{aligned}$$

Proposition 4.20 implies that this map is surjective.

Remark 4.22

A purely topological version of the action of Remark 4.11,

$$\begin{aligned} {\textrm{H}}^3(X) \times {\text {T{-}Corr}}^{{\textrm{top}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X) \end{aligned}$$

exists, and it obviously acts in the fibers of the map ${\text {T{-}Corr}}^{{\textrm{top}}}(X) \rightarrow \textrm{Bun}_{{\mathbb {T}}^{n}}(X) \times \textrm{Bun}_{{\mathbb {T}}^{n}}(X)$. Bunke–Rumpf–Schick have investigated a similar action in [11, §7.2].

4.4 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 [5] and Bouwknegt–Hannabuss–Mathai in [6]. 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 [15] based on [5, 6], and we will use this here.

Definition 4.23

A T-background with H-flux over X is a principal ${\mathbb {T}}^{n}$-bundle E over X together with a closed 3-form $H\in \Omega ^3(E)$ with integral periods.

Every geometric T-background $(E,g,\mathcal {G})$ induces one with H-flux where the metric g is forgotten and H is the curvature of $\mathcal {G}$. Conversely, every T-background with H-flux (E, H) can be upgraded to a geometric T-background by choosing some ${\mathbb {T}}^{n}$-invariant metric and some bundle gerbe with connection of curvature H.

For the following definition, we consider the correspondence space $E \times _X \hat{E}$, and at each point $(e,\hat{e})$, projecting to some $x\in X$, the subspaces $V_e,\hat{V}_{\hat{e}} \subseteq T_{e,\hat{e}}(E \times _X \hat{E}) = T_eE \times _{T_xX} T_{\hat{e}}\hat{E}$ obtained as the image of the maps

$$\begin{aligned}&i: {\mathbb {R}}^{n} {\mathop {\rightarrow }\limits ^{\cong }}V_eE \hookrightarrow T_eE \times _{T_xX} T_{\hat{e}}\hat{E},\quad v \mapsto (T_1R_e(v),0){} & {} V_e := \textrm{Im}(i) \\&\hat{i}: {\mathbb {R}}^{n} {\mathop {\rightarrow }\limits ^{\cong }}\hat{V}_{\hat{e}}E \hookrightarrow T_eE \times _{T_xX} T_{\hat{e}}\hat{E},\quad w \mapsto (0,T_1R_{\hat{e}}(w)){} & {} \hat{V}_{\hat{e}} := \textrm{Im}(\hat{i}). \end{aligned}$$

Definition 4.24

A T-duality correspondence with H-flux consists of two T-backgrounds with H-flux (E, H) and $(\hat{E},\hat{H})$ and a $\mathbb {T}^{2n}$-invariant 2-form $F\in \Omega ^2(E \times _X \hat{E})$ such that

• (a) ${\textrm{pr}}^{*}H-\hat{{\textrm{pr}}}^{*}\hat{H} = {\textrm{d}}F$.

• (b) The restriction of $F_{e,\hat{e}}$ to $V_e \times \hat{V}_{\hat{e}}$ is non-degenerate, for all $(e,\hat{e})\in E \times _X \hat{E}$.

Now we are in position to show that geometric T-duality reduces to T-duality with H-flux. This is a result of Kunath’s PhD thesis [34, Thm. 5.10].

Proposition 4.25

Suppose $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$ are in geometric T-duality correspondence $\mathcal {D}$. Then, $F:= \rho _{g,\hat{g}}$ defined in (4.1) is a T-duality correspondence with H-flux.

Proof

As remarked in Sect. 4.1, the 2-form $\rho _{g,\hat{g}}$ is ${\mathbb {T}}^{2n}$-invariant. The first condition is proved in Remark 4.3. For the second condition, we have

$$\begin{aligned} F_{e,\hat{e}}((T_1R_e(v),0),(0,T_1R_{\hat{e}}(w)))&= \hat{\omega }_{\hat{e}}(0)\cdot \omega _e(0)-\hat{\omega }_{\hat{e}}(T_1R_{\hat{e}}(w))\cdot \omega _e(T_1R_e(v))\\ {}&= -w\cdot v; \end{aligned}$$

i.e., we obtain (minus) the standard scalar product of ${\mathbb {R}}^{n}$, which is non-degenerate.

Remark 4.26

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 $c_1(E) \cup c_1(\hat{E}) \in {\textrm{H}}^4(X,{\mathbb {Z}})$, while a T-duality correspondence with H-flux only implies the triviality of that class in de Rham cohomology.

5 Local Perspective to Geometric T-Duality

We may see condition (T3) of Definition 4.9 as enforcing a geometric T-duality correspondence to be locally trivial. Just as for locally trivial fiber 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 Sect. 5.1. In Sect. 5.2 we organize local data in a more systematic way, establishing the table in Fig. 2 of Sect. 1. Sections 5.3 to 5.5 are devoted to a full proof of a bijection between the set ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$ of equivalence classes of geometric T-duality correspondences and a set ${\textrm{Loc}}^{{\textrm{geo}}}(X)$ of equivalence classes of local data. In Sect. 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.

5.1 Extraction of Local Data

We suppose that we have a geometric T-duality correspondence $\mathcal {D}$ as in Definition 4.9, between geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$ over X. We assume then that X is covered by open sets $U_i$ over which condition (T3) holds, and that corresponding bundle trivializations $\varphi _i,\hat{\varphi }_i$, bundle gerbe trivializations $\mathcal {T}_i,\hat{\mathcal {T}}_i$ and 2-isomorphisms $\xi _i$ are chosen for all $U_i$, where $\xi _i$ are the connection-preserving 2-isomorphisms

$$\begin{aligned} \xi _i: \hat{\mathcal {T}}_i\circ \Phi _i^{*}\mathcal {D} \circ \mathcal {T}_i^{-1} \Rightarrow {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}. \end{aligned}$$

(5.1)

Let $a_{ij}:U_i \cap U_j \rightarrow {\mathbb {T}}^{n}$ be the transition functions of E, which are determined by the trivializations $\varphi _i$ and $\varphi _j$, i.e., $\varphi _j(x,a)\cdot a_{ij}(x)=\varphi _i(x,a)$. It will soon become necessary to choose and fix lifts of these transition functions along ${\mathbb {R}}^{n} \rightarrow {\mathbb {T}}^{n}$, which is always possible after eventually passing to a refinement of the open cover. The former cocycle condition then reveals “winding numbers” $m_{ijk} \in {\mathbb {Z}}^{n}$ such that

$$\begin{aligned} a_{ij} + a_{jk} +m_{ijk}&= a_{ik} \end{aligned}$$

(5.2)

and these integers $m_{ijk}$ themselves satisfy the usual Čech cocycle condition. We will also denote by $a_{ij}$ the corresponding map

$$\begin{aligned} (U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow (U_i \cap U_j) \times {\mathbb {T}}^{n} :(x,a)\mapsto (x,a+a_{ij}(x)) \end{aligned}$$

that multiplies by $a_{ij}(x)$; note that this map satisfies $\varphi _i = \varphi _j \circ a_{ij}$. Next, we consider the composite

of bundle gerbe isomorphisms over $(U_i \cap U_j)\times {\mathbb {T}}^{n}$, which corresponds by Proposition 2.4 to a principal ${\mathbb {T}}$-bundle $L_{ij}$ over $(U_i \cap U_j) \times {\mathbb {T}}^{n}$ with connection of curvature $a_{ij}^{*}B_j - B_i$.

The same works on the dual side, resulting in transition functions $\hat{a}_{ij}:U_i \cap U_j \rightarrow {\mathbb {R}}^{n}$, winding numbers $\hat{m}_{ijk} \in {\mathbb {Z}}^{n}$ satisfying

$$\begin{aligned} \hat{a}_{ij} + \hat{a}_{jk} +\hat{m}_{ijk}&= \hat{a}_{ik}, \end{aligned}$$

(5.3)

and principal ${\mathbb {T}}$-bundles $\hat{L}_{ij}$ over $(U_i \cap U_j) \times {\mathbb {T}}^{n}$ with connection of curvature $\hat{a}_{ij}^{*}\hat{B}_j - \hat{B}_i$.

Before we proceed, we remark that the local trivializations $\varphi _i,\hat{\varphi }_i$ also define local ${\mathbb {T}}^{n}$-invariant metrics $g_i:= \varphi _i^{*}g$ and $\hat{g}_i:=\hat{\varphi }_i^{*}\hat{g}$ on $U_i \times {\mathbb {T}}^{n}$. Due to the ${\mathbb {T}}^{n}$-invariance of g and $\hat{g}$, we have

$$\begin{aligned} a_{ij}^{*}g_j=g_i \quad \text { and }\quad \hat{a}_{ij}^{*}\hat{g}_j=\hat{g}_i. \end{aligned}$$

(5.4)

As seen in Proposition 4.14, the pairs $(g_i,B_i)$ and $(\hat{g}_i,\hat{B}_i)$ satisfy the Buscher rules.

Lemma 5.1

The principal ${\mathbb {T}}$-bundles $L_{ij}$ and $\hat{L}_{ij}$ are trivializable. Thus, there exist $A_{ij},\hat{A}_{ij} \in \Omega ^1((U_i \cap U_j) \times {\mathbb {T}}^{n})$ and connection-preserving isomorphisms

$$\begin{aligned} \lambda _{ij}: L_{ij} \rightarrow \textbf{I}_{A_{ij}} \quad \text { and }\quad \hat{\lambda }_{ij}: \hat{L}_{ij} \rightarrow \textbf{I}_{\hat{A}_{ij}} \end{aligned}$$

over $(U_i \cap U_j) \times {\mathbb {T}}^{n}$. In particular, we have

$$\begin{aligned} a_{ij}^{*}B_j - B_i= {\textrm{d}} A_{ij} \quad \text { and }\quad \hat{a}_{ij}^{*}\hat{B}_j - \hat{B}_i = {\textrm{d}}\hat{A}_{ij}. \end{aligned}$$

(5.5)

Proof

We assume that all non-empty double intersections $U_i \cap U_j$ are contractible; this can again be achieved by passing to a refinement. Then, the first Chern classes of $L_{ij}$ and $\hat{L}_{ij}$ must be pullbacks from ${\mathbb {T}}^{n}$. We have ${\textrm{H}}^2({\mathbb {T}}^{n},{\mathbb {Z}}) \cong \mathfrak {so}(n,{\mathbb {Z}})$, the group of skew-symmetric integral $(n\times n)$-matrices, and this isomorphism can be realized explicitly using the Poincaré bundle $\mathcal {P}$ over ${\mathbb {T}}^2$: we send a matrix $D\in \mathfrak {so}(n,{\mathbb {Z}})$ to the principal ${\mathbb {T}}$-bundle

$$\begin{aligned} \mathcal {P}_D := \bigotimes _{1\le \alpha <\beta \le n} {\textrm{pr}}_{\alpha \beta }^{*}\mathcal {P}^{D_{\alpha \beta }}, \end{aligned}$$

see [42, §B]. Thus, there exist unique matrices $D_{ij},\hat{D}_{ij}\in \mathfrak {so}(n,{\mathbb {Z}})$ and (non-unique) bundle isomorphisms $L_{ij}\cong {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{D_{ij}}$ and $\hat{L}_{ij}\cong {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{\hat{D}_{ij}}$. Taking connections into account, there exist 1-forms $A_{ij},\hat{A}_{ij} \in \Omega ^1((U_i \cap U_j) \times {\mathbb {T}}^{n})$ and connection-preserving isomorphisms

$$\begin{aligned} \lambda _{ij}: L_{ij} \rightarrow {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{D_{ij}} \otimes \textbf{I}_{A_{ij}} \quad \text { and }\quad \hat{\lambda }_{ij}: \hat{L}_{ij} \rightarrow {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{\hat{D}_{ij}} \otimes \textbf{I}_{\hat{A}_{ij}} \end{aligned}$$

over $(U_i \cap U_j) \times {\mathbb {T}}^{n}$.

We show next that $D_{ij}=\hat{D}_{ij}=0$, implying the claim of the lemma. This will be a consequence of the geometric T-duality correspondence, and so we need to work over $(U_i\cap U_j) \times \ {\mathbb {T}}^{2n}$. We consider the following maps:

$$\begin{aligned} {\textrm{pr}}&: (U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow (U_i \cap U_j) \times {\mathbb {T}}^{n}:(x,a,\hat{a}) \mapsto (x,a) \\ \hat{{\textrm{pr}}}&: (U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow (U_i \cap U_j) \times {\mathbb {T}}^{n}:(x,a,\hat{a}) \mapsto (x,\hat{a}) \\ \tilde{a}_{ij}&: (U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow (U_i \cap U_j) \times {\mathbb {T}}^{2n}: (x,a,\hat{a})\mapsto (x,a+a_{ij}(x),\hat{a} + \hat{a}_{ij}(x)) \\ \Phi _i&: U_i \times {\mathbb {T}}^{2n} \rightarrow E \times _X \hat{E}: (x,a,\hat{a}) \mapsto (\varphi _i(x,a),\hat{\varphi }_i(x,\hat{a})), \end{aligned}$$

and construct with them the following diagram of bundle gerbes with connections and connection-preserving isomorphisms over $(U_i \cap U_j) \times {\mathbb {T}}^{2n}$:

(5.6)

The unlabeled 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 2.4, a connection-preserving bundle isomorphism

$$\begin{aligned} \eta _{ij}: \tilde{a}_{ij}^{*}{\textrm{pr}}^{*}_{{\mathbb {T}}^{2n}}\mathcal {P}\otimes {\textrm{pr}}^{*}{\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{D_{ij}}\otimes \textbf{I}_{{\textrm{pr}}^{*} A_{ij}} \cong \hat{{\textrm{pr}}}^{*}{\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{\hat{D}_{ij}} \otimes \textbf{I}_{\hat{{\textrm{pr}}}^{*}\hat{A}_{ij}} \otimes {\textrm{pr}}^{*}_{{\mathbb {T}}^{2n}}\mathcal {P}\end{aligned}$$

(5.7)

over $(U_i\cap U_j)\times {\mathbb {T}}^{2n}$.

We shall forget the connections (and thus all trivial bundles) for a moment. Due to the equivariance of the Poincaré bundle discussed in Sect. 2.2, the lifts $a_{ij}$ and $\hat{a}_{ij}$ determine an isomorphism $\tilde{a}_{ij}^{*}{\textrm{pr}}_{{\mathbb {T}}^{2n}}\mathcal {P}\cong {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}$. Using this in (5.7), we are in the situation that all bundles are pulled back along the projection $(U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}^{2n}$. Hence, these bundles must already have been isomorphic before pullback, and we conclude that there exists a bundle isomorphism

$$\begin{aligned} \mathcal {P}\otimes {\textrm{pr}}^{*}\mathcal {P}_{D_{ij}} \cong \hat{{\textrm{pr}}}^{*}\mathcal {P}_{\hat{D}_{ij}} \otimes \mathcal {P}. \end{aligned}$$

over ${\mathbb {T}}^{2n}$. Hence, there also exists a bundle isomorphism

$$\begin{aligned} {\textrm{pr}}^{*}\mathcal {P}_{D_{ij}} \cong \hat{{\textrm{pr}}}^{*}\mathcal {P}_{\hat{D}_{ij}}, \end{aligned}$$

and this shows that both bundles separately are trivializable. This implies $D_{ij}=\hat{D}_{ij}=0$.

Remark 5.2

The principal ${\mathbb {T}}$-bundle $L_{ij}$ and $\hat{L}_{ij}$ can be regarded as part of the gluing data for the bundle gerbes $\mathcal {G}$ and $\hat{\mathcal {G}}$, 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 $\mathcal {G}$ and $\hat{\mathcal {G}}$ are in the second step of the filtration of ${\textrm{H}}^3(E,{\mathbb {Z}})$ that comes from the Serre spectral sequence, see [11] and [42, §2.1].

Next we will spend some time on finding trivializations $\lambda _{ij}$ and $\hat{\lambda }_{ij}$ with particular covariant derivatives $A_{ij}$ and $\hat{A}_{ij}$. We start with arbitrary choices as they exist by Lemma 5.1 and will then perform three revisions of the isomorphisms $\lambda$ and $\hat{\lambda }$, and accordingly shift the 1-forms $A_{ij}$ and $\hat{A}_{ij}$, finally arriving at (5.18).

We will only discuss $A_{ij}$, the treatment of $\hat{A}_{ij}$ is analogous. We remark that due to Lemma 3.5, (3.11), the 2-form $a_{ij}^{*}B_j - B_i$ is ${\mathbb {T}}^{n}$-invariant; moreover, we have

$$\begin{aligned} ( a_{ij}^{*}B_j - B_i)_{1+2}= (a_{ij}^{*}B_j - B_i)_{1}-\hat{a}_{ij}^{*}\theta \dot{\wedge }\theta _2 \end{aligned}$$

(5.8)

over $(U_i \cap U_j) \times {\mathbb {T}}^{2n}$. Here, we use the notation introduced in Sect. 2.2: an index $(..)_{\alpha }$ means a pullback from the $\alpha$-th ${\mathbb {T}}^{n}$-factor, and the index $(..)_{1+2}$ means a pullback along the addition of two ${\mathbb {T}}^{n}$-factors. (5.5) and (5.8) imply

$$\begin{aligned} {\textrm{d}}((A_{ij})_2-(A_{ij})_1)=-{\textrm{d}}(\hat{a}_{ij}\theta _{2-1}). \end{aligned}$$

This shows that we have a closed 1-form

$$\begin{aligned} \alpha _{ij}&:=(A_{ij})_2-(A_{ij})_1+\hat{a}_{ij}\theta _{2-1} \in \Omega ^1_{cl}((U_i \cap U_j) \times {\mathbb {T}}^{2n} ). \end{aligned}$$

(5.9)

Since the de Rham cohomology class of $\alpha _{ij}$ can only have contributions from the torus, and these contributions must be linear combinations of the generators $[\theta ] \in {\textrm{H}}^1(S^1,{\mathbb {R}})$, there exists a smooth map $\beta _{ij}:(U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {R}}$ and vectors $p_{ij},q_{ij}\in {\mathbb {R}}^{n}$ such that

$$\begin{aligned} \alpha _{ij} = {\textrm{d}}\beta _{ij} + p_{ij}\theta _1 + q_{ij}\theta _2. \end{aligned}$$

(5.10)

Moreover, since the definition of $\alpha _{ij}$ is skew-symmetric with respect to the exchange of the two ${\mathbb {T}}^{n}$-factors, we have $q_{ij}=-p_{ij}$. We may now shift the isomorphism $\lambda _{ij}$ by the smooth map

$$\begin{aligned} (U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}:(x,a)\mapsto -p_{ij}a. \end{aligned}$$

Its derivative is $-p_{ij}\theta$; thus, $A_{ij}$ becomes replaced by $A_{ij}+p_{ij}\theta$, and (5.10) is replaced by just

$$\begin{aligned} \alpha _{ij}= {\textrm{d}}\beta _{ij} . \end{aligned}$$

(5.11)

In particular, we have shown that $\lambda _{ij}$ can be chosen such that $\alpha _{ij}$ is trivial in de Rham cohomology. The left hand side is still skew-symmetric, and so we have ${\textrm{d}}(\beta _{ij} + s^{*}\beta _{ij})=0$, where s is the map that swaps the ${\mathbb {T}}^{n}$ factors. This means that $c_{ij}:=\beta _{ij}(x,a,b) + \beta _{ij}(x,b,a)$ is a constant function. Shifting $\beta _{ij}$ by $-\frac{1}{2}c_{ij}$, we can achieve that $c_{ij}=0$, i.e., achieve that $\beta _{ij}$ is skew-symmetric in a and b.

Over $(U_i \cap U_j) \times {\mathbb {T}}^{3n}$ one can deduce from (5.9) the cocycle condition

$$\begin{aligned} ({\textrm{d}}\beta _{ij})_{1,3} = ({\textrm{d}}\beta _{ij})_{1,2} + ({\textrm{d}}\beta _{ij})_{2,3}. \end{aligned}$$

This shows that there exists a constant $c_{ij}\in {\mathbb {R}}$ such that

$$\begin{aligned} \beta _{ij}(x,a,c)= \beta _{ij}(x,b,c)+\beta _{ij}(x,a,b) + c_{ij} \end{aligned}$$

for all $a,b,c\in {\mathbb {T}}^{n}$. Putting $a=b=c$ shows that $c_{ij}=0$. Putting $b=0$ implies that

$$\begin{aligned} \beta _{ij}(x,a,c)= \beta _{ij}(x,0,c)+\beta _{ij}(x,a,0). \end{aligned}$$

Thus, we may define $\tilde{\beta }_{ij}: (U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {R}}$ by $\tilde{\beta }_{ij}(x,a):= \beta _{ij}(x,a,0)$ and obtain, using the skew-symmetry of $\beta _{ij}$,

$$\begin{aligned} \beta _{ij}(x,a,b) = \tilde{\beta }_{ij}(x,a) - \tilde{\beta }_{ij}(x,b). \end{aligned}$$

We are now in position to make a second revision of the choice of the isomorphism $\lambda _{ij}$, and shift it by the smooth map $(U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}:(x,a)\mapsto -\tilde{\beta }_{ij}(x,a)$. This shifts $A_{ij}$ by ${\textrm{d}}\tilde{\beta }_{ij}$. Then, (5.11) is replaced by $\alpha _{ij}=0$, and (5.9) results in

$$\begin{aligned} (A_{ij})_2-(A_{ij})_1&=-\hat{a}_{ij}\theta _{2-1}. \end{aligned}$$

(5.12)

On the dual side, we obtain analogously

$$\begin{aligned} (\hat{A}_{ij})_{2}- (\hat{A}_{ij})_1&=-a_{ij}\theta _{2-1}. \end{aligned}$$

(5.13)

Next we have to bring $A_{ij}$ and $\hat{A}_{ij}$ together, and consider for this purpose the connection-preserving isomorphism $\eta _{ij}$ of (5.7). By Lemma 5.1, it simplifies to a connection-preserving isomorphism

$$\begin{aligned} \eta _{ij} : \tilde{a}_{ij}^{*}{\textrm{pr}}^{*}_{{\mathbb {T}}^{2n}}\mathcal {P}\otimes \textbf{I}_{{\textrm{pr}}^{*} A_{ij}} \rightarrow \textbf{I}_{\hat{{\textrm{pr}}}^{*}\hat{A}_{ij}} \otimes {\textrm{pr}}^{*}_{{\mathbb {T}}^{2n}}\mathcal {P}. \end{aligned}$$

(5.14)

As a result of the fixed lifts $a_{ij}$ and $\hat{a}_{ij}$, we obtain canonically a connection-preserving isomorphism

$$\begin{aligned} R_{ij} : \tilde{a}_{ij}^{*}{\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\rightarrow {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\otimes \textbf{I}_{\psi _{ij}} \end{aligned}$$

(5.15)

over $(U_i \cap U_j) \times {\mathbb {T}}^{2n}$, see (2.8). Here, $\psi _{ij} \in \Omega ^1((U_i \cap U_j) \times {\mathbb {T}}^{2n})$ with

$$\begin{aligned} \psi _{ij} :=- a_{ij}{\textrm{d}}\hat{a}_{ij} - a_{ij}\hat{{\textrm{pr}}}^{*}\theta + \hat{a}_{ij}{\textrm{pr}}^{*}\theta , \end{aligned}$$

(5.16)

Under the isomorphism $R_{ij}$ of (5.15), we obtain from (5.14) a connection-preserving bundle isomorphism

$$\begin{aligned} \textbf{I}_{\psi _{ij}} \otimes \textbf{I}_{{\textrm{pr}}^{*} A_{ij}} \cong \textbf{I}_{\hat{{\textrm{pr}}}^{*}\hat{A}_{ij}}, \end{aligned}$$

which in turn corresponds via the bijection (2.2) to a smooth map $h_{ij}:(U_i \cap U_j) \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} \hat{{\textrm{pr}}}^{*}\hat{A}_{ij} = {\textrm{pr}}^{*} A_{ij}+\psi _{ij} + h_{ij}^{*}\theta . \end{aligned}$$

(5.17)

Lemma 5.3

The maps $h_{ij}$ are independent of the ${\mathbb {T}}^{2n}$-factor.

Proof

Considering (5.17) over $(U_i \cap U_j) \times {\mathbb {T}}^{4n}$ twice,

$$\begin{aligned} (\hat{A}_{ij})_4&= (A_{ij})_2+ (\psi _{ij})_{2,4} + (h_{ij}^{*}\theta )_{2,4} \\ (\hat{A}_{ij})_3&= (A_{ij})_1+ (\psi _{ij})_{1,3} + (h_{ij}^{*}\theta )_{1,3}, \end{aligned}$$

taking their difference, and using (5.12) to (5.16) yields

$$\begin{aligned} (h_{ij}^{*}\theta )_{2,4} = (h_{ij}^{*}\theta )_{1,3}. \end{aligned}$$

This implies that

$$\begin{aligned} z_{ij} := h_{ij}(x,b,\hat{b})^{-1}\cdot h_{ij}(x,a,\hat{a})\in {\mathbb {T}}\end{aligned}$$

is a constant. Putting $a=b$ and $\hat{a}=\hat{b}$ shows that $z_{ij}=0$. We obtain $h_{ij}(x,b,\hat{b})= h_{ij}(x,a,\hat{a})$. This shows the claim.

We now make one last revision of the choice of the isomorphism $\hat{\lambda }_{ij}$, and shift it by $h_{ij}$. This changes $\hat{A}_{ij}$ by $h_{ij}^{*}\theta$, and hence turns (5.17) into

$$\begin{aligned} \hat{{\textrm{pr}}}^{*}\hat{A}_{ij} = {\textrm{pr}}^{*} A_{ij}+\psi _{ij}. \end{aligned}$$

(5.18)

Note that (5.12) and (5.13) continue to hold, as a change by a 1-form that does not depend on ${\mathbb {T}}^{n}$ cancels itself on both sides.

The definition of the principal ${\mathbb {T}}$-bundle $L_{ij}$ induces a canonical connection-preserving bundle isomorphism

$$\begin{aligned} L_{ij} \otimes a_{ij}^{*}L_{jk} \cong L_{ik} \end{aligned}$$

over $(U_i \cap U_j \cap U_k) \times {\mathbb {T}}^{n}$. Under the trivialization $\lambda _{ij}$, it corresponds to a smooth map

$$\begin{aligned} c_{ijk}: (U_i \cap U_j \cap U_k) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}\end{aligned}$$

such that

$$\begin{aligned} A_{ik}=A_{ij} + a_{ij}^{*}A_{jk} + c_{ijk}^{*}\theta . \end{aligned}$$

(5.19)

Further, by going to a quadruple intersection, it is straightforward to see that we obtain a cocycle condition

$$\begin{aligned} a_{ij}^{*}c_{jkl}\cdot c_{ijl} = c_{ijk}\cdot c_{ikl}. \end{aligned}$$

(5.20)

The same holds on the dual side, leading to a smooth map $\hat{c}_{ijk}$ satisfying

$$\begin{aligned} \hat{A}_{ik}=\hat{A}_{ij} + \hat{a}_{ij}^{*}\hat{A}_{jk} + \hat{c}_{ijk}^{*}\theta \end{aligned}$$

(5.21)

and the cocycle condition

$$\begin{aligned} a_{ij}^{*}\hat{c}_{jkl}\cdot \hat{c}_{ijl} = \hat{c}_{ijk}\cdot \hat{c}_{ikl}. \end{aligned}$$

(5.22)

Lemma 5.4

The following equation of maps $(U_i \cap U_j\cap U_k) \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}$ holds:

$$\begin{aligned} \hat{{\textrm{pr}}}^{*} \hat{c}_{ijk} = {\textrm{pr}}^{*}c_{ijk}\cdot f_{ijk}, \end{aligned}$$

where $f_{ijk}$ is defined by the expression

$$\begin{aligned} f_{ijk}: (U_i \cap U_j\cap U_k) \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}: (x,a,\hat{a}) \mapsto \hat{m}_{ijk}a-m_{ijk}(\hat{a} +\hat{a}_{ik}(x))-a_{jk}(x)\hat{a}_{ij}(x). \end{aligned}$$

Proof

We put the diagrams (5.6) for ij and jk, respectively, next to each other. In the middle, two occurrences of $\tilde{a}_{jk}^{*}\xi _j$ cancel, and we obtain the following equality of connection-preserving 2-isomorphisms:

(5.23)

Our choice of isomorphisms $L_{ij}\cong \textbf{I}_{A_{ij}}$ and $\hat{L}_{ij}\cong \textbf{I}_{\hat{A}_{ij}}$ is such that we have an equality

Substituting this in (5.23) we collect on the left hand side an isomorphism $R_{jk} \circ \tilde{a}_{jk}^{*}R_{ij}$ 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.

5.2 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:

• 1. Riemannian, ${\mathbb {T}}^{n}$-invariant metrics $g_i$ and $\hat{g}_i$ on $U_i \times {\mathbb {T}}^{n}$,

• 2. 2-forms $B_i,\hat{B}_i \in \Omega ^2(U_i \times {\mathbb {T}}^{n})$,

• 3. 1-forms $A_{ij},\hat{A}_{ij} \in \Omega ^1((U_i \cap U_j) \times {\mathbb {T}}^{n})$,

• 4. smooth maps $a_{ij},\hat{a}_{ij}:U_i \cap U_j \rightarrow {\mathbb {R}}^{n}$,

• 5. $m_{ijk},\hat{m}_{ijk}\in {\mathbb {Z}}^{n}$, and

• 6. smooth maps $c_{ijk},\hat{c}_{ijk}:(U_i \cap U_j \cap U_k) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}$.

These local data are subject to the following conditions (LD1) to (LD9).

• (LD1) The pair $(a_{ij},m_{ijk})$ is local data for a principal ${\mathbb {T}}^{n}$-bundle E over X, i.e.,

  $$\begin{aligned} a_{ik}&= m_{ijk}+a_{ij}+a_{jk}\\ m_{jkl}+m_{ijl}&= m_{ikl}+m_{ijk}. \end{aligned}$$

  We remark that the second line follows from the first; it is only listed for convenience.

• (LD2) The pair $(\hat{a}_{ij},\hat{m}_{ijk})$ is local data for a principal ${\mathbb {T}}^{n}$-bundle $\hat{E}$ over X, i.e.,

  $$\begin{aligned} \hat{a}_{ik}&= \hat{m}_{ijk}+\hat{a}_{ij}+\hat{a}_{jk} \\ \hat{m}_{jkl}+\hat{m}_{ijl}&= \hat{m}_{ikl}+\hat{m}_{ijk}. \end{aligned}$$

• (LD3) The metrics $g_i$ yield a metric on E, i.e.,

  $$\begin{aligned} a_{ij}^{*}g_j = g_i. \end{aligned}$$

• (LD4) The metrics $\hat{g}_i$ yield a metric on $\hat{E}$, i.e.,

  $$\begin{aligned} \hat{a}_{ij}^{*}\hat{g}_j = \hat{g}_i. \end{aligned}$$

• (LD5) The triple $(B_i,A_{ij},c_{ijk})$ is local data for a bundle gerbe with connection over E, i.e.,

  $$\begin{aligned} a_{ij}^{*}B_j&= B_i+ {\textrm{d}} A_{ij} \\ A_{ik}&=A_{ij} + a_{ij}^{*}A_{jk} + c_{ijk}^{*}\theta \\ a_{ij}^{*}c_{jkl}\cdot c_{ijl}&= c_{ijk}\cdot c_{ikl} \end{aligned}$$

• (LD6) The triple $(\hat{B}_i,\hat{A}_{ij},\hat{c}_{ijk})$ is local data for a bundle gerbe with connection over $\hat{E}$, i.e.,

  $$\begin{aligned} \hat{a}_{ij}^{*}\hat{B}_j&= \hat{B}_i+ {\textrm{d}} \hat{A}_{ij} \\ \hat{A}_{ik}&=\hat{A}_{ij} + \hat{a}_{ij}^{*}\hat{A}_{jk} + \hat{c}_{ijk}^{*}\theta \\ \hat{a}_{ij}^{*}\hat{c}_{jkl}\cdot \hat{c}_{ijl}&= \hat{c}_{ijk}\cdot \hat{c}_{ikl} \end{aligned}$$

• (LD7) The pairs $(g_i,B_i)$ and $(\hat{g}_i,\hat{B}_i)$ satisfy the Buscher rules.

• (LD8) The second order Buscher rules are satisfied:

  $$\begin{aligned} \hat{{\textrm{pr}}}^{*}\hat{A}_{ij}&= {\textrm{pr}}^{*}A_{ij} -a_{ij}\hat{{\textrm{pr}}}^{*}\theta + \hat{a}_{ij}{\textrm{pr}}^{*}\theta -a_{ij}\hat{a}_{ij}^{*}\theta . \end{aligned}$$

• (LD9) The third order Buscher rules are satisfied:

  $$\begin{aligned} \hat{c}_{ijk}(x,\hat{a})+m_{ijk}(\hat{a}_{ik}(x)+\hat{a})= c_{ijk}(x,a) +\hat{m}_{ijk}a-a_{ij}(x)\hat{a}_{jk}(x). \end{aligned}$$

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 obtained in the context of topological T-duality (Sect. 5.6) and differential T-duality (Sect. 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 5.5

Let $\omega _i,\hat{\omega }_i \in \Omega ^1(U_i \times {\mathbb {T}}^{n},{\mathbb {R}}^{n})$ be the connections on the trivial bundle $U_i \times {\mathbb {T}}^{n}$ that are induced by the metrics $g_i$ and $\hat{g}_i$, respectively, under Theorem 2.7. We remark that by (LD3) and (LD4) the bundle isomorphisms $a_{ij}$ and $\hat{a}_{ij}$ are isometries, and hence connection-preserving by Remark 2.8. Thus, by bijection (2.2), the connections transform under the transition functions as

$$\begin{aligned} \omega _i =\omega _j+a_{ij}^{*}\theta \quad \text { and }\quad \hat{\omega }_i =\hat{\omega }_j+\hat{a}_{ij}^{*}\theta . \end{aligned}$$

The connections in turn correspond to 1-forms $A_i,\hat{A}_i\in \Omega ^1(U_i,{\mathbb {R}}^{n})$, via $\omega _i=(A_i)_1+\theta _2$ and $\hat{\omega }_i= (\hat{A}_i)_1+\theta _2$, which then transform as

$$\begin{aligned} A_i = A_j + a_{ij}^{*}\theta \quad \text { and }\quad \hat{A}_i = \hat{A}_j + \hat{a}_{ij}^{*}\theta . \end{aligned}$$

By (LD7) and Lemma 3.4, the equivariance rules of Lemma 3.5 apply to $B_i$ and $\hat{B}_i$, i.e.

$$\begin{aligned} (B_i)_{1,2+3}&= (B_i)_{1,2}+ (\hat{A}_i)_1\wedge \theta _3 \end{aligned}$$

(5.24)

$$\begin{aligned} (\hat{B}_i)_{1,2+3}&= (\hat{B}_i)_{1,2}+ (A_i)_1 \wedge \theta _3 \end{aligned}$$

(5.25)

over $U_i \times {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. In particular, $\hat{B}_i$ and $B_i$ are ${\mathbb {T}}^{n}$-invariant. We may further consider the 3-forms $K_i:= \omega _i \dot{\wedge }\hat{F} - {\textrm{d}} B_i$ and $\hat{K}_i:= \hat{\omega }_i \dot{\wedge }F-{\textrm{d}}\hat{B}_i$ on $U_i \times {\mathbb {T}}^{n}$, where F and $\hat{F}$ are the globally defined curvatures of the connections $\omega$ and $\hat{\omega }$, respectively. Using (LD7) one can show that $K_i=\hat{K}_i$ and that they are the pullback of a globally defined 3-form $K\in \Omega ^3(X)$ along $U_i \times {\mathbb {T}}^{n} \rightarrow X$.

Remark 5.6

Similarly as proved in Sect. 5.1, (LD8) implies

$$\begin{aligned} (A_{ij})_2-(A_{ij})_1&=-\hat{a}_{ij}\theta _{2-1}. \\ (\hat{A}_{ij})_{2}- (\hat{A}_{ij})_1&=-a_{ij}\theta _{2-1}. \end{aligned}$$

In particular, $A_{ij}$ and $\hat{A}_{ij}$ are ${\mathbb {T}}^{n}$-invariant.

Remark 5.7

We notice that in (LD9) the right hand side is independent of $\hat{a}$, 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 in $\hat{a}$, and these two constants are equal. Explicitly, if we define

$$\begin{aligned} t_{ijk}:U_i \cap U_j \cap U_k \rightarrow {\mathbb {T}}\end{aligned}$$

to be this constant, then we get

$$\begin{aligned} -\hat{c}_{ijk}(x,\hat{a})-m_{ijk}(\hat{a}_{ik}(x)+\hat{a}) =t_{ijk}(x)= -c_{ijk}(x,a)-\hat{m}_{ijk}a +a_{ij}(x)\hat{a}_{jk}(x) \nonumber \\ \end{aligned}$$

(5.26)

for all $a,\hat{a}\in {\mathbb {T}}^{n}$. We deduce from this the equivariance rules

$$\begin{aligned} c_{ijk}(x,a+a')&= c_{ijk}(x,a)-\hat{m}_{ijk}a' \end{aligned}$$

(5.27)

$$\begin{aligned} \hat{c}_{ijk}(x,\hat{a}+\hat{a}')&=\hat{c}_{ijk}(x,\hat{a})-m_{ijk}\hat{a}' \end{aligned}$$

(5.28)

Remark 5.8

The Buscher rules (LD7) to (LD9) determine $\hat{g}_i$, $\hat{B}_i$, $\hat{A}_{ij}$, and $\hat{c}_{ijk}$ uniquely. If $\hat{g}_i$, $\hat{B}_i$, $\hat{A}_{ij}$, and $\hat{c}_{ijk}$ exist and satisfy (LD7) to (LD9), one can in fact show that $\hat{g}_i$ is a ${\mathbb {T}}^{n}$-invariant Riemannian metric satisfying (LD4), and that $(\hat{B}_i,\hat{A}_{ij},\hat{c}_{ijk})$ 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

$$\begin{aligned} (g_i,\hat{g}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \\ (g_i',\hat{g}_i',B_i',\hat{B}_i',A_{ij}',\hat{A}_{ij}',a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',c_{ijk}',\hat{c}_{ijk}') \end{aligned}$$

with respect to the same open cover $\{U_i\}$. These are considered to be equivalent, if there exist:

• 1. 1-forms $C_i,\hat{C}_i \in \Omega ^1(U_i \times {\mathbb {T}}^{n}),$

• 2. smooth maps $p_i,\hat{p}_i: U_i \rightarrow {\mathbb {R}}^{n}$,

• 3. numbers $z_{ij},\hat{z}_{ij}\in {\mathbb {Z}}^{n}$, and

• 4. smooth maps $d_{ij},\hat{d}_{ij}:(U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}$,

such that the following conditions (LD-E1) to (LD-E8) are satisfied. Abusing notation, we consider in the following the functions $p_i,\hat{p}_i$ eventually as maps $p_i,\hat{p}_i: U_i \times {\mathbb {T}}^{n} \rightarrow U_i \times {\mathbb {T}}^{n}$ given by $(x,a)\mapsto (x,a+p_i(x))$ and $(x,\hat{a})\mapsto (x,\hat{a}+\hat{p}_i(x))$, respectively.

• (LD-E1) The bundles E and $E'$ corresponding to $(a_{ij},m_{ijk})$ and $(a'_{ij},m'_{ijk})$ are isomorphic:

  $$\begin{aligned} a_{ij}' + p_i&= z_{ij}+p_j+a_{ij} \\ m_{ijk}'+z_{ij}+z_{jk}&= z_{ik} +m_{ijk} \end{aligned}$$

  We remark that the second line follows from the first and (LD1); it is only listed for convenience.

• (LD-E2) The bundles $\hat{E}$ and $\hat{E}'$ corresponding to $(\hat{a}_{ij},\hat{m}_{ijk})$ and $(\hat{a}'_{ij},\hat{m}'_{ijk})$ are isomorphic:

  $$\begin{aligned} \hat{a}_{ij}' + \hat{p}_i&= \hat{z}_{ij}+\hat{p}_j+\hat{a}_{ij} \\ \hat{m}_{ijk}'+\hat{z}_{ij}+\hat{z}_{jk}&= \hat{z}_{ik}+\hat{m}_{ijk} \end{aligned}$$

• (LD-E3) Under the bundle isomorphism of (LD-E1), the metrics g and $g'$ corresponding to $g_i$ and $g_i'$ are identified:

  $$\begin{aligned} p_i^{*}g_i' = g_i \end{aligned}$$

• (LD-E4) Under the bundle isomorphism of (LD-E2), the metrics $\hat{g}$ and $\hat{g}'$ corresponding to $\hat{g}$ and $\hat{g}'$ are identified:

  $$\begin{aligned} \hat{p}_i^{*}\hat{g}_i' = \hat{g}_i \end{aligned}$$

• (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 $(B_i',A_{ij}',c_{ijk}')$:

  $$\begin{aligned} p_i^{*}B_i'&= B_i + {\textrm{d}}C_i \\ p_i^{*}A_{ij}'&= A_{ij} - C_i + a_{ij}^{*} C_j + d_{ij}^{*}\theta \\ p_i^{*}c_{ijk}'&= c_{ijk} +d_{ik}- d_{ij} - a_{ij}^{*}d_{jk} \end{aligned}$$

• (LD-E6) The pair $(\hat{C}_i,\hat{d}_{ij})$ is a connection-preserving 1-isomorphism between the bundle gerbes corresponding to $(\hat{B}_i,\hat{A}_{ij},\hat{c}_{ijk})$ and $(\hat{B}_i',\hat{A}_{ij}',\hat{c}_{ijk}')$:

  $$\begin{aligned} \hat{p}_i^{*}\hat{B}_i'&= \hat{B}_i + {\textrm{d}}\hat{C}_i \\ \hat{p}_i^{*}\hat{A}_{ij}'&= \hat{A}_{ij} - \hat{C}_i +\hat{a}_{ij}^{*} \hat{C}_j + \hat{d}_{ij}^{*}\theta \\ \hat{p}_i^{*}\hat{c}_{ijk}'&= \hat{c}_{ijk} +\hat{d}_{ik}- \hat{d}_{ij} - \hat{a}_{ij}^{*}\hat{d}_{jk} \end{aligned}$$

• (LD-E7) The following equality of 1-forms on $U_i \times {\mathbb {T}}^{2n}$ holds:

  $$\begin{aligned} \hat{{\textrm{pr}}}^{*}\hat{C}_i - {\textrm{pr}}^{*}C_i =-p_i\hat{{\textrm{pr}}}^{*}\theta -p_i{\textrm{d}}\hat{p}_i +\hat{p}_i{\textrm{pr}}^{*}\theta \end{aligned}$$

• (LD-E8) The following equality holds for all $(x,a,\hat{a})\in (U_i \cap U_j) \times {\mathbb {T}}^{2n}$:

  $$\begin{aligned} d_{ij} (x,a)+\hat{z}_{ij}a-z_{ij}(\hat{p}_i(x)+ \hat{a}_{ij}'(x))+ \hat{a}_{ij}'(x)p_i(x) \\ = \hat{d}_{ij}(x,\hat{a})+z_{ij}\hat{a}+\hat{p}_j(x)a_{ij}(x). \end{aligned}$$

Remark 5.9

Let $\omega _i,\omega _i' \in \Omega ^1(U_i \times {\mathbb {T}}^{n},{\mathbb {R}}^{n})$ be the connections on the trivial bundle $U_i \times {\mathbb {T}}^{n}$ that are induced by the metrics $g_i$ and $g_i'$, respectively, under Theorem 2.7. We remark that the bundle isomorphism $p_{i}$ is an isometry, and hence connection-preserving by Remark 2.8. Thus, the connections transform under the functions $p_{i}$ as $\omega _i = \omega '_i + p_i^{*}\theta$. The connections in turn correspond to 1-forms $A_i,A'_i\in \Omega ^1(U_i,{\mathbb {R}}^{n})$, via $\omega _i=(A_i)_1+\theta _2$ and $\omega '_i=(A'_i)_1+\theta _2$, which then, according to (2.2), transform as

$$\begin{aligned} A_i = A_i' + p_{i}^{*}\theta . \end{aligned}$$

(5.29)

Analogous formulas hold on the dual side, i.e.,

$$\begin{aligned} \hat{A}_i = \hat{A}_i' + \hat{p}_{i}^{*}\theta . \end{aligned}$$

(5.30)

Remark 5.10

From (LD-E7) one can derive the following equivariance rules over $U_i \times {\mathbb {T}}^{2n}$:

$$\begin{aligned} (C_i)_2 - (C_i)_1&= \hat{p}_i\theta _{1-2} \\ (\hat{C}_i)_2 - (\hat{C}_i)_1&= p_i\theta _{1-2} \end{aligned}$$

Remark 5.11

We notice that in (LD-E8) the left hand side is independent of $\hat{a}$, and the right hand side is independent of a. In other words, the right hand side is constant in $\hat{a}$, and the left hand side is constant in a, and these two constants are equal. If we define

$$\begin{aligned} e_{ij}: U_i \cap U_j \rightarrow {\mathbb {T}}\end{aligned}$$

to be this constant, then we get, for all $a,\hat{a}\in {\mathbb {T}}^{n}$, the equality

$$\begin{aligned} -d_{ij} (x,a)-\hat{z}_{ij}a+z_{ij}(\hat{p}_i(x)+ \hat{a}_{ij}'(x))- \hat{a}_{ij}'(x)p_i(x)=e_{ij} (x)\\ =-\hat{d}_{ij}(x,\hat{a})-z_{ij}\hat{a}-\hat{p}_j(x)a_{ij}(x). \end{aligned}$$

From this, we can deduce the following equivariance properties:

$$\begin{aligned} d_{ij}(x,a+a')&=d_{ij} (x,a)-\hat{z}_{ij}a' \end{aligned}$$

(5.31)

$$\begin{aligned} \hat{d}_{ij}(x,\hat{a}+\hat{a}')&= \hat{d}_{ij}(x,\hat{a})-z_{ij}\hat{a}' \end{aligned}$$

(5.32)

The set of equivalence classes of geometric T-duality cocycles with respect to an open cover $\{U_i\}$ is denoted by ${\textrm{Loc}}^{{\textrm{geo}}}(\{U_i\})$. A refinement $\{V_j\} \rightarrow \{U_i\}$ of open covers evidently induces a restriction map ${\textrm{Loc}}^{{\textrm{geo}}}(\{U_i\}) \rightarrow {\textrm{Loc}}^{{\textrm{geo}}}(\{V_j\})$, turning ${\textrm{Loc}}^{{\textrm{geo}}}$ into a direct system w.r.t. to refinements.

Definition 5.12

The direct limit of ${\textrm{Loc}}^{{\textrm{geo}}}(\{U_i\})$ over refinements of open covers is denoted by ${\textrm{Loc}}^{{\textrm{geo}}}(X)$.

With this precise definition of local data at hand, we will prove in the following two sections that ${\textrm{Loc}}^{{\textrm{geo}}}(X)$ indeed classifies geometric T-duality correspondences over X.

5.3 Reconstruction of a Geometric T-Duality Correspondence

In the following, we describe a procedure that constructs from a geometric T-duality cocycle

$$\begin{aligned} (g_i,\hat{g}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \end{aligned}$$

a geometric T-duality correspondence in the sense of Definition 4.9. First of all, the maps $a_{ij}$ and $\hat{a}_{ij}$ become (after exponentiation) ${\mathbb {T}}^{n}$-valued transition functions, and we let E and $\hat{E}$ be the corresponding principal ${\mathbb {T}}^{n}$-bundles. Note that these come with canonical trivializations $\varphi _i$ and $\hat{\varphi }_i$ over $U_i$, which induce the given transition functions. Due to (LD3) and (LD4), the locally defined metrics $g_i$ and $\hat{g}_i$ yield metrics on E and $\hat{E}$, respectively, which are Riemannian and ${\mathbb {T}}^{n}$-invariant.

Next we construct the bundle gerbe $\mathcal {G}$ over E. We define the surjective submersion $\pi :Y \rightarrow E$ by putting

$$\begin{aligned} Y := \coprod _{i\in I} U_i \times {\mathbb {T}}^{n} \end{aligned}$$

and $\pi |_{U_i \times {\mathbb {T}}^{n}}:= \varphi _i$. Over Y we consider the 2-form B defined by $B|_{U_i \times {\mathbb {T}}^{n}}:= B_i$. The fiber products over E can be identified in the following way:

$$\begin{aligned} Y^{[k]} \cong \coprod _{(i_1,\ldots ,i_k) \in I^{k}} Y_{i_1,\ldots ,i_k} \quad \text { with }\quad Y_{i_1,\ldots ,i_k} := (U_{i_1}\cap \cdots \cap U_{i_k} )\times {\mathbb {T}}^{n}, \end{aligned}$$

(5.33)

where the projection maps ${\textrm{pr}}_j: Y^{[k]} \rightarrow Y$ become, under this identification,

$$\begin{aligned} {\textrm{pr}}_j|_{Y_{i_1,\ldots ,i_k}}(x,a) = (i_j,x,a+a_{i_1i_j}(x)). \end{aligned}$$

(5.34)

We remark that the more general projections ${\textrm{pr}}_{j_1,\ldots ,j_l}:Y^{[k]} \rightarrow Y^{[l]}$ can then be described using (5.34) in each component of the range separately.

On $Y^{[2]}$ we define the 1-form A by $A|_{Y_{ij}}:= A_{ij}$; then, the first line of (LD5) implies ${\textrm{pr}}_2^{*}B-{\textrm{pr}}_1^{*}B={\textrm{d}}A$. We may interpret A as a connection on the trivial principal ${\mathbb {T}}$-bundle L over $Y^{[2]}$, so that ${\textrm{d}}A$ is its curvature. Finally, we define an isomorphism

$$\begin{aligned} \mu : {\textrm{pr}}_{12}^{*}L \otimes {\textrm{pr}}_{23}^{*}L \rightarrow {\textrm{pr}}_{13}^{*}L \end{aligned}$$

over $Y^{[3]}$ as multiplication by the smooth map $-c: Y \rightarrow {\mathbb {T}}$, i.e., $-c|_{Y_{ijk}}:= -c_{ijk}$. The second line of (LD5) implies that $\mu$ is connection-preserving, and the third line implies that it satisfies the cocycle condition. This finishes the construction of the bundle gerbe $\mathcal {G}$.

Note that the pullback $\varphi _i^{*}\mathcal {G}$ comes with a canonical trivialization $\mathcal {T}_i: \varphi _i^{*}\mathcal {G} \rightarrow \mathcal {I}_{B_i}$ induced by the section

On the dual side, the construction of $\hat{\mathcal {G}}$ is completely analogous, using (LD6). In particular, we use the same manifold Y, but with the projection $\hat{\pi }:Y \rightarrow \hat{E}$ defined by $\hat{\pi }|_{Y_i}:=\hat{\varphi }_i$. In particular, $\hat{\varphi }_i^{*}\hat{\mathcal {G}}$ comes with a canonical trivialization $\hat{\mathcal {T}}_i: \hat{\varphi }_i^{*}\hat{\mathcal {G}} \rightarrow \mathcal {I}_{\hat{B}_i}$.

It remains to construct the connection-preserving isomorphism $\mathcal {D}$ on correspondence space. We may consider the commutative diagram

where

$$\begin{aligned} Z := \coprod _{i\in I} Z_i \quad \text { with }\quad Z_i:= U_i \times {\mathbb {R}}^{2n}, \end{aligned}$$

and the maps are defined by $\zeta (i,x,a,\hat{a}):= (\varphi _i(x,a),\hat{\varphi }_i(x,\hat{a}))$ as well as ${\textrm{pr}}'(i,x,a,\hat{a}):=(i,x,a)$ and $\hat{{\textrm{pr}}}'(i,x,a,\hat{a}):= (i,x,\hat{a})$. The fiber products of $\zeta :Z \rightarrow E \times _X \hat{E}$ can be identified as

$$\begin{aligned} Z^{[k]} \cong \coprod _{i_1,\ldots ,i_k} Z_{i_1,\ldots ,i_k} \quad \text { with }\quad Z_{i_1,\ldots ,i_k}:= U_{i_1}\cap \cdots \cap U_{i_k} \times {\mathbb {R}}^{2n} \times \underbrace{{\mathbb {Z}}^{2n}\times \cdots \times {\mathbb {Z}}^{2n}}_{(k-1){\text { times}}} \end{aligned}$$

under a diffeomorphism

$$\begin{aligned} ((i_1,x,a_1,\hat{a}_1),\ldots ,(i_k,x,a_k,\hat{a}_k)) \mapsto (i_1,\ldots ,i_k,x,a_1,\hat{a}_1,m_2,\hat{m}_2,\ldots ,m_k,\hat{m}_k), \end{aligned}$$

where the integers are defined by $a_{p}=a_{1}+a_{i_1i_p}(x)+m_{p}$ for $2\le p \le k$, and similarly for the $\hat{m}_p$.

The bundle gerbes $\mathcal {G}$ and $\hat{\mathcal {G}}$ pull back to correspondence space and become bundle gerbes with surjective submersion $\zeta$. Thus, we can construct the isomorphism $\mathcal {D}$ working over Z. For this, we need to find a smooth map $z:Z^{[2]} \rightarrow {\mathbb {T}}$ and a 1-form $\omega \in \Omega ^1(Z)$ such that

$$\begin{aligned} \hat{{\textrm{pr}}}'^{*}\hat{B} + \zeta ^{*}\rho _{g,\hat{g}}&= {\textrm{pr}}'^{*}B+ {\textrm{d}}\omega{} & {} {\text { over }}Z=U_i \times {\mathbb {R}}^{2n} \end{aligned}$$

(5.35)

$$\begin{aligned} (\hat{{\textrm{pr}}}'^{[2]})^{*}\hat{A}+{\textrm{pr}}_1^{*}\omega&= ({\textrm{pr}}'^{[2]})^{*}A+{\textrm{pr}}_2^{*}\omega +z^{*}\theta{} & {} {\text { over }}Z^{[2]} \end{aligned}$$

(5.36)

$$\begin{aligned} (\hat{{\textrm{pr}}}'^{[3]})^{*}\hat{c} +{\textrm{pr}}_{12}^{*} z+{\textrm{pr}}_{23}^{*}z&= {\textrm{pr}}_{13}^{*}z + ({\textrm{pr}}'^{[3]})^{*}c{} & {} {\text { over }}Z^{[3]} \end{aligned}$$

(5.37)

hold. We define $\omega _i \in \Omega ^1(U_i \times {\mathbb {R}}^{2n})$ by

$$\begin{aligned} \omega _i=- a{\textrm{d}}\hat{a} \end{aligned}$$

(5.38)

where $(a,\hat{a})$ are the coordinates of ${\mathbb {R}}^{2n}$, and define $\omega$ by $\omega |_{Z_i}:= \omega _i$. Moreover, we define $z_{ij}:Z_{ij} \rightarrow {\mathbb {T}}$ by

$$\begin{aligned} z_{ij}(x,a,\hat{a},m_2,\hat{m}_2):=m_2\hat{a}+ \hat{a}_{ij}(x) m_2+ \hat{a}_{ij}(x)a, \end{aligned}$$

(5.39)

and define z by $z|_{Z_{ij}}:= z_{ij}$.

Lemma 5.13

Our definitions (5.38) and (5.39) satisfy (5.35) to (5.37).

Proof

Equation (5.35) follows from (LD7), as ${\textrm{d}}\omega = \Omega$. For the remaining equations, it is now important to understand the various projections $Z^{[k]} \rightarrow Z^{[l]}$, under above identifications. We have:

$$\begin{aligned} {\textrm{pr}}_{1}(i,j,x,a,\hat{a})&=(i,x,a,\hat{a}) \\ {\textrm{pr}}_{2}(i,j,x,a,\hat{a})&=(j,x,a+a_{ij}(x)+m_2,\hat{a}+\hat{a}_{ij}(x)+\hat{m}_2) \end{aligned}$$

From this, we can calculate ${\textrm{pr}}_1^{*}\omega$ and ${\textrm{pr}}_2^{*}\omega$; together with (LD8) this gives (5.36).

Finally, we have:

$$\begin{aligned} {\textrm{pr}}_{12}(i,j,k,x,a, \hat{a},m_2,\hat{m}_2,m_3,\hat{m}_3)&=(i,j,x,a,\hat{a},m_2,\hat{m}_2) \\ {\textrm{pr}}_{23}(i,j,k,x,a, \hat{a},m_2,\hat{m}_2,m_3,\hat{m}_3)&= (j,k,x,a_{ij}(x)+a+m_2,\hat{a}_{ij}(x)+\hat{a}+\hat{m}_2, \\&\quad m_{ijk}+m_3-m_2,\hat{m}_{ijk}+\hat{m}_3-\hat{m}_2) \\ {\textrm{pr}}_{13}(i,j,k,x,a, \hat{a},m_2,\hat{m}_2,m_3,\hat{m}_3)&= (i,k,x,a,\hat{a},m_3,\hat{m}_3) \end{aligned}$$

Then, a direct calculation shows that

$$\begin{aligned} ({\textrm{pr}}_{12}^{*}z + {\textrm{pr}}_{23}^{*}z - {\textrm{pr}}_{13}^{*}z)|_{Z_{ijk}} = m_{ijk}(\hat{a}_{ik}(x)+\hat{a})+ \hat{a}_{jk}(x)a_{ij}(x)- \hat{m}_{ijk}a. \end{aligned}$$

This is, via (LD9), the claimed equality (5.37).

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 4.9 follow from (LD7) via Lemmas 3.3 and 3.4. For (T3), consider one of the open sets $U_i$, over which we have the trivializations $\varphi _i$ and $\hat{\varphi }_i$, and the trivializations $\mathcal {T}_i: \varphi _i^{*}\mathcal {G} \rightarrow \mathcal {I}_{B_i}$ and $\hat{\mathcal {T}}_i:\hat{\varphi }_i^{*}\hat{\mathcal {G}} \rightarrow \mathcal {I}_{\hat{B}_i}$ mentioned above.

Lemma 5.14

The principal ${\mathbb {T}}$-bundle with connection over $U_i \times {\mathbb {T}}^{2n}$ that corresponds to the connection-preserving bundle gerbe isomorphism

is given w.r.t. the covering $Z_i \rightarrow U_i \times {\mathbb {T}}^{2n}$ by the connection 1-form $\omega _i\in \Omega ^1(Z_i)$ and the transition function $z_{ii}: Z_i^{[2]} \rightarrow {\mathbb {T}}$.

Proof

All bundle gerbes and bundle gerbe isomorphisms that appear in the composition above just involve trivial principal ${\mathbb {T}}$-bundles. The composition has to be computed over a common refinement of all involved surjective submersions; here, $Z_i \rightarrow U_i \times {\mathbb {T}}^{2n}$ is sufficient. The trivializations contribute, since we work over a single open set $U_i$, the trivial functions $c_{iii}=1$ and $\hat{c}_{iii}=1$. It remains the contribution of $\Phi _i^{*}\mathcal {D}$, which is $z_{ii}$. For the connections, it is similar: the trivializations contribute $A_{ii}=0$ and $\hat{A}_{ii}=0$, and $\Phi _i^{*}\mathcal {D}$ contributes $\omega _i$.

It remains to notice that $z_{ii}(x,a,\hat{a},m,\hat{m})=\hat{a}m$. This function, as well as the 1-form $\omega _i$, are obviously pulled back along the following map of coverings:

Comparing with (2.6) and (2.7), we see that $z_{ii}$ and $\omega _i$ are the local data of the Poincaré bundle and its connection, w.r.t. the section $\chi _l:{\mathbb {R}}^{2n} \rightarrow {\mathbb {T}}^{2n}$. This shows that (T3) is satisfied.

Remark 5.15

Under reconstruction, the 3-forms $K\in \Omega ^3(X)$ from Remarks 5.5 and 4.3(b) coincide.

5.4 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 Sect. 5.3 is compatible with equivalences between correspondences (Definition 4.5) and cocycles (Sect. 5.2). For this purpose, we consider two geometric T-duality cocycles

$$\begin{aligned} (g_i,\hat{g}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \\ (g_i',\hat{g}_i',B_i',\hat{B}_i',A_{ij}',\hat{A}_{ij}',a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',c_{ijk}',\hat{c}_{ijk}') \end{aligned}$$

and an equivalence between them provided by a tuple $(C_i,\hat{C}_i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},d_{ij},$ $\hat{d}_{ij})$. Moreover, we let $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$ and $((E',g',\mathcal {G}'),(\hat{E}',\hat{g}',\hat{\mathcal {G}'}),\mathcal {D}')$ be the geometric T-duality correspondences reconstructed from the two cocycles.

The functions $p_i$ and $\hat{p}_i$ define bundle isomorphisms $p: E \rightarrow E'$ and $\hat{p}: \hat{E} \rightarrow \hat{E}'$ due to (LD-E1) and (LD-E2). It is straightforward to see using (LD-E3) and (LD-E4) that p and $\hat{p}$ are isometric. Concerning the bundle gerbe $\mathcal {G}$ and $\mathcal {G}'$, we have a commutative diagram

with $p'(i,x,a):= (i,x,a+p_i(x))$, i.e., $p'|_{Y_{i}}=p_i$. Thus, we may construct a bundle gerbe isomorphism $\mathcal {A}:\mathcal {G} \rightarrow p^{*}\mathcal {G}'$ using the common refinement

of their surjective submersions. We define the 1-form $C \in \Omega ^1(Y)$ by setting $C|_{U_i \times {\mathbb {T}}^{n}}:= C_i$, and consider the trivial bundle $Q:=\textbf{I}_{C}$ over Y. Then, the first ingredient of the isomorphism $\mathcal {A}$ is the equation $p'^{*}B' = B + \textrm{curv}(Q)$, which follows immediately from the first equation in (LD-E5). The next part is to provide a connection-preserving bundle isomorphism

$$\begin{aligned} \alpha : L \otimes {\textrm{pr}}_2^{*}Q \rightarrow {\textrm{pr}}_1^{*}Q \otimes (p'^{[2]})^{*}L' \end{aligned}$$

over $Y^{[2]}$. Since all bundles are trivial ($L=\textbf{I}_A$ and $L'=\textbf{I}_{A'}$), this is the same as a smooth map $d:Y^{[2]} \rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} {\textrm{pr}}_1^{*}C+(p'^{[2]})^{*}A' = A+{\textrm{pr}}_2^{*}C + d^{*}\theta . \end{aligned}$$

Over $Y_{ij}=(U_i \cap U_j) \times {\mathbb {T}}^{n}$, this is a smooth map $d_{ij}: (U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} C_i + p_i^{*}A'_{ij} = A_{ij} + a_{ij}^{*} C_j + d_{ij}^{*}\theta ; \end{aligned}$$

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

of bundle isomorphisms over $Y^{[3]}$ is commutative. Restricting to $Y_{ijk}$, this means that

$$\begin{aligned} d_{ik} + c_{ijk} = \rho _i^{*}c_{ijk}' + d_{ij} + a_{ij}^{*}d_{jk}, \end{aligned}$$

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 $\xi$ of Definition 4.5. We consider a commutative diagram

where $P:=p\times p'$, and $P': Z \rightarrow Z$ is defined by

$$\begin{aligned} P'|_{Z_i} :=\tilde{p}_i(x,i,a,\hat{a}) := (x,i,a+p_i(x),\hat{a}+\hat{p}_i(x)). \end{aligned}$$

The 2-isomorphism $\xi$ is given by a function $w: Z \rightarrow {\mathbb {T}}$ satisfying:

$$\begin{aligned} {\textrm{pr}}^{*}C + P'^{*}\omega '&= \omega + \hat{{\textrm{pr}}}^{*}\hat{C} +w^{*}\theta \end{aligned}$$

(5.40)

$$\begin{aligned} {\textrm{pr}}^{*}d +(P'^{[2]})^{*}z'&= z +\hat{{\textrm{pr}}}^{*} \hat{d} + {\textrm{pr}}_1^{*}w-{\textrm{pr}}_2^{*}w \end{aligned}$$

(5.41)

Here, $\omega ,\omega '$ are the 1-forms (5.38) from the reconstruction of $\mathcal {D}$ and $\mathcal {D}'$, respectively, and $z,z'$ are the corresponding ${\mathbb {T}}$-valued functions (5.39).

Lemma 5.16

The function $w(i,x,a,\hat{a}):= -\hat{p}_i(x)a$ satisfies (5.40) and (5.41).

Proof

We set $w_i:= w|_{Z_i}$. Employing definitions, we find

$$\begin{aligned} \tilde{p}_i^{*}\omega _i' - \omega _i&=-p_i{\textrm{d}}\hat{a}-a{\textrm{d}}\hat{p}_i -p_i{\textrm{d}}\hat{p}_i \\ w_i^{*}\theta&= -\hat{p}_i {\textrm{d}} a - a{\textrm{d}}\hat{p}_i, \end{aligned}$$

under which (5.40) becomes (LD-E7). In order to treat (5.41) we need to compute the induced map $P'^{[2]}: Z^{[2]} \rightarrow Z^{[2]}$, resulting in

$$\begin{aligned} (i,j,x,a,\hat{a},m_2,\hat{m}_2)\mapsto (i,j,x,a+p_i(x),\hat{a}+\hat{p}_i(x),m_2-z_{ij},\hat{m}_2-\hat{z}_{ij}). \end{aligned}$$

Using this, (5.41) becomes equivalent to

$$\begin{aligned}{} & {} d_{ij} (x,a)+ z'_{ij}(x,a+p_i(x),\hat{a}+\hat{p}_i(x),m_2-z_{ij},\hat{m}_2-\hat{z}_{ij}) \nonumber \\{} & {} \quad = z_{ij}(x,a,\hat{a},m_2,\hat{m}_2) + \hat{d}_{ij}(x,\hat{a}) + w_i(x,a,\hat{a})\\{} & {} \qquad - w_j(x,a+a_{ij}(x)+m_2,\hat{a}+\hat{a}_{ij}(x)+\hat{m}_2). \end{aligned}$$

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.

Summarizing the work of Sects. 5.3 and 5.4, we have constructed a well-defined map ${\textrm{Loc}}^{{\textrm{geo}}}(\{U_i\}) \rightarrow {\text {T{-}Corr}}^{{\textrm{geo}}}(X)$. It is straightforward to see that this map is invariant under refinements of open covers, and hence induces a map

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{geo}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{geo}}}(X). \end{aligned}$$

(5.42)

In the next section we show that it is a bijection.

5.5 Local-to-Global Equivalence

In this section, we prove the following result.

Proposition 5.17

The map (5.42) is a bijection,

We begin with showing surjectivity. Given a geometric T-duality correspondence $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$, we extract local data as explained in Sect. 5.1, using trivializations $\varphi _i,\hat{\varphi }_i$ of the ${\mathbb {T}}^{n}$-bundles, trivializations $\mathcal {T}_i,\hat{\mathcal {T}}_i$ of the bundle gerbes, and 2-isomorphisms $\xi _i$ as in (5.1). Let

$$\begin{aligned} (g_i,\hat{g}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \end{aligned}$$

be the local data obtained by these choices. Under reconstruction we obtain a new geometric T-duality correspondence $((E',g',\mathcal {G}'),(\hat{E}',\hat{g}',\hat{\mathcal {G}'}),\mathcal {D}')$.

Obviously, bundle isomorphisms $\psi : E' \rightarrow E$ and $\hat{\psi }: \hat{E}' \rightarrow \hat{E}$ are given by $\psi ([i,x,a]):= \varphi _i(x,a)$ and $\hat{\psi }([i,x,a]):= \hat{\varphi }_i(x,a)$. Concerning the bundle gerbes, our extraction procedure exhibits $\mathcal {G}$ as canonically isomorphic to a bundle gerbe defined as follows:

• 1. Its surjective submersion is $\pi : Y \rightarrow E$, where $\displaystyle Y=\coprod _{i\in I} U_i \times {\mathbb {T}}^{n}$ and $\pi (i,x,a):= \varphi _i(x,a)$.

• 2. Its curving $B\in \Omega ^2(Y)$ defined by $B|_{U_i \times {\mathbb {T}}^{n}}:= B_i$.

• 3. Its principal ${\mathbb {T}}$-bundle is $L_{ij}$, which is in turn isomorphic to $\textbf{I}_{A_{ij}}$ under the isomorphisms $\lambda _{ij}$, see Lemma 5.1.

• 4. Its bundle gerbe product $\textbf{I}_{A_{ij}} \otimes a_{ij}^{*}\textbf{I}_{A_{jk}} \rightarrow \textbf{I}_{A_{ik}}$ is induced by the map $c_{ijk}$, see (5.19).

Since we have a commutative diagram

pulling back along the diffeomorphism $\psi$ leaves this structure as it is, yielding a bundle gerbe with connection $\mathcal {G}'$ over $E'$. We observe that $\mathcal {G}'$ is precisely the bundle gerbe reconstructed from the data $(B_i,A_{ij},c_{ijk})$. This way, we obtain a connection-preserving isomorphism $\mathcal {A}: \mathcal {G}' \rightarrow \psi ^{*}\mathcal {G}$. Analogously, we treat the dual side, and obtain another connection-preserving isomorphism $\hat{\mathcal {A}}:\hat{\mathcal {G}}' \rightarrow \hat{\psi }^{*}\hat{\mathcal {G}}$.

It remains to treat the correspondence isomorphism $\mathcal {D}$. (5.1) says that it becomes—under the isomorphisms $\mathcal {A}$ and $\hat{\mathcal {A}}$—2-isomorphic to an isomorphism $\mathcal {D}'$ defined over $Z:= \coprod U_i \times {\mathbb {T}}^{2n}$, with bundle the Poincaré bundle ${\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}$, and over $Z_{ij}$ the bundle isomorphism $\eta _{ij}$ of (5.14). In more detail, this is a connection-preserving line bundle isomorphism

$$\begin{aligned} \eta _{ij}: {\textrm{pr}}^{*}\textbf{I}_{A_{ij}} \otimes \tilde{a}_{ij}^{*}{\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\rightarrow {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\otimes \hat{{\textrm{pr}}}^{*}\textbf{I}_{\hat{A}_{ij}} \end{aligned}$$

which was composed of the isomorphism

$$\begin{aligned} R_{ij}: \tilde{a}_{ij}{\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\rightarrow {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\otimes \psi _{ij} \end{aligned}$$

from (5.15) and the equality $\hat{{\textrm{pr}}}^{*}\hat{A}_{ij} = {\textrm{pr}}^{*} A_{ij}+\psi _{ij}$ from (5.18). In order to compare $\mathcal {D}'$ with the reconstructed correspondence isomorphism, we change the covering of $\mathcal {D}'$ along

$$\begin{aligned} Z':=\coprod _{i \in I} U_i \times {\mathbb {R}}^{2n} \rightarrow Z. \end{aligned}$$

One can then trivialize the Poincaré bundle using the section $\chi _l: {\mathbb {R}}^{2n} \rightarrow \mathcal {P}$, see Sect. 2.2. This results into a 2-isomorphic 1-morphism $\mathcal {D}''$. As the covariant derivative of $\chi _l$ is the 1-form $\omega := -a{\textrm{d}}\hat{a}$ on ${\mathbb {R}}^{2n}$, the principal ${\mathbb {T}}$-bundle of $\mathcal {D}''$ is $\textbf{I}_{\omega }$. Its isomorphism is the composite

where the projections ${\textrm{pr}}_1,{\textrm{pr}}_2: Z'^{[2]} \rightarrow Z'$ are as in the proof of Lemma 5.13. Using the formulas (2.5) and (2.10) we can calculate this isomorphism explicitly:

where $z_{ij}$ was defined in (5.39). This shows that the bundle isomorphism of $\mathcal {D}''$ is multiplication with $z_{ij}$. Hence, $\mathcal {D}''$ is precisely the reconstructed isomorphism, proving the surjectivity in Proposition 5.17.

It remains to prove injectivity of reconstruction. For this purpose, we look at two geometric T-duality cocycles,

$$\begin{aligned} (g_i,\hat{g}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \\ (g_i',\hat{g}_i',B_i',\hat{B}_i',A_{ij}',\hat{A}_{ij}',a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',c_{ijk}',\hat{c}_{ijk}') \end{aligned}$$

consider the corresponding reconstructed geometric T-duality correspondences $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$ and $((E',g',\mathcal {G}'),(\hat{E}',\hat{g}',\hat{\mathcal {G}'}),\mathcal {D}')$, and assume that these are equivalent in the sense of Definition 4.5. Thus, there exist isometric bundle isomorphisms $p:E \rightarrow E'$ and $\hat{p}: \hat{E} \rightarrow \hat{E}'$, connection-preserving bundle gerbe isomorphisms $\mathcal {A}:\mathcal {G} \rightarrow p^{*}\mathcal {G}'$ and $\hat{\mathcal {A}}:\hat{\mathcal {G}} \rightarrow \hat{p}^{*}\hat{\mathcal {G}}'$, and a connection-preserving 2-isomorphism

where $P:= p \times \hat{p}:E \times _X \hat{E} \rightarrow E' \times _X \hat{E}'$. It is straightforward to see that the isomorphisms p and $\hat{p}$ induce smooth maps $p_i,\hat{p}_i:U_i \rightarrow {\mathbb {R}}^{n}$ and $z_{ij},\hat{z}_{ij} \in {\mathbb {Z}}^{n}$ satisfying (LD-E1) to (LD-E4). Note that the surjective submersions of all 4 bundle gerbes have the same domain $Y = \coprod Y_i$, with $Y_i:=U_i \times {\mathbb {T}}^{n}$, and the bundle isomorphisms p and $\hat{p}$ lift to $Y^{[k]}$ as the component-wise defined maps

$$\begin{aligned} p_{i_1}: (U_{i_1} \cap \cdots \cap U_{i_k}) \times {\mathbb {T}}^{n} \rightarrow (U_{i_1} \cap \cdots \cap U_{i_k}) \times {\mathbb {T}}^{n}:(x,a) \mapsto (x,a+p_{i_1}(x)), \end{aligned}$$

and the analogous $\hat{p}_i$. We may thus assume that the isomorphisms $\mathcal {A}$ and $\hat{\mathcal {A}}$ consist of principal ${\mathbb {T}}$-bundles Q and $\hat{Q}$ with connections over Y. Their restrictions to $Y_i$ will be denoted by $Q_i$ and $\hat{Q}_i$, respectively. The curvatures are $\textrm{curv}(Q_i) = p_i^{*}B_i'-B_i$ and $\textrm{curv}(\hat{Q}_i)=\hat{p}_i^{*}\hat{B}_i'-B_i'$, and their connection-preserving bundle isomorphisms over $Y^{[2]}\cong \coprod Y_{ij}$ are component-wise

$$\begin{aligned} \chi _{ij}: \textbf{I}_{A_{ij}} \otimes {\textrm{pr}}_2^{*}Q_{j} \rightarrow {\textrm{pr}}_1^{*}Q_{i} \otimes \textbf{I}_{ p_i^{*}A'_{ij}} \end{aligned}$$

and an analogous $\hat{\chi }_{ij}$.

As explained in the proof of Lemma 5.1, there exist bundle isomorphisms $Q_i\cong {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{F_i}$ and $\hat{Q}_i\cong {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\mathcal {P}_{\hat{F}_i}$, where $F,\hat{F} \in \mathfrak {so}(n,{\mathbb {Z}})$. The isomorphism $\chi _{ij}$ shows that $F_i=F_j$ and $\hat{F}_i =\hat{F}_j$, so that we can omit the indices. The 2-isomorphism $\xi$ induces over $(U_i \cap U_j) \times {\mathbb {T}}^{2n}$ an isomorphism

$$\begin{aligned} {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\otimes {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\hat{{\textrm{pr}}}^{*}\mathcal {P}_{\hat{F}} \cong {\textrm{pr}}^{*}_{{\mathbb {T}}^{2n}}{\textrm{pr}}^{*}\mathcal {P}_{F} \otimes {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}\end{aligned}$$

which then implies $F=\hat{F}=0$. Thus, there exist 1-forms $C_i,\hat{C}_i\in \Omega ^1(Y_i)$ and connection-preserving bundle isomorphisms $\kappa _i: Q_i \rightarrow \textbf{I}_{C_i}$ and $\hat{\kappa }_i: \hat{Q}_i\rightarrow \textbf{I}_{\hat{C}_i}$. The isomorphisms $\chi _{ij}$ and $\hat{\chi }_{ij}$ then induce functions $d_{ij},\hat{d}_{ij}: (U_i \cap U_j) \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}$ such that (LD-E5) and (LD-E6) are satisfied.

Note that we have ${\textrm{d}}C_i=\textrm{curv}(Q_i) = p_i^{*}B_i'-B_i$. From this, (5.24) to (5.30) one can then derive

$$\begin{aligned} ({\textrm{d}}C_i)_{1,3} - ({\textrm{d}}C_i)_{1,2} = -{\textrm{d}}(\hat{p}_i\theta _{3-2}) \end{aligned}$$

over $U_i \times {\mathbb {T}}^{n} \times {\mathbb {T}}^{n}$. Now we proceed similar as in Sect. 5.1. We have a closed 1-form $\alpha _{i}\in \Omega ^1_{cl}(U_i \times {\mathbb {T}}^{2n})$ defined by $\alpha _i:= (C_i)_{1,3}- (C_i)_{1,2}+\hat{p}_i \theta _{3-2}$. Since the de Rham cohomology of $U_i \times {\mathbb {T}}^{2n}$ only has torus contributions, there exist a smooth map $\beta _i: U_i \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {R}}$ and vectors $r_i,s_i\in {\mathbb {R}}^{n}$ such that

$$\begin{aligned} \alpha _i = {\textrm{d}}\beta _{i} + r_i \theta _2 + s_i\theta _3. \end{aligned}$$

Moreover, since the definition of $\alpha _{i}$ 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 $\kappa _i$ by the smooth map $U_i \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}: (x,a)\mapsto r_i a$. This shifts $C_i$ by $r_i\theta$ and shows that

$$\begin{aligned} (C_i)_{1,3}- (C_i)_{1,2}+\hat{p}_i\theta _{3-2}={\textrm{d}}\beta _{i}. \end{aligned}$$

Again, the left hand side is skew-symmetric, so that

$$\begin{aligned} {\textrm{d}}(\beta _{i}+s^{*}\beta _{i})=0, \end{aligned}$$

where s swaps the two ${\mathbb {T}}^{n}$-factors. Thus, $c_i:=\beta _{i}+s^{*}\beta _{i}\in {\mathbb {R}}$ is a constant. Shifting $\beta _i$, we can achieve that this constant is zero, and that $\beta _i$ is skew-symmetric; moreover, defining $\tilde{\beta }_i: U_i \times {\mathbb {T}}^{n} \rightarrow {\mathbb {R}}$ by $\tilde{\beta }_i(x,a):= \beta _i(x,a,0)$, we obtain

$$\begin{aligned} \beta _i(x,a,b) = \tilde{\beta }_i(x,a)-\tilde{\beta }_i(x,b). \end{aligned}$$

We may now shift $\kappa _i$ by the function $(x,a) \mapsto \tilde{\beta }_i(x,a)$, getting the formula

$$\begin{aligned} (C_i)_{1,3} = (C_i)_{1,2}-\hat{p}_i\theta _{3-2}. \end{aligned}$$

(5.43)

On the dual side, we obtain analogously

$$\begin{aligned} (\hat{C}_i)_{1,3} =(\hat{C}_i)_{1,2}-p_i\theta _{3-2}. \end{aligned}$$

(5.44)

We continue by looking at the local description of the 2-isomorphism $\xi$. We pull back to the space $Z=\coprod U_i \times {\mathbb {T}}^{2n}$, where, as $\mathcal {D}$ and $\mathcal {D}'$ are obtained by reconstruction, they consist of the trivial bundles with connections $\omega _{i},\omega '_i$ and of the bundle morphisms $z_{ij},z'_{ij}$ defined in (5.38) and (5.39). Note that $\omega _i'=\omega _i$, whereas $z_{ij}$ and $z'_{ij}$ are different. Thus, the 2-isomorphism $\xi$ consists of smooth maps $w_{i}: U_i \times {\mathbb {R}}^{2n}\rightarrow {\mathbb {T}}$ such that

$$\begin{aligned} \tilde{p}_{i}^{*}\omega + {\textrm{pr}}^{*}C_i={\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\omega + \hat{{\textrm{pr}}}^{*}\hat{C}_i + w_{i}^{*}\theta \end{aligned}$$

(5.45)

and

$$\begin{aligned} \tilde{p}_i^{*}z_{ij}'+{\textrm{pr}}^{*} d_{ij} = \hat{{\textrm{pr}}}^{*}\hat{d}_{ij}(x,\hat{a})+z_{ij}-w_j + w_i \end{aligned}$$

(5.46)

also see (5.40) and (5.41). We study now the dependence of $w_i$ on the first and the second ${\mathbb {R}}^{n}$-factor. From (5.45) and (5.43), one can show that

$$\begin{aligned} (w_i^{*}\theta )_{1,3,4} - (w_i^{*}\theta )_{1,2,4} = {\textrm{d}}(\hat{p}_i (a-a')) \end{aligned}$$

over $(x,a,a',\hat{a})\in U_i \times {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}$ holds. Similarly, (5.45) and (5.44) imply that

$$\begin{aligned} (w_i^{*}\theta )_{1,2,4} - (w_i^{*}\theta )_{1,2,3} =0. \end{aligned}$$

In particular, defining $\tilde{w}_i: U_i \rightarrow {\mathbb {T}}$ by $\tilde{w}_i(x):= w_i(x,0,0)$, we have

$$\begin{aligned} w_i^{*}\theta =\tilde{w}_i^{*}\theta -{\textrm{d}}(a\hat{p}_i) \end{aligned}$$

over $(x,a,\hat{a})\in U_i \times {\mathbb {R}}^{2n}$. Thus, there exists $z_i\in {\mathbb {T}}$ such that

$$\begin{aligned} w_i(x,a,\hat{a}) = \tilde{w}_i(x)- a\hat{p}_i(x) + z_i. \end{aligned}$$

Putting $a=\hat{a}=0$ shows that $z_i=0$. We make a final revision of the isomorphism $\kappa _i$ by the function $\tilde{w}_i$. This changes $C_i$ to $C_i + \tilde{w}_i^{*}\theta$, and changes $w_i$ to just

$$\begin{aligned} w_i(x,a,\hat{a}):= -a\hat{p}_i(x). \end{aligned}$$

(5.47)

Now, (5.45) becomes exactly (LD-E7). Finally, we consider (5.46). Using the definitions of $z_{ij}$ and $z'_{ij}$ from (5.39), and using (5.47), (5.46) becomes (LD-E8); see the comments at the end of Sect. 5.4. This shows that the geometric T-duality cocycles we started with are equivalence, and completes the proof of injectivity of Proposition 5.17.

5.6 Local Perspective to Topological T-Duality

In this section, we deduce from the local perspective to geometric T-duality obtained in Sects. 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

$$\begin{aligned} (a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \end{aligned}$$

of data as in Sect. 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 $(z_{ij},\hat{z}_{ij},p_i,\hat{p}_i,d_{ij},\hat{d}_{ij})$ as in Sect. 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 ${\textrm{Loc}}^{{\textrm{top}}}(X)$.

Applying the reconstruction procedure of Sects. 5.3 and 5.4 to only the topological data establishes a map

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{top}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X). \end{aligned}$$

(5.48)

In principle, it could be argued similarly as in Sect. 5.5 that this map is a bijection. However, we will prove this in a different way using the non-abelian differential cohomology ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D})$ of the T-duality 2-group $\mathbb{T}\mathbb{D}$, and a result of [42], see Proposition 5.20.

The T-duality 2-group $\mathbb{T}\mathbb{D}$ has been introduced in [42, §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 ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D})$, see [42, Rem. 3.7]. An element in ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D})$ is represented with respect to an open cover $\{U_i\}$ by a $\mathbb{T}\mathbb{D}$-cocycle, a tuple $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$, where the first four quantities are exactly as in geometric T-duality cocycles, and $t_{ijk}:U_i \cap U_j \cap U_k \rightarrow {\mathbb {T}}$ are smooth functions. The cocycle conditions are (LD1) and (LD2), and

$$\begin{aligned} t_{ikl}+t_{ijk}-m_{ijk}\hat{a}_{kl} = t_{ijl}+t_{jkl}. \end{aligned}$$

(5.49)

Two $\mathbb{T}\mathbb{D}$-cocycles $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$ and $(a'_{ij},\hat{a}'_{ij},m'_{ijk},\hat{m}'_{ijk},t'_{ijk})$ are equivalent if there exists a tuple $(z_{ij},\hat{z}_{ij},p_i,\hat{p}_i,\tilde{e}_{ij})$, with the first four quantities just as in the case of an equivalence between geometric T-duality cocycles, and smooth functions $\tilde{e}_{ij}:U_i \cap U_j \cap U_k \rightarrow {\mathbb {T}}$, satisfying (LD-E1) and (LD-E2), and

$$\begin{aligned} t_{ijk}'+\tilde{e}_{ij}-\hat{a}'_{jk}z_{ij}+\tilde{e}_{jk} = \tilde{e}_{ik}+t_{ijk}-\hat{p}_km_{ijk}. \end{aligned}$$

(5.50)

Then, ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D})$ is a direct limit of equivalence classes of $\mathbb{T}\mathbb{D}$-cocycles over refinement of open covers. We recall the following result.

Proposition 5.18

[42, Prop. 3.9]. There is a bijection

We will now describe a map

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{top}}}(X) \rightarrow {\textrm{H}}^1(X,\mathbb{T}\mathbb{D}) \end{aligned}$$

(5.51)

and prove that it is a bijection, see Lemma 5.19. Let $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk})$ be a topological T-duality cocycle, representing an element in ${\textrm{Loc}}^{{\textrm{top}}}(X)$. In Remark 5.7, we have already defined the function

$$\begin{aligned} t_{ijk}(x) := -\hat{c}_{ijk}(x,0)-m_{ijk}\hat{a}_{ik}(x). \end{aligned}$$

(5.52)

A straightforward calculation using (5.28) shows that $t_{ijk}$ indeed satisfies (5.49).

Given an equivalence between two topological T-duality cocycles

$$\begin{aligned} (a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \\ (a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',c_{ijk}',\hat{c}_{ijk}'), \end{aligned}$$

established by a tuple $(z_{ij},\hat{z}_{ij},p_i,\hat{p}_i,d_{ij},\hat{d}_{ij})$, we consider a slight modification of the function $e_{ij}$ defined in Remark 5.11, namely, we set

$$\begin{aligned} \nonumber \tilde{e}_{ij}(x) :\!&= e_{ij}(x)-z_{ij}(\hat{a}_{ij}(x)+\hat{p}_j(x))+\hat{p}_j(x)a_{ij}(x) \\&= - \hat{d}_{ij}(x,0) -z_{ij}(\hat{a}_{ij}(x)+\hat{p}_j(x)) \end{aligned}$$

(5.53)

Equivalently, using (LD-E8), we could write

$$\begin{aligned} \tilde{e}_{ij}(x)&=-d_{ij} (x,0)- \hat{a}_{ij}'(x)p_i(x)+\hat{p}_j(x)a_{ij}(x). \end{aligned}$$

(5.54)

One can then check (using (5.53) and the first equations of (LD-E1) and (LD-E2)) that $\tilde{e}_{ij}$ satisfies (5.50), i.e., it establishes an equivalence between the $\mathbb{T}\mathbb{D}$-cocycles. This completes the construction of the map (5.51).

Lemma 5.19

The map (5.51) establishes a bijection,

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{top}}}(X) \cong {\textrm{H}}^1(X,\mathbb{T}\mathbb{D}). \end{aligned}$$

Proof

We construct an inverse map. If $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$ is a $\mathbb{T}\mathbb{D}$-cocycle, then we may restore $c_{ijk}$ and $\hat{c}_{ijk}$ from formula (5.26) in Remark 5.7, namely

$$\begin{aligned} c_{ijk}(x,a)&:= -t_{ijk}(x)-\hat{m}_{ijk}a+a_{ij}(x)\hat{a}_{jk}(x) \end{aligned}$$

(5.55)

$$\begin{aligned} \hat{c}_{ijk}(x,\hat{a})&:=-t_{ijk}(x)-m_{ijk}(\hat{a}_{ik}(x)+\hat{a}). \end{aligned}$$

(5.56)

This satisfies obviously (LD9); and it is straightforward to show using (5.49) that the last equations of (LD5) and (LD6) are satisfied. Hence, we obtain a topological T-duality cocycle. Moreover, this is strictly inverse to (5.52).

Next we consider an equivalence between $\mathbb{T}\mathbb{D}$-cocycles $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk}, t_{ijk})$ and $(a'_{ij},\hat{a}'_{ij},m'_{ijk},\hat{m}'_{ijk},t'_{ijk})$, established by a tuple $(p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},\tilde{e}_{ij})$. We will then define

$$\begin{aligned} e_{ij}(x) := \tilde{e}_{ij}(x) +z_{ij}(\hat{a}_{ij}(x)+\hat{p}_j(x))-\hat{p}_j(x)a_{ij}(x) \end{aligned}$$

(5.57)

and recover $d_{ij}$ and $\hat{d}_{ij}$ via Remark 5.11, namely,

$$\begin{aligned} d_{ij} (x,a)&:=-e_{ij} (x) -\hat{z}_{ij}a+z_{ij}(\hat{p}_i(x)+ \hat{a}_{ij}'(x))- \hat{a}_{ij}'(x)p_i(x) \end{aligned}$$

(5.58)

$$\begin{aligned} \hat{d}_{ij}(x,\hat{a})&:=-e_{ij} (x)-z_{ij}\hat{a}-\hat{p}_j(x)a_{ij}(x)-\hat{p}_j(x)p_j(x). \end{aligned}$$

(5.59)

This satisfies (LD-E8) by definition, and the last equations of (LD-E5) and (LD-E6) follow from a straightforward computation.

Proposition 5.20

The maps from (5.51) and (5.48) and Proposition 5.18 fit into a commutative diagram

in which all maps are bijections.

Proof

The commutativity of the diagram needs to be checked using the definition of the map ${\textrm{H}}^1(X,\mathbb{T}\mathbb{D}) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X)$ from [42]; this can be done in a straightforward way. Then, Proposition 5.18 and Lemma 5.19 show that all maps are bijections.

6 Differential T-Duality

In this section, we investigate the relation between geometric T-duality as discussed in Sect. 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” [35]. 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.

6.1 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 [35] will be described later in Sect. 6.3.

Definition 6.1

A differential T-background over X is a triple $(E,\omega ,\mathcal {G})$ consisting of a principal ${\mathbb {T}}^{n}$-bundle E with connection $\omega$ over X and a bundle gerbe $\mathcal {G}$ with connection over E. Two differential T-backgrounds $(E,\omega ,\mathcal {G})$ and $(E',\omega ',\mathcal {G}')$ over X are equivalent if there exists a connection-preserving bundle isomorphism $p: E \rightarrow E'$ and a connection-preserving bundle gerbe isomorphism $\mathcal {G} \cong p^{*}\mathcal {G}'$. The set of equivalence classes of differential T-backgrounds is denoted by ${{\mathrm {T-BG}}}^{{\textrm{diff}}}(X)$.

Obviously, every geometric T-background $(E,g,\mathcal {G})$ induces a differential T-background $(E,\omega ,\mathcal {G})$, where $\omega$ is the Kaluza–Klein connection of g. By Theorem 2.7, this establishes in fact a bijection

$$\begin{aligned} {{\mathrm {T-BG}}}^{{\textrm{geo}}}(X) \cong {{\mathrm {T-BG}}}^{{\textrm{diff}}}(X) \times \textrm{RieM}(X) \times C^{\infty }(X,\textrm{PDS}({\mathbb {R}}^{n})), \end{aligned}$$

where $\textrm{RieM}(X)$ is the set of all Riemannian metrics on X, and $\textrm{PDS}({\mathbb {R}}^{n})$ is the manifold of all positive-definite symmetric bilinear forms on ${\mathbb {R}}^{n}$. We see that differential T-backgrounds are almost as good as geometric T-backgrounds, up to independent global information.

Given two differential T-backgrounds $(E,\omega ,\mathcal {G})$ and $(\hat{E},\hat{\omega },\hat{\mathcal {G}})$ over X, we consider again the correspondence space $E \times _X \hat{E}$ and the ${\mathbb {T}}^{2n}$-invariant 2-form

$$\begin{aligned} \rho _{\omega ,\hat{\omega }} := \hat{{\textrm{pr}}}^{*}\hat{\omega } \dot{\wedge }{\textrm{pr}}^{*}\omega \in \Omega ^2(E \times _X \hat{E}). \end{aligned}$$

Definition 6.2

A differential T-duality correspondence between two differential T-backgrounds $(E,\omega ,\mathcal {G})$ and $(\hat{E},\hat{\omega },\hat{\mathcal {G}})$ is a connection-preserving isomorphism

$$\begin{aligned} \mathcal {D}: {\textrm{pr}}^{*}\mathcal {G} \rightarrow \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \otimes \mathcal {I}_{\rho _{\omega ,\hat{\omega }}} \end{aligned}$$

over $E \times _X \hat{E}$, such that every point $x\in X$ has an open neighborhood $U \subseteq X$ over which condition (T3) of Definition 4.9 is satisfied.

Here, it is understood that the 2-form $\rho _{g,\hat{g}}$ that appears in (T3) is replaced by $\rho _{\omega ,\hat{\omega }}$. We shall fix the following obvious observation.

Proposition 6.3

Suppose $\mathcal {D}$ is a geometric T-duality correspondence between geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$. Then, $\mathcal {D}$ is a differential T-duality correspondence between the induced differential T-backgrounds $(E,\omega ,\mathcal {G})$ and $(\hat{E},\hat{\omega }, \hat{\mathcal {G}})$.

We also have the following converse result.

Proposition 6.4

Suppose $\mathcal {D}$ is a differential T-duality correspondence between differential T-backgrounds $(E,\omega ,\mathcal {G})$ and $(\hat{E},\hat{\omega },\hat{\mathcal {G}})$. Then, there exist metrics g on E and $\hat{g}$ on $\hat{E}$ whose Kaluza–Klein connections are $\omega$ and $\hat{\omega }$, respectively, such that $\mathcal {D}$ is a geometric T-duality correspondence between $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$.

Proof

We choose a Riemannian metric $g'$ on X. Let $h: {\mathbb {R}}^{n} \times {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}$ denote the standard inner product. We define g to be the ${\mathbb {T}}^{n}$-invariant metric on E corresponding to the triple $(\omega ,g',h)$ under Theorem 2.7, and we define $\hat{g}$ to be the metric on $\hat{E}$ corresponding to $(\hat{\omega },g',h)$. We have $\rho _{\omega ,\hat{\omega }}=\rho _{g,\hat{g}}$, so that $\mathcal {D}$ has the correct structure of a geometric T-duality correspondence. Finally, we observe that it satisfies all three conditions, (T1) to (T3).

Equivalences between differential T-duality correspondences are defined analogous to Definition 4.5. The set of equivalence classes of differential T-duality correspondences is denoted by ${\text {T{-}Corr}}^{{\textrm{diff}}}(X)$. Proposition 6.4 establishes a map

$$\begin{aligned} {\text {T{-}Corr}}^{{\textrm{geo}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{diff}}}(X), \end{aligned}$$

and Proposition 6.4 shows that this map is surjective. In fact, there is a bijection

$$\begin{aligned} {\text {T{-}Corr}}^{{\textrm{geo}}}(X) \cong {\text {T{-}Corr}}^{{\textrm{diff}}}(X) \times \textrm{RieM}(X) \times C^{\infty }(X,\textrm{PDS}({\mathbb {R}}^{n})), \end{aligned}$$

under which a geometric T-duality correspondence $((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})$ corresponds to the triple $(((E,\omega ,\mathcal {G}),(\hat{E},\hat{\omega },\hat{\mathcal {G}}),\mathcal {D}),g',h)$, where the metrics g and $\hat{g}$ correspond under Theorem 2.7 to the triples $(\omega ,g',h)$ and $(\hat{\omega },g',h^{-1})$, respectively.

The following result is more difficult to show, and its proof relies on the local formalism developed in Sect. 5 and extended to differential T-duality below in Sect. 6.2.

Proposition 6.5

Suppose $(E,\mathcal {G})$ and $(\hat{E},\hat{\mathcal {G}})$ are topological T-backgrounds, and $\mathcal {D}$ is a topological T-duality correspondence between them. Suppose further that $\omega$ and $\hat{\omega }$ are connections on E and $\hat{E}$, respectively. Then, there exist connections on $\mathcal {G}$, $\hat{\mathcal {G}}$, and $\mathcal {D}$, such that $\mathcal {D}$ becomes a differential T-duality correspondence between $(E,\omega ,\mathcal {G})$ and $(\hat{E},\hat{\omega },\hat{\mathcal {G}})$.

Proof

Proposition 6.10 in combination with Lemmas 5.19 and 6.9.

The obvious composition of Proposition 6.4 and 6.5, about lifting topological T-duality correspondences to geometric ones, is stated as Proposition 4.20 in Sect. 4.3. On the level of equivalence classes, it is clear that the map ${\text {T{-}Corr}}^{{\textrm{geo}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X)$ from Sect. 4.3 factors as

$$\begin{aligned} {\text {T{-}Corr}}^{{\textrm{geo}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{diff}}}(X) \rightarrow {\text {T{-}Corr}}^{{\textrm{top}}}(X), \end{aligned}$$

where both maps are surjective.

6.2 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 Sect. 5.2 by replacing the metrics $g_i$ and $\hat{g}_i$ by 1-forms $A_i,\hat{A}_i\in \Omega ^1(U_i,{\mathbb {R}}^{n})$, and replacing conditions (LD3) and (LD4) by the following new conditions:

• (LD1’) $A_j = A_i - a_{ij}^{*}\theta$

• (LD2’) $\hat{A}_j = \hat{A}_i - \hat{a}_{ij}^{*}\theta$.

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:

• (LD-E1’) $A'_i = A_i - p_{i}^{*}\theta$

• (LD-E2’) $\hat{A}'_i = \hat{A}_i - \hat{p}_{i}^{*}\theta$.

The corresponding set of equivalence classes, and its direct limit over refinements of open covers will be denoted by ${\textrm{Loc}}^{{\textrm{diff}}}(X)$. Enforced by Theorem 2.7, and using Remarks 5.5 and 5.9, there is a bijection

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{geo}}}(X) \cong {\textrm{Loc}}^{{\textrm{diff}}}(X) \times \textrm{RieM}(X) \times C^{\infty }(X,\textrm{PDS}({\mathbb {R}}^{n})), \end{aligned}$$

obtained by replacing the metrics $g_i$ and $\hat{g}_i$ by the local connection 1-forms $A_i,\hat{A}_i$ of their Kaluza–Klein connections.

The reconstruction procedure described in Sects. 5.3 and 5.4, together with the proof of Proposition 5.17, goes through with obvious small modifications, so that we infer the following result.

Proposition 6.6

Reconstruction is a bijection,

Next we set differential T-duality in relation to the differential non-abelian cohomology of the T-duality 2-group $\mathbb{T}\mathbb{D}$, whose investigation was started recently by Kim-Saemann [32]. Differential non-abelian cohomology in general has been studied by Breen-Messing [8] and further developed in [43, 46, 47]. A common phenomenon in higher gauge theory is the appearance of several versions of connection-data, which, in my review in [51, §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 $\Gamma$, related by maps

$$\begin{aligned} \hat{{\textrm{H}}}^1(X,\Gamma )^{ff} \rightarrow \hat{{\textrm{H}}}^1(X,\Gamma )^{reg} \rightarrow \hat{{\textrm{H}}}^1(X,\Gamma )^{gen} \end{aligned}$$

that commute with the projections to the (non-differential) non-abelian cohomology ${\textrm{H}}^1(X,\Gamma )$.

Additionally, Kim-Saemann have invented a formalism of adjusted differential cohomology [32, 33]. It requires to equip the Lie 2-group $\Gamma$ with an additional structure, called an adjustment $\kappa$. Together with an adjustment, there is another version of non-abelian differential cohomology denoted $\hat{{\textrm{H}}}^1(X,\Gamma _{\kappa })$. It comes equipped with a map $\hat{{\textrm{H}}}^1(X,\Gamma _{\kappa }) \rightarrow \hat{{\textrm{H}}}^1(X,\Gamma )^{gen}$, and the choice $\kappa =0$ reproduces $\hat{{\textrm{H}}}^1(X,\Gamma _0)=\hat{{\textrm{H}}}^1(X,\Gamma )^{reg}$. Relevant for us will be the adjusted differential cohomology $\hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa })$ of the Lie 2-group $\mathbb{T}\mathbb{D}$.

In order to explain it on the basis of [51, §2.2] and [32], we need to express the Lie 2-group $\mathbb{T}\mathbb{D}$ and its associated Lie 2-algebra as crossed modules (of Lie groups and Lie algebras, respectively). The crossed module of $\mathbb{T}\mathbb{D}$ consists of the Lie group homomorphism

$$\begin{aligned} \tau : H \rightarrow G,\qquad H:={\mathbb {T}}\times {\mathbb {Z}}^{2n}, \qquad G:= {\mathbb {R}}^{2n},\qquad \tau (t,m,\hat{m}) := (m, \hat{m}) \end{aligned}$$

and the action $\alpha : G \times H \rightarrow H$ defined by $\alpha ((a,\hat{a}), (t,m,\hat{m})):= (t-\hat{a}m,m,\hat{m})$, see [42, §3.2]. The corresponding crossed module of Lie algebras is trivial: it consists of the induced Lie algebra homomorphism, $\tau _{*}=0$, and the induced action of the Lie algebra $\mathfrak {g}$ of G on the Lie algebra $\mathfrak {h}$ of H, $\alpha _{*}=0$. Of relevance is further the differential of the action of a fixed element of G, $\alpha _{g}: H \rightarrow H$, which is here $(\alpha _{a,\hat{a}})_{*}={\textrm{id}}_{{\mathbb {R}}}$, and the differential of the map

$$\begin{aligned} \tilde{\alpha }_h: G \rightarrow H: g \mapsto h^{-1}\alpha (g,h), \end{aligned}$$

which is here $(\tilde{\alpha }_{t,m,\hat{m}})_{*}(a,\hat{a}) = -\hat{a}m$.

With these expressions at hand, we can recall the definition of $\hat{{\textrm{H}}}^1 (X,\mathbb{T}\mathbb{D})^{gen}$ on the basis of [51, §2.2]. Thus, a generalized differential $\mathbb{T}\mathbb{D}$-cocycle consists of a $\mathbb{T}\mathbb{D}$-cocycle $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$ as in Sect. 5.6, and additionally of 1-forms $A_i,\hat{A}_i\in \Omega ^1(U_i,{\mathbb {R}}^{n})$, a 2-form $R_i \in \Omega ^2(U_i)$, and a 1-form $\varphi _{ij}\in \Omega ^1(U_i \cap U_j)$ such that (LD1’) and (LD2’) and

$$\begin{aligned} \varphi _{ik} -\hat{A}_k m_{ijk}&= \varphi _{jk}+\varphi _{ij} -t_{ijk}^{*}\theta \end{aligned}$$

(6.1)

are satisfied. Indeed, for an equivalence between generalized differential $\mathbb{T}\mathbb{D}$-cocycles

$$\begin{aligned} (A_i,\hat{A}_i,R_i,a_{ij},\hat{a}_{ij},\varphi _{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})\\(A_i',\hat{A}_i',R_i',a_{ij}',\hat{a}_{ij}',\varphi _{ij}',m'_{ijk},\hat{m}'_{ijk},t'_{ijk}) \end{aligned}$$

we require a tuple $(\phi _i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},\tilde{e}_{ij})$, where $\phi _i\in \Omega ^1(U_i)$, and $(p_i,\hat{p}_i,z_{ij},\hat{z}_{ij}, \tilde{e}_{ij})$ is, as in Sect. 5.6, an equivalence between the $\mathbb{T}\mathbb{D}$-cocycles $(a_{ij}',\hat{a}'_{ij},m'_{ijk}, \hat{m}'_{ijk},t'_{ijk})$ and $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$, i.e., it satisfies (LD-E1) and (LD-E2) and (5.50). Additionally, we require (LD-E1’) and (LD-E2’) and

$$\begin{aligned} \varphi '_{ij} + \phi _i- z_{ij}\hat{A}_j'&=\phi _j+\varphi _{ij}-\tilde{e}_{ij}^{*}\theta . \end{aligned}$$

(6.2)

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 $\kappa$ for $\mathbb{T}\mathbb{D}$. In general, an adjustment is a map $\kappa : G \times \mathfrak {g} \rightarrow \mathfrak {h}$, and in case of $\mathbb{T}\mathbb{D}$ Saemann-Kim [32] use

$$\begin{aligned} \kappa ((a, \hat{a}),(b,\hat{b})) := a\hat{b}. \end{aligned}$$

Then, an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle satisfies, in addition to the conditions listed above, the condition

$$\begin{aligned} R_j + {\textrm{d}}\varphi _{ij} =R_i+a_{ij}\hat{F}, \end{aligned}$$

(6.3)

where $\hat{F}\in \Omega ^2(X)$ is defined by $\hat{F}|_{U_i} = {\textrm{d}}\hat{A}_i$. Moreover, for an equivalence between adjusted differential $\mathbb{T}\mathbb{D}$-cocycles, we additionally require the condition

$$\begin{aligned} R_i'+{\textrm{d}}\phi _i = R_i + p_i \hat{F}. \end{aligned}$$

(6.4)

Remark 6.7

The 3-curvature of an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle is, by definition,

$$\begin{aligned} K:={\textrm{d}}R_i + A_i \wedge \hat{F} \in \Omega ^3(X). \end{aligned}$$

(6.5)

Having recalled the definition of the $\kappa$-adjusted differential cohomology of $\mathbb{T}\mathbb{D}$, we are in position to construct a map

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{diff}}}(X) \rightarrow \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }). \end{aligned}$$

(6.6)

Given a differential T-duality cocycle $(A_i,\hat{A}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk}, \hat{m}_{ijk},c_{ijk},\hat{c}_{ijk})$, we consider the underlying $\mathbb{T}\mathbb{D}$-cocycle $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$, where $t_{ijk}$ was defined in Remark 5.7, namely,

$$\begin{aligned} t_{ijk}(x) := -c_{ijk}(x,0)+a_{ij}(x)\hat{a}_{jk}(x). \end{aligned}$$

This coincides with the expression given in (5.52), using (LD9). We add the given 1-forms $A_i$ and $\hat{A}_i$, so that (LD1’) and (LD2’) are satisfied as before. Let $\sigma : U_i \rightarrow U_i \times {\mathbb {T}}^{n}$ be the zero section, $\sigma (x):=(x,0)$. The 2-form $R_i$ is then defined by

$$\begin{aligned} R_i := -\sigma ^{*}B_i, \end{aligned}$$

(6.7)

and the 1-form $\varphi _{ij}$ is defined by

$$\begin{aligned} \varphi _{ij} := \sigma ^{*}A_{ij}+a_{ij}\hat{A}_j. \end{aligned}$$

(6.8)

It remains to check condition (6.1) for generalized differential cocycles and the additional condition (6.3) for adjusted differential cocycles. These are straightforward calculations; the first involving (LD4’) and (LD5) and Remark 5.6, the second involving (LD5) and (3.11).

Let us now suppose that we have an equivalence between two differential T-duality cocycles, established by a tuple $(C_i,\hat{C}_i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},d_{ij},\hat{d}_{ij})$. We recall from Sect. 5.6 that the functions $p_i,\hat{p}_i:U_i \rightarrow {\mathbb {R}}^{2n}$ and $\tilde{e}_{ij}:U_i \cap U_j \rightarrow {\mathbb {T}}$, defined in (5.54) by

$$\begin{aligned} \tilde{e}_{ij} :=-d_{ij} (x,0)- \hat{a}_{ij}'(x)p_i(x)+\hat{p}_j(x)a_{ij}(x) \end{aligned}$$

establish an equivalence between the underlying two $\mathbb{T}\mathbb{D}$-cocycles. Additionally, conditions (LD-E1’) and (LD-E2’) remain valid. It remains to provide 1-forms $\phi _i \in \Omega ^1(U_i)$ satisfying (6.2) and (6.4). We set

$$\begin{aligned} \phi _i := \sigma ^{*}C_i + p_i \hat{A}_i'. \end{aligned}$$

(6.9)

Checking (6.4) is straightforward using (5.24)and (LD-E5). (6.2) is a bit more difficult to verify; one can first derive from (LD-E5) and Remarks 5.6 and 5.10 the formula

$$\begin{aligned} \sigma ^{*}A'_{ij}+ \sigma ^{*}C_i= \sigma ^{*}A_{ij} +\sigma ^{*}C_j+\hat{a}_{ij}'{\textrm{d}}p_i-\hat{p}_j{\textrm{d}}a_{ij} + \sigma ^{*}d_{ij}^{*}\theta . \end{aligned}$$

(6.10)

This formula together with (LD-E1) and (LD4’) proves (6.2). This completes the construction of the map (6.6).

Remark 6.8

We recall from Remark 5.5 that every geometric T-duality cocycle comes equipped with a globally defined 3-form $K\in \Omega ^3(X)$, which corresponds to the 3-form of a geometric T-duality correspondence, see Remarks 4.3(b) and 5.15. Under the map ${\textrm{Loc}}^{{\textrm{geo}}}(X) \rightarrow {\textrm{Loc}}^{{\textrm{diff}}}(X)$, the same 3-form can be obtained from a differential T-duality cocycle, namely

$$\begin{aligned} K|_{U_i} = A_i \dot{\wedge }\hat{F} - \sigma ^{*}{\textrm{d}} B_i. \end{aligned}$$

Under the map (6.6), ${\textrm{Loc}}^{{\textrm{diff}}}(X) \rightarrow \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa })$, the 3-form K is precisely the curvature of Remark 6.7.

Lemma 6.9

The map (6.6) is a bijection,

$$\begin{aligned} {\textrm{Loc}}^{{\textrm{diff}}}(X) \cong \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }). \end{aligned}$$

Proof

We suppose that we have an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle

$$\begin{aligned} ( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}). \end{aligned}$$

First, we reproduce, as in the proof of Lemma 5.19, the topological part of a differential T-duality cocycle, i.e., we define $c_{ijk}$ and $\hat{c}_{ijk}$ as in (5.55) and (5.56).

We further revert the assignments made in the definition of (6.6) using Lemma 3.5, and set

$$\begin{aligned} B_i := -(R_i)_1 + (\hat{A}_i)_1\wedge \theta _2 \end{aligned}$$

on $U_i \times {\mathbb {T}}^{n}$. Similarly, using Remark 5.6, we set

$$\begin{aligned} A_{ij} := (\varphi _{ij})_1-a_{ij}(\hat{A}_j)_1-\hat{a}_{ij}\theta _2. \end{aligned}$$

One can then check using (6.3) and (6.1) that the first and second lines of (LD5) are satisfied (the third line is already checked in Lemma 5.19). Finally, we define $\hat{B}_i$ and $\hat{A}_{ij}$ such that the Buscher rules (LD7) and (LD8) are satisfied. As mentioned in Remark 5.8, 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,

$$\begin{aligned} (A_i,\hat{A}_i,B_i,\hat{B}_i,A_{ij},\hat{A}_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},c_{ijk},\hat{c}_{ijk}) \\(A_i',\hat{A}_i',B_i',\hat{B}_i',A_{ij}',\hat{A}_{ij}',a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',c_{ijk}',\hat{c}_{ijk}') \end{aligned}$$

become equivalent after passing to $\hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa })$. That is, there exists a tuple $(\phi _i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},\tilde{e}_{ij})$ satisfying (LD-E1’) and (LD-E2’) (6.2) and (6.4), as well as the usual (non-differential) cocycle conditions (LD-E1) and (LD-E2) and (5.50). We have seen in the proof of Lemma 5.19 how to obtain $d_{ij}$ and $\hat{d}_{ij}$ such that the third lines of (LD-E5) and (LD-E6) and (LD-E8) are satisfied. It remains to provide 1-forms $C_i,\hat{C}_i \in \Omega ^1(U_i \times {\mathbb {R}}^{n})$ such that the first two lines of (LD-E5) and (LD-E6), and (LD-E7) hold. We set

$$\begin{aligned} C_i&:= (\phi _i)_1 -p_i (\hat{A}_i')_1-\hat{p}_i \theta _2 \\ \hat{C}_i&:= (\phi _i)_1 -p_i (\hat{A}_i)_1 -p_i\theta _2 \end{aligned}$$

on $U_i \times {\mathbb {T}}^{n}$. The first line reverts (6.9), and the second is chosen such that (LD-E7) holds. The first line of (LD-E5) con now be verified using (5.24) to (6.4), and the second line of (LD-E5) can be verified using Remark 5.6 and (6.2) and (6.8). 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 $\mathbb{T}\mathbb{D}$ has the advantage that the presentation with differential $\mathbb{T}\mathbb{D}$-cocycles is less redundant than the one with differential T-duality cocycles: instead of two 2-forms $B_i$ and $\hat{B}_i$ there is only a single 2-form $R_i$, instead of $A_{ij}$ and $\hat{A}_{ij}$ there is only $\varphi _{ij}$, and instead of $c_{ijk}$ and $\hat{c}_{ijk}$ 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 $U_i \times {\mathbb {T}}^{n}$ and their intersections. The following two results show that (adjusted) differential cohomology is very efficient for calculations. The first, Proposition 6.10, delivers the core ingredient to the proofs of our main results Theorems 1.2 and 1.3.

Proposition 6.10

Every $\mathbb{T}\mathbb{D}$-cocycle can be lifted to an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle, i.e., the map

$$\begin{aligned} \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow {\textrm{H}}^1(X,\mathbb{T}\mathbb{D}) \end{aligned}$$

is surjective.

Proof

Given a $\mathbb{T}\mathbb{D}$-cocycle $(a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$, by the well-known existence of connections on principal bundles we find 1-forms $A_i,\hat{A}_i\in \Omega ^1(U_i,{\mathbb {R}}^{n})$ satisfying (LD1’) and (LD2’). We write (6.1) as

$$\begin{aligned} (\delta \varphi )_{ijk} = t_{ijk}^{*}\theta -\hat{A}_k m_{ijk}, \end{aligned}$$

where $\delta$ denotes the Čech coboundary operator. It is easy to check using (5.49) that the right hand side is a Čech 2-cocycle; then, by the exactness of the Čech complex with values in the sheaf $\Omega ^1$ it follows that $\varphi _{ij}$ exist such that (6.1) is satisfied. Finally, we write (6.3) as

$$\begin{aligned} (\delta R)_{ij} =a_{ij}\hat{F}- {\textrm{d}}\varphi _{ij} \end{aligned}$$

and check again that the right hand side is a Čech 1-cocycle. This shows that $R_i$ exists such that (6.3) is satisfied.

Our second result concerns the action (Remarks 4.7 and 4.11) of the group of isomorphism classes of bundle gerbes with connection, $\textrm{Grb}^{\nabla }(X)$, on the set of equivalence classes of geometric T-duality correspondences, ${\text {T{-}Corr}}^{{\textrm{geo}}}(X)$. We recall that this action was induced by

$$\begin{aligned} (\mathcal {H} \;,\; ((E,g,\mathcal {G}),(\hat{E},\hat{g},\hat{\mathcal {G}}),\mathcal {D})) \mapsto ((E,g,\mathcal {G} \otimes p^{*}\mathcal {H}),(\hat{E},\hat{g},\hat{\mathcal {G}} \otimes \hat{p}^{*}\mathcal {H}),\mathcal {D} \otimes {\textrm{id}})). \nonumber \\ \end{aligned}$$

(6.11)

Since the action does not concern the metrics, there is a corresponding action on differential T-duality correspondences, which, under the bijections of Proposition 5.17 and Lemma 6.9, becomes an action

$$\begin{aligned} \hat{{\textrm{H}}}^3(X) \times \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }). \end{aligned}$$

(6.12)

It is straightforward to obtain a formula for (6.12): a Deligne 2-cocycle acts on an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle by

$$\begin{aligned}{} & {} (B_i,A_{ij},c_{ijk}) \cdot ( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \\{} & {} \quad :=( A_i,\hat{A}_i,R_i+B_i,\varphi _{ij}+A_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}+c_{ijk}). \end{aligned}$$

Next we consider the projection

$$\begin{aligned} {\text {T{-}Corr}}^{{\textrm{geo}}}(X) \rightarrow \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X) \times \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X) \end{aligned}$$

from a geometric T-duality correspondence to the two principal ${\mathbb {T}}^{n}$-bundles E and $\hat{E}$, which can be equipped with the Kaluza–Klein connections $\omega ,\hat{\omega }$ induced from the metrics g and $\hat{g}$, respectively. This projection is obviously invariant under the action (6.11). The same projection exists for differential T-duality correspondences, and then in adjusted differential cohomology,

$$\begin{aligned} \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X) \times \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X). \end{aligned}$$

There, it is induced by the formula

$$\begin{aligned} ( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \mapsto ((A_i,a_{ij}),(\hat{A}_i,\hat{a}_{ij})). \end{aligned}$$

Summarizing, we have a commutative diagram

Finally, we note that a pair $((E,\omega ),(\hat{E},\hat{\omega }))$ of isomorphism classes of bundles with connection has a well-defined pair $(F,\hat{F}) \in \Omega ^2(X) \times \Omega ^2(X)$ of curvatures. We consider the subgroup

$$\begin{aligned} \mathcal {F}_{F,\hat{F}} := \{ \mathcal {I}_{y \hat{F} + \hat{y} F} \;|\;y,\hat{y}\in {\mathbb {R}}\} \subseteq \textrm{Grb}^{\nabla }(X). \end{aligned}$$

This is a non-trivial subgroup, as $\mathcal {I}_{B} \cong \mathcal {I}_{C}$ holds if and only if $C-B$ is a closed 2-form with integral periods. Now, F and $\hat{F}$ are closed 2-forms with integral periods, but allowing real multiplies spoils integrality.

Proposition 6.11

The action of (6.12),

$$\begin{aligned} \hat{{\textrm{H}}}^3(X) \times \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }), \end{aligned}$$

has the following properties:

• (i) It acts transitively in the fibers of the projection $\hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}} (X) \times \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X)$.

• (ii) The stabilizer of each element in the fiber over $(\xi ,\hat{\xi })\in \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X) \times \textrm{Bun}^{\nabla }_{{\mathbb {T}}^{n}}(X)$ with curvature pair $(F,\hat{F})$ is the subgroup $\mathcal {F}_{F,\hat{F}}$.

In particular, the quotient $\hat{{\textrm{H}}}^3(X)/\mathcal {F}_{F,\hat{F}}$ acts freely and transitively on the fiber over $(\xi ,\hat{\xi })$.

Proof

We first show that $\mathcal {F}_{F,\hat{F}}$ stabilizes. For this, we have to provide an equivalence

$$\begin{aligned}&( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \\&\quad \sim ( A_i,\hat{A}_i,R_i+y\hat{F} + \hat{y} F,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \end{aligned}$$

of adjusted differential cocycles. We set $p_i:= y$ and $\hat{p}_i:=- \hat{y}$, as well as $z_{ij}=\hat{z}_{ij}=0$. Moreover, we put $\phi _i:=- \hat{y} A_i$, and $\tilde{e}_{ij}:=-\hat{y} a_{ij}$. Now, (LD-E1’) and (LD-E2’) hold since $p_i$ and $\hat{p}_i$ are constant. (6.2) and (6.4) follow directly from the definitions. Finally, (5.50) becomes

$$\begin{aligned} \tilde{e}_{ij}+\tilde{e}_{jk} = \tilde{e}_{ik}+\hat{y}m_{ijk} \end{aligned}$$

and thus follows from (LD1).

Next we show that no other group elements stabilize. For this, we suppose that a Deligne 2-cocycle $(B_i,A_i,c_{ijk})$ acts trivially, i.e., that we have an equivalence between adjusted differential $\mathbb{T}\mathbb{D}$-cocycles

$$\begin{aligned}{} & {} ( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \\{} & {} \quad \sim ( A_i,\hat{A}_i,R_i+B_i,\varphi _{ij}+A_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}+c_{ijk}). \end{aligned}$$

Let $(\phi _i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},\tilde{e}_{ij})$ be a tuple expressing this equivalence. We start by looking at (LD-E1’) and (LD-E2’), which here result in ${\textrm{d}}p_i={\textrm{d}}\hat{p}_i=0$; in other words, these functions are constant. We further have $z_{ij} = p_i-p_j$. This means that $[z_{ij}] \in {\textrm{H}}^1(X,{\mathbb {Z}}^{n})$ goes to zero under the map to ${\textrm{H}}^1(X,{\mathbb {R}}^{n})$. But this map is injective, as the relevant part of the long exact sequence is

and the second arrow is surjective. Thus, there exist $z_i\in {\mathbb {Z}}^{n}$ such that $z_{ij}=z_i-z_j$. Observe that $p_i-z_i=p_j-z_j$, i.e., there is a real number $y\in {\mathbb {R}}$ such that $y=p_i-z_i$. Analogously, we treat $\hat{z}_{ij}$, getting $\hat{y}\in {\mathbb {R}}$ such that $\hat{y} = \hat{p}_i - \hat{z}_i$. We consider now

$$\begin{aligned} f_{ij}(x):=\tilde{e}_{ij}(x) + \hat{y} a_{ij} +\hat{a}_{ij}z_i; \end{aligned}$$

then, one can show using (5.50) that $f_{ij}$ trivializes $c_{ijk}$, i.e.,

$$\begin{aligned} f_{ik}-f_{ij}-f_{jk}=c_{ijk}. \end{aligned}$$

Next we define $H_i \in \Omega ^1(U_i)$ by $H_i:= -\phi _i+z_i \hat{A}_i+\hat{y} A_i$. Then we compute, using (6.2) and the fact that $\hat{p}_j$ is constant

$$\begin{aligned} A_{ij}=H_i - H_j - f_{ij}^{*}\theta . \end{aligned}$$

Finally, we get from (6.4) that

$$\begin{aligned} B_i={\textrm{d}}H_i+y\hat{F}-\hat{y} F. \end{aligned}$$

Summarizing, the last three equations show that there exist $y,\hat{y}\in {\mathbb {R}}$ such that $(B_i,A_{ij},c_{ijk}) \sim (y\hat{F}-\hat{y} F,0,0)$, i.e., $(B_i,A_{ij},c_{ijk})\in \mathcal {F}_{F,\hat{F}}$.

It remains to prove the transitivity statement. For this, we suppose that we have two differential cocycles

$$\begin{aligned} (A_i,\hat{A}_i,R_i,\phi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \\ (A'_i,\hat{A}'_i,R'_i,\phi '_{ij},a'_{ij},\hat{a}'_{ij},m'_{ijk},\hat{m}'_{ijk},t'_{ijk}) \end{aligned}$$

and given equivalences $(p_i,z_{ij})$ and $(\hat{p}_i,\hat{z}_{ij})$ between the cocycles of the projected principal ${\mathbb {T}}^{n}$-bundles with connection, $(A_i,a_{ij},m_{ijk})$ and $(A'_i,a'_{ij},m_{ijk}')$, and $(\hat{A}_i,\hat{a}_{ij},\hat{m}_{ijk})$ and $(\hat{A}'_i,\hat{a}'_{ij},\hat{m}_{ijk}')$, respectively. We have to find a Deligne 2-cocycle $(B_i,A_{ij},c_{ijk})$ such that

$$\begin{aligned}{} & {} ( A_i,\hat{A}_i,R_i+B_i,\varphi _{ij}+A_{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}+c_{ijk})\nonumber \\{} & {} \quad \sim (A'_i,\hat{A}'_i,R'_i,\varphi '_{ij},a'_{ij},\hat{a}'_{ij},m'_{ijk},\hat{m}'_{ijk},t'_{ijk}). \end{aligned}$$

(6.13)

This is achieved by the definitions

$$\begin{aligned} B_i&:= R_i'-R_i-p_i\hat{F} \\ A_{ij}&:=-\varphi _{ij}+\varphi '_{ij}-z_{ij}\hat{A}_j \\ c_{ijk}&:= t'_{ijk}-t_{ijk}+\hat{p}_km_{ijk}'-z_{ij}\hat{a}_{jk} \end{aligned}$$

It is indeed straightforward to check using (LD-E1) and (LD-E2) and (6.1) and (5.49) that $(B_i,A_{ij},c_{ijk})$ is a Deligne 2-cocycle. In order to establish the equivalence (6.13), we set $\phi _i:= 0$. (6.4) is then obviously satisfied. The next part of the equivalence is (6.2), which here reads

$$\begin{aligned} z_{ij}\hat{A}_j- z_{ij}\hat{A}_j'&=-\tilde{e}_{ij}^{*}\theta . \end{aligned}$$

This is satisfied by putting $\tilde{e}_{ij}:= -z_{ij}\hat{p}_j$. The last equivalence conditions is now (5.50), which follows immediately from (LD-E1) and (LD-E2).

6.3 Kahle–Valentino’s T-Duality Pairs

In this section we discuss the relation between differential T-duality correspondences as introduced in Definition 6.2 and differential T-duality pairs considered by Kahle–Valentino [35].

The setting of Kahle–Valentino [35] is different as it does not explicitly involve string backgrounds. Their discussion is also limited to the case of torus dimension $n=1$. At the basis of their formalism is a groupoid version of differential cohomology, of which below we recall a slightly simplified version. We consider differential cohomology groupoids $\mathcal {H}^p(X)$, so that the set of isomorphism classes of objects of $\mathcal {H}^{n}(X)$ is the ordinary differential cohomology group $\hat{{\textrm{H}}}^n(X)$. Differential cohomology groupoids are supposed to be equipped with cup product functors

$$\begin{aligned} \cup : \mathcal {H}^p(X) \times \mathcal {H}^q(X) \rightarrow \mathcal {H}^{p+q}(X). \end{aligned}$$

Moreover, they come equipped with a functor

$$\begin{aligned} \mathcal {I}: \Omega ^{p-1}(X)_{dis} \rightarrow \mathcal {H}^{p}(X), \end{aligned}$$

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 $\xi \in \mathcal {H}^p(X)$ is a differential form $K \in \Omega ^{p-1}(X)$ and an isomorphism $\tau : \xi \rightarrow \mathcal {I}_{K}$ in $\mathcal {H}^{p}(X)$. The set $\hat{{\textrm{H}}}^{p-1}(X)$ acts on the set of all geometric trivializations of $\xi$, where $[\eta ]\in \hat{{\textrm{H}}}^2(X)$ sends $\tau$ to $\tau +\eta$, and K gets shifted by the “curvature” of $\eta$. 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 [35, §A.2]. The objects of $\mathcal {H}^p(X)$ are Deligne $(p-1)$-cocycles $\xi$, and the morphisms $\xi _1 \rightarrow \xi _2$ are equivalence classes $[\eta ]$ of $(p-2)$-cochains $\eta$ satisfying $\xi _2=\xi _1+\textrm{D}\eta$, where $\textrm{D}$ denotes the Deligne differential, and $\eta _1\sim \eta _2$ if there exists a $(p-3)$-cochain $\beta$ with $\eta _2=\eta _1+\textrm{D}\beta$. 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 $\mathcal {I}$ is the usual inclusion $\varphi \mapsto (\varphi ,0,.,,,0)$ of a globally defined differential form as a “topologically trivial” Deligne cocycle. For $p=2$, the groupoid $\mathcal {H}^2(X)$ is equivalent to the groupoid of principal ${\mathbb {T}}$-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 by $\hat{{\textrm{H}}}^1(X)=C^{\infty }(X,{\mathbb {T}})$ is the action of smooth ${\mathbb {T}}$-valued functions on sections.

Definition 6.12

A differential T-duality pair consists of two objects $\xi ,\hat{\xi } \in \mathcal {H}^2(X)$ and a geometric trivialization $\tau : \xi \cup \hat{\xi } \rightarrow \mathcal {I}_K$.

Kahle–Valentino claim in [35, §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 6.13

Two differential T-duality pairs $(\xi ,\hat{\xi },K,\tau )$ and $(\xi ',\hat{\xi }',K',\tau ')$ over X are equivalent if $K'=K$ and there exist isomorphisms $p:\xi \rightarrow \xi '$ and $\hat{p}:\hat{\xi } \rightarrow \hat{\xi }'$ in $\mathcal {H}^2(X)$ such that the diagram

in $\mathcal {H}^4(X)$ is commutative. The set of equivalence classes of differential T-duality pairs is denoted by ${\textrm{TDP}}(X)$.

Note that the projection to the objects $\xi ,\hat{\xi }$ gives a well-defined map

$$\begin{aligned} {\textrm{TDP}}(X) \rightarrow \hat{{\textrm{H}}}^2(X) \times \hat{{\textrm{H}}}^2(X). \end{aligned}$$

Below, we will prove the following result.

Proposition 6.14

There is a canonical bijection between equivalence classes of differential T-duality correspondences and equivalence classes of differential T-duality pairs,

such that the diagram

is commutative.

Because of the cup product, it is necessary to work with extended Deligne cohomology, i.e., in degree p with the sheaf complex

$$\begin{aligned} {\mathbb {Z}}\rightarrow \underline{{\mathbb {R}}} \rightarrow \underline{\Omega }^1 \rightarrow \underline{\Omega }^2 \rightarrow \cdots \rightarrow \underline{\Omega }^{p}, \end{aligned}$$

(6.14)

whereas before we worked with the quasi-isomorphic complex . In order to be more precise, let us denote the complex (6.14) by $\mathcal {D}^q(p)$, so that, for instance, $\mathcal {D}^{-1}(p)={\mathbb {Z}}$ and . The Deligne coboundary operator on the corresponding Čech double complex $C^r(\mathcal {D}^{q}(p))$ is defined to be $\textrm{D}^{r,q}:= (-1)^{q+1}\delta ^{r} + {\textrm{d}}^{q}$, where ${\textrm{d}}^{-1}$ is the inclusion . The cup product of extended Deligne cocycles

$$\begin{aligned} \xi&=(A^{p-1}_i,A^{p-2}_{i_1i_2},\ldots ,A^{0}_{i_1,\ldots ,i_p},m_{i_1,\ldots ,i_{p+1}}) \quad \text { and }\quad \\ \hat{\xi }&=(\hat{A}^{q-1}_i,\hat{A}^{q-2}_{i_1i_2},\ldots ,\hat{A}^{0}_{i_1,\ldots ,i_q},\hat{m}_{i_1,\ldots ,i_{q+1}}) \end{aligned}$$

is defined in the usual way [12, §1.5][24, Sec. 2.2] by

$$\begin{aligned} \xi \cup \hat{\xi }:= & {} (A^{p-1}_{i_1}\wedge {\textrm{d}}\hat{A}_{i_1}^{q-1},\ldots ,\;A^0_{i_1,\ldots ,i_p}\wedge {\textrm{d}}\hat{A}^{q-1}_{i_{p+1}}, \nonumber \\{} & {} \quad m_{i_1,\ldots ,i_{p+1}}\hat{A}^{q-1}_{i_{p+1}},\ldots ,\;m_{i_1,\ldots ,i_{p+1}}\hat{A}^0_{i_{p+1},\ldots ,i_{p+q}},\;\nonumber \\{} & {} \quad m_{i_1,\ldots ,i_{p+1}}\hat{m}_{i_{p+1},\ldots ,i_{p+q+1}}).\nonumber \\ \end{aligned}$$

(6.15)

Of most importance for us is the cup product of two objects $\xi ,\hat{\xi }\in \mathcal {H}^2(X)$. Namely, for $\xi =(A_{i},a_{ij},m_{ijk})$ and $\hat{\xi }=(\hat{A}_i,\hat{a}_{ij},\hat{m}_{ijk})$ we obtain

$$\begin{aligned} \xi \cup \hat{\xi }=(A_{i}\wedge \hat{F}\;,\;a_{ij}\hat{F}\;,\;m_{ijk}\hat{A}_{k}\;,\;m_{ijk}\hat{a}_{kl}\;,\;m_{ijk}\hat{m}_{klp}). \end{aligned}$$

(6.16)

Unfortunately, I have not been able to find a description for the cup product of morphisms in $\mathcal {H}^p(X)$. Kahle–Valentino just claim in [35, §A.2] that the cup product “extends” to morphism, but do not explain how, whereas the obvious attempt, namely to apply formula (6.15) to cochains, does not work. Concretely, we need the cup product of two morphisms $[\eta ]:\xi \rightarrow \xi '$ and that $[\hat{\eta }]:\hat{\xi } \rightarrow \hat{\xi }'$ in $\mathcal {H}^2(X)$, i.e., $\xi '=\xi +\textrm{D}\eta$ and $\hat{\xi }'=\hat{\xi } + \textrm{D}\hat{\eta }$. Suppose $\eta =(p_i,z_{ij})$ and $\hat{\eta }=(\hat{p}_i,\hat{z}_{ij})$. The only way that I was able to produce a 3-cochain $\eta \cup \hat{\eta }$ such that $\xi '\cup \hat{\xi }' =\xi \cup \hat{\xi } +\textrm{D}(\eta \cup \hat{\eta })$ is

$$\begin{aligned} \eta \cup \hat{\eta } = (p_i\hat{F}\;,\;z_{ij}\hat{A}_j\;,\; z_{ij}\hat{a}_{jk}-m'_{ijk}\hat{p}_k\;,\;m'_{ijk}\hat{z}_{kl}+z_{ij}\hat{m}_{jkl}). \end{aligned}$$

(6.17)

In the following, I assume that this is the correct cup product of morphisms in $\mathcal {H}^2(X)$.

Exploring the notion of a differential T-duality pair further, we spell out in the following what a geometric trivialization of $\xi \cup \hat{\xi }$ from (6.16) is. It consists of:

• (a) a 3-form $K\in \Omega ^3(X)$

• (b) 2-forms $R_i \in \Omega ^2(U_i)$, such that

  $$\begin{aligned} A_i \wedge \hat{F} = K + {\textrm{d}}R_i. \end{aligned}$$

  (6.18)

• (c) 1-forms $\varphi _{ij}\in \Omega ^1(U_i \cap U_j)$ such that

  $$\begin{aligned} a_{ij}\hat{F}=-R_j+R_i + {\textrm{d}}\varphi _{ij}. \end{aligned}$$

  (6.19)

• (d) functions $b_{ijk}:U_i \cap U_j \cap U_k \rightarrow {\mathbb {R}}$ such that

  $$\begin{aligned} m_{ijk}\hat{A}_k = \varphi _{ij}+\varphi _{jk}-\varphi _{ik}+{\textrm{d}}b_{ijk}. \end{aligned}$$

  (6.20)

• (e) numbers $q_{ijkl}\in {\mathbb {Z}}$ satisfying

  $$\begin{aligned} m_{ijk}\hat{a}_{kl} = q_{ijkl} + b_{ijk}+b_{ikl}-b_{ijl}-b_{jkl} \end{aligned}$$

  (6.21)

  and

  $$\begin{aligned} m_{ijk}\hat{m}_{klp}=q_{ijkl}-q_{ijkp}+q_{ijlp}-q_{iklp}+q_{jklp}. \end{aligned}$$

  (6.22)

At this point, it makes sense to discuss the action of $\hat{{\textrm{H}}}^3(X)$ on differential T-duality pairs, which is induced by the above-mentioned action of $\hat{{\textrm{H}}}^3(X)$ on all geometric trivializations of $\xi \cup \hat{\xi }$. Here, this action takes the form

$$\begin{aligned} \hat{{\textrm{H}}}^3(X) \times {\textrm{TDP}}(X) \rightarrow {\textrm{TDP}}(X) \end{aligned}$$

and is given, using the above description of geometric trivializations, by the formula

$$\begin{aligned}&((B_i,A_{ij},c_{ijk},s_{ijkl}), (K,R_i,\varphi _{ij},b_{ijk},q_{ijkl}))\\&\quad \mapsto (K+{\textrm{d}}B_i,R_i+B_i,\varphi _{ij}+A_{ij},b_{ijk}+c_{ijk},q_{ijkl}+s_{ijkl}). \end{aligned}$$

Note that K is shifted by the globally defined 3-form $H={\textrm{d}}B_i$, the curvature. It is clear that this action restricts to the fibers of the map ${\textrm{TDP}}(X) \rightarrow \hat{{\textrm{H}}}^2(X) \times \hat{{\textrm{H}}}^2(X)$ and is transitive in each fiber.

As in Sect. 6.2, see Proposition 6.11, we consider the pair $(F,\hat{F}) \in \Omega ^2(X) \times \Omega ^2(X)$ determined by an element of $\hat{{\textrm{H}}}^2(X) \times \hat{{\textrm{H}}}^2(X)$, and the subgroup $\mathcal {F}_{F,\hat{F}} \subseteq \hat{{\textrm{H}}}^3(X)$.

Lemma 6.15

The subgroup $\mathcal {F}_{F,\hat{F}}$ acts trivially, and the quotient $\hat{{\textrm{H}}}^3(X)/\mathcal {F}_{F,\hat{F}}$ acts freely and transitively in the fiber over $(\xi ,\hat{\xi })$.

Proof

We have to show that $(K,R_i,\varphi _{ij},b_{ijk},q_{ijkl})$ and $(K,R_i+\hat{y} F + y\hat{F},\varphi _{ij}, b_{ijk},q_{ijkl})$ define the same morphism. We consider the automorphism of $\xi =(A_{i},a_{ij},m_{ijk})$ given by (y, 0), this works as $\textrm{D}(y,0)=(0,0,0)$, and similarly, the automorphism of $\hat{\xi }$ given by $(\hat{y},0)$. According to (6.17), we have

$$\begin{aligned} (y,0)\cup (\hat{y},0)=(y\hat{F}\;,\;0\;,\; -m_{ijk}\hat{y},\;0) \end{aligned}$$

We can change this by the coboundary of $(-\hat{y} A_i,-\hat{y}a_{ij},0)$, which is $(-\hat{y} F,0, \hat{y} m_{ijk},0)$. Thus,

$$\begin{aligned}{}[(y,0)\cup (\hat{y},0)] = [(y \hat{F}-\hat{y} F,0,0,0)]. \end{aligned}$$

This proves that $\mathcal {F}$ acts trivially. Conversely, if

$$\begin{aligned} (K,R_i,\varphi _{ij},b_{ijk},q_{ijkl}) \quad \text { and }\quad (K+{\textrm{d}}B_i,R_i+B_i,\varphi _{ij}+A_{ij},b_{ijk}+c_{ijk},q_{ijkl}+s_{ijkl}) \end{aligned}$$

are equivalent, we have to show that $(B_i,A_{ij},c_{ijk},s_{ijkl})\sim (\hat{y} F-y\hat{F},0,0,0)$. The proof of this is very similar to the one given in Proposition 6.11, and omitted for brevity.

Finally, we are in a position to give the proof of Proposition 6.14. Under Proposition 5.17 and Lemma 6.9, it remains to provide a bijection

$$\begin{aligned} \hat{{\textrm{H}}}^1(X,\mathbb{T}\mathbb{D}_{\kappa }) \rightarrow {\textrm{TDP}}(X). \end{aligned}$$

(6.23)

Let $( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk})$ be an adjusted differential $\mathbb{T}\mathbb{D}$-cocycle. We set $\xi =(A_i,a_{ij},-m_{ijk})$ and $\hat{\xi }=(\hat{A}_i,\hat{a}_{ij},-\hat{m}_{ijk})$: this ensures that the diagram in Proposition 6.14 will commute, while the signs accounts for the different conventions used in non-abelian cohomology and Deligne cohomology. We define

$$\begin{aligned} K:= {\textrm{d}}R_i + A_i \wedge \hat{F}; \end{aligned}$$

(6.24)

this is the 3-curvature of Remark 6.7, and hence a globally defined 3-form. Thus, passing to $-R_i$, we have (6.18). We may then use the given 1-form $\varphi _{ij}$, and note that (6.3) results into (6.19). Next, we choose real-valued functions $b_{ijk}$ that represent the given ${\mathbb {T}}$-valued functions $-t_{ijk}$; then, (6.1) results into (6.20). Finally, we consider (5.49),

$$\begin{aligned} t_{ikl}+t_{ijk}-m_{ijk}\hat{a}_{kl} = t_{ijl}+t_{jkl}, \end{aligned}$$

which is an equation of ${\mathbb {T}}$-valued functions. Substituting the lifts $b_{ijk}$ reveals $q_{ijkl}\in {\mathbb {Z}}$ such that

$$\begin{aligned} -b_{ikl}-b_{ijk}-m_{ijk}\hat{a}_{kl} = -b_{ijl}-b_{jkl} + q_{ijkl}, \end{aligned}$$

this is (6.21). Finally, (6.22) is a straightforward calculation. Summarizing, $(K,R_i,\varphi _{ij},b_{ijk},q_{ijkl})$ is a geometric trivialization of $\xi \cup \hat{\xi }$.

Next we consider an equivalence between adjusted differential cocycles

$$\begin{aligned}&( A_i,\hat{A}_i,R_i,\varphi _{ij},a_{ij},\hat{a}_{ij},m_{ijk},\hat{m}_{ijk},t_{ijk}) \quad \text { and }\quad \\&\quad ( A_i',\hat{A}_i',R_i',\varphi _{ij}' ,a_{ij}',\hat{a}_{ij}',m_{ijk}',\hat{m}_{ijk}',t_{ijk}') \end{aligned}$$

established by a tuple $(\phi _i,p_i,\hat{p}_i,z_{ij},\hat{z}_{ij},\tilde{e}_{ij})$. Then, $\eta :=(-p_i,z_{ij})$ and $\hat{\eta }:=(-\hat{p}_i,\hat{z}_{ij})$ are morphisms in $\mathcal {H}^2(X)$ between $\xi$ and $\xi ':=( A_i',a'_{ij},-m'_{ijk})$, and $\hat{\xi }$ and $\hat{\xi }'=(\hat{A}'_i,\hat{a}_{ij}',-\hat{m}'_{ijk})$, respectively. We have to show that

$$\begin{aligned} (K,R_i,\varphi _{ij},b_{ijk},q_{ijkl})\sim (K',R_i',\varphi _{ij}',b_{ijk}',q_{ijkl}')+\eta \cup \hat{\eta }. \end{aligned}$$

We claim that both cochains differ in fact by the coboundary of $(\phi _i,z_{ij}\hat{p}_j+f_{ij},-r_{ijk}+z_{ij}\hat{z}_{jk})$, where $f_{ij}$ is a real-valued lift of $\tilde{e}_{ij}$, and $r_{ijk}\in {\mathbb {Z}}$ are the numbers that emerge from the ${\mathbb {T}}$-valued cocycle condition (5.50) under this lift. The claim is straightforward to check using (6.17).

By now we have constructed a well-defined map (6.23), such that it preserves the fibers of the projections to $\hat{{\textrm{H}}}^2(X) \times \hat{{\textrm{H}}}^2(X)$. It is easy to see that our map (6.23) is equivariant w.r.t. to the actions of $\hat{{\textrm{H}}}^3(X)/\mathcal {F}_{F,\hat{F}}$ in each fiber. Since these actions are free and transitive on both sides (Lemma 6.15 and Proposition 6.11), it follows that (6.23) is a bijection. This proves Proposition 6.14.

7 Examples of Geometric T-Duality

We first consider in Sect. 7.1 the situation of a general principal ${\mathbb {T}}^{n}$-bundle E, a general metric, and trivial B-field, and present a construction of a T-dual geometric T-background. In Sect. 7.2 we specialize to the case that E is the Hopf fibration, in which we explicitly compute the dual metric and dual bundle gerbe. In Sect. 7.3 we keep the Hopf fibration but consider a non-trivial B-field, whose Dixmier-Douady class is a generator of ${\textrm{H}}^3(S^3,{\mathbb {Z}})$. We prove that this geometric T-background is self-dual.

7.1 A Torus Bundle with Trivial B-Field

We consider a geometric T-background $(E,g,\mathcal {G})$ over a smooth manifold X, whose bundle gerbe is the trivial one, i.e., $\mathcal {G}=\mathcal {I}_0$. In this section, we explicitly construct a geometric T-duality correspondence whose left leg is $(E,g,\mathcal {I}_0)$.

We let $(\omega ,g',h)$ be the triple corresponding to g under Theorem 2.7, and we let $F \in \Omega ^2(X)$ be the curvature of the connection $\omega$. We consider the trivial bundle $\hat{E}:= X \times {\mathbb {T}}^{n}$, and equip it with the trivial connection, $\hat{\omega }:= \theta$. We let $\hat{g}$ be the invariant metric on $\hat{E}$ that corresponds to the triple $(\hat{\omega },g',h^{-1})$. Next we construct the bundle gerbe $\hat{\mathcal {G}}$ over $\hat{E}$.

The surjective submersion is $Y:=E \times {\mathbb {T}}^{n} \rightarrow X \times {\mathbb {T}}^{n}$. The curving is

$$\begin{aligned} \Psi := {\textrm{pr}}_E^{*}\omega \wedge {\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\theta \in \Omega ^2(Y). \end{aligned}$$

The 2-fold fiber product is $Y^{[2]} = E^{[2]} \times {\mathbb {T}}^{n}$. Note that we have a smooth map $g:E^{[2]} \rightarrow {\mathbb {T}}^{n}$, $e_2=e_1g(e_1,e_2)$, and ${\textrm{pr}}_2^{*}\omega = {\textrm{pr}}_1^{*}\omega + g^{*}\theta$. Thus, we see that

$$\begin{aligned} {\textrm{pr}}_2^{*}\Psi - {\textrm{pr}}_1^{*}\Psi =g^{*}\theta \dot{\wedge }{\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\theta \end{aligned}$$

on $Y^{[2]}$. Comparing with Remark 2.5, the right hand side is the curvature of the pullback of the Poincaré bundle $\mathcal {P}$ along the map $\tilde{g}: E^{[2]} \times {\mathbb {T}}^{n} \rightarrow {\mathbb {T}}^{2n}: (e_1,e_2,a)\mapsto (a,g(e_1,e_2))$. Thus, we readily define

$$\begin{aligned} P:=\tilde{g}^{*}\mathcal {P}\end{aligned}$$

as the principal ${\mathbb {T}}$-bundle with connection of $\hat{\mathcal {G}}$. Over $Y^{[3]}=E^{[3]} \times {\mathbb {T}}^{n}$, we have an isomorphism

$$\begin{aligned}{} & {} ({\textrm{pr}}_{23}^{*}P\otimes {\textrm{pr}}_{12}^{*}P)_{(e_1,e_2,e_3,a)} = \mathcal {P}_{a,g(e_2,e_3)} \otimes \mathcal {P}_{a,g(e_1,e_2)} \\{} & {} {\mathop {\longrightarrow }\limits ^{\varphi _r}} \mathcal {P}_{a,g(e_2,e_3)g(e_1,e_2)}=\mathcal {P}_{a,g(e_1,e_3)}=({\textrm{pr}}_{13}^{*}P)_{(e_1,e_2,e_3,a)}, \end{aligned}$$

where $\varphi _r$ was defined in Sect. 2.2. This isomorphism satisfies the associativity condition over $Y^{[4]}$ due to the commutativity of the analog of (2.3) for $\varphi _r$.

Remark 7.1

If $n=1$, then $\hat{\mathcal {G}}$ is precisely the cup product bundle gerbe ${\textrm{pr}}_X^{*}E \cup {\textrm{pr}}_{{\mathbb {T}}}$, where ${\textrm{pr}}_X: \hat{E} \rightarrow X$ and ${\textrm{pr}}_{{\mathbb {T}}}: \hat{E} \rightarrow {\mathbb {T}}^1$ are the projections; explicitly, ${\textrm{pr}}_X^{*}E$ is a principal ${\mathbb {T}}$-bundle over $\hat{E}$ with connection, and ${\textrm{pr}}_{{\mathbb {T}}}$ is a ${\mathbb {T}}$-valued function on $\hat{E}$. A description of the cup product of such structures, resulting in a bundle gerbe with connection, has been given by Johnson in [31]. Our construction above (for $n=1$) reproduced exactly that description. Johnson also proved that the cup product of a principal ${\mathbb {T}}$-bundle with connection and a ${\mathbb {T}}$-valued function coincides with the cup product in Deligne cohomology [31].

We will now construct a geometric T-duality correspondence between the geometric T-backgrounds $(E,g,\mathcal {G})$ and $(\hat{E},\hat{g},\hat{\mathcal {G}})$. On correspondence space $E \times _X \hat{E}$ we need to find a connection-preserving isomorphism $\mathcal {D}:{\textrm{pr}}^{*}\mathcal {G}\rightarrow \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \otimes \mathcal {I}_{\rho _{g,\hat{g}}}$, where

$$\begin{aligned} \rho _{g,\hat{g}} = \hat{{\textrm{pr}}}^{*}\hat{\omega } \dot{\wedge }{\textrm{pr}}^{*}\omega = -\Psi . \end{aligned}$$

We note that $\hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}}$ is trivializable since its surjective submersion has a section $\sigma$ along $\hat{{\textrm{pr}}}$, namely, the identity, $\sigma ={\textrm{id}}_{E \times {\mathbb {T}}^{n}}$:

It induces a trivialization $\mathcal {S}: \hat{{\textrm{pr}}}^{*}\hat{\mathcal {G}} \rightarrow \mathcal {I}_{\Psi }$, and $\mathcal {D}$ may be defined as

Thus, $\mathcal {D}$ is a geometric correspondence. It remains to check that it is a geometric T-duality correspondence.

Conditions (T1) and (T2) of Definition 4.9 hold by construction of the metric $\hat{g}$. In order to check condition (T3), we consider an open subset $U \subseteq X$ that admits a trivialization $\varphi : U \times {\mathbb {T}}^{n} \rightarrow E|_U$. On the dual side, we choose the identity trivialization, $\hat{\varphi }={\textrm{id}}$. We put $B:= 0$ and $\mathcal {T}:= {\textrm{id}}$, as a trivialization of $\mathcal {I}_0=\varphi ^{*}\mathcal {G} \rightarrow \mathcal {I}_B$. Note that $\hat{\varphi }^{*}\hat{\mathcal {G}}=\hat{\mathcal {G}}|_{U \times {\mathbb {T}}^{n}}$. Thus, the surjective submersion of $\hat{\varphi }^{*}\hat{\mathcal {G}}$ has a global section, $\tau :=(\varphi ,{\textrm{pr}}_{{\mathbb {T}}^{n}}):U \times {\mathbb {T}}^{n} \rightarrow E \times {\mathbb {T}}^{n}$. It induces a trivialization $\hat{\mathcal {T}}: \hat{\varphi }^{*}\hat{\mathcal {G}} \rightarrow \mathcal {I}_{\tau ^{*}\Psi }$. We put

$$\begin{aligned} \hat{B} := \tau ^{*}\Psi = \varphi ^{*}\omega \dot{\wedge }{\textrm{pr}}_{{\mathbb {T}}^{n}}^{*}\theta . \end{aligned}$$

Now we work over $U \times {\mathbb {T}}^{2n}$, where we find the diagram

whose rectangular part is commutative, but the sections differ. This means that the induced trivializations $\Phi ^{*}\mathcal {S}$ and $\hat{{\textrm{pr}}}^{*}\hat{\mathcal {T}}$ differ by the ${\mathbb {T}}$-bundle with connection

$$\begin{aligned} (\sigma \circ \Phi , \tau \circ \hat{{\textrm{pr}}})^{*}P; \end{aligned}$$

a discussion of this fact can be found in [50, Lem. 3.2.3]. We readily compute the map

$$\begin{aligned} k:=\tilde{g} \circ (\sigma \circ \Phi , \tau \circ \hat{{\textrm{pr}}}): U \times {\mathbb {T}}^{2n} \rightarrow {\mathbb {T}}^{2n}:(x,a,\hat{a}) \mapsto (\hat{a},\hat{a}-a). \end{aligned}$$

We note that $k^{*}\mathcal {P}\cong \mathcal {P}_{3,3-2}\cong \mathcal {P}_{3,3} \otimes \mathcal {P}_{3,-2}\cong \mathcal {P}_{2,3}$, using the results of Sect. 2.2. The 2-isomorphism $\hat{{\textrm{pr}}}^{*}\hat{\mathcal {T}} \cong \Phi ^{*}\mathcal {S} \otimes k^{*}\mathcal {P}$ implies that the relevant isomorphism of (T3),

corresponds to the principal ${\mathbb {T}}$-bundle $k^{*}\mathcal {P}\cong {\textrm{pr}}_{{\mathbb {T}}^{2n}}^{*}\mathcal {P}$. This completes the proof the we have a geometric T-duality correspondence. In particular, by Proposition 4.14, the Buscher rules hold locally.

7.2 The Hopf Fibration with a Trivial B-Field

In this section, we apply the construction of the previous Sect. 7.1 to the example where the torus bundle E is the Hopf fibration $E:=S^3 \rightarrow S^2$. This reproduces a result from the PhD thesis of Kunath [34, §3.4, §4.4], where that case has been discussed separately.

We denote the round metric on the n-sphere by $g_n$; the metric on E is $g=g_{3}$, which is indeed ${\mathbb {T}}$-invariant. Then, the dual torus bundle is $\hat{E}:=S^2 \times {\mathbb {T}}$. This was probably the first observation of a topology change, and made in [1]. There, the following result has been proved, by applying locally the Buscher rules. Here, we re-derive it by applying the general procedure of Sect. 7.1.

Lemma 7.2

The dual metric is $\hat{g}=\frac{1}{4}g_{2} \oplus g_{1}$.

Proof

We claim that $g'=\frac{1}{4}g_2$ and $h=g_1$. Then, Remark 2.9 applied to $\hat{E}$ and the trivial connection $A=0$ yields the lemma. The claim can be proved in an explicit model for the Hopf fibration. We model $p:S^3 \rightarrow S^2$ as the restriction to unit vectors of the map

$$\begin{aligned} {\mathbb {R}}^4 \rightarrow {\mathbb {R}}^3: (x_0,x_1,x_2,x_3) \mapsto (2(x_0x_2+x_1x_3),2(x_1x_2-x_0x_3),x_0^2+x_1^2-x_2^2-x_3^2). \end{aligned}$$

The action of $z\in {\mathbb {T}}$ sends $(x_0,x_1,x_2,x_3)$ to $(x_0',x_1',x_2',x_3')$, where

$$\begin{aligned} x_0' + {\textrm{i}}x_1' := z\cdot (x_0+{\textrm{i}}x_1) \quad \text { and }\quad x_2' + {\textrm{i}}x_3' := z\cdot (x_2+{\textrm{i}}x_3). \end{aligned}$$

The tangent space $T_xS^3$ at $x\in S^3\subseteq {\mathbb {R}}^4$ is $x^{\perp }\subseteq {\mathbb {R}}^4$. The round metric $g_3$ is given by the standard inner product on ${\mathbb {R}}^4$, i.e., $g_3(v,w):=v\cdot w$. (One can see now directly that it is ${\mathbb {T}}$-invariant.) The differential of the bundle projection p at $x=(x_0,x_1,x_2,x_3)$ is

$$\begin{aligned} T_xp =2 \begin{pmatrix}x_2 &{}\quad x_3 &{}\quad x_0 &{}\quad x_1 \\ -x_3 &{}\quad x_2 &{}\quad x_1 &{}\quad -x_0 \\ x_0 &{}\quad x_1 &{}\quad -x_2 &{}\quad -x_3 \\ \end{pmatrix}. \end{aligned}$$

One computes

$$\begin{aligned} V_x=\textrm{Kern}(T_xp)=\left\langle (-x_1,x_0,-x_3,x_2) \right\rangle \end{aligned}$$

and thus,

$$\begin{aligned} h_{p(x)}(r,s):=g_3((-rx_1,rx_0,-rx_3,rx_2),(-sx_1,sx_0,-sx_3,sx_2))=rs. \end{aligned}$$

In particular, this metric does not depend on the base point p(x). We observe that $T_xp\cdot T_xp^{\textrm{tr}}=4E_4$, where $E_4$ denotes the unit matrix. We know that $T_xp|_{H_x}:H_x\rightarrow T_{p(x)}X$ is an isomorphism, and so $\frac{1}{4}T_xp^{\textrm{tr}}$ is a right inverse. Thus,

$$\begin{aligned} g(v,w):=\textstyle g_3(\frac{1}{4}T_xp^{\textrm{tr}}(v),\frac{1}{4}T_xp^{\textrm{tr}}(w))=\frac{1}{4}vw=\frac{1}{4}g_2(v,w). \end{aligned}$$

This proves the claim.

By Remark 7.1, the dual bundle gerbe $\hat{\mathcal {G}}$ is the cup product $\hat{\mathcal {G}} = {\textrm{pr}}_{S^2}^{*}E \cup {\textrm{pr}}_{S^1}$ of the principal ${\mathbb {T}}$-bundle ${\textrm{pr}}_{S^2}^{*}E$ and the ${\mathbb {T}}$-valued function ${\textrm{pr}}_{S^1}$. Summarizing, we have the following result.

Proposition 7.3

Let $E:= S^3 \rightarrow S^2$ be the Hopf fibration, $g:= g_3$ be the round metric, and $\mathcal {G}=\mathcal {I}_0$ be the trivial bundle gerbe. Then, there exists a geometric T-duality correspondence between $(E,g,\mathcal {G})$ and the geometric T-background $(S^2 \times S^1,\frac{1}{4}g_2\oplus g_1,\hat{\mathcal {G}})$, where $\hat{\mathcal {G}} = {\textrm{pr}}_{S^2}^{*}E \cup {\textrm{pr}}_{S^1}$ is the cup product bundle gerbe. In particular, the Dixmier-Douady class of $\hat{\mathcal {G}}$ is a cup product in singular cohomology,

$$\begin{aligned} \textrm{DD}(\hat{\mathcal {G}}) = {\textrm{pr}}_{S^2}^{*}c_1 \cup {\textrm{pr}}_{S^1}^{*}\theta , \end{aligned}$$

where $c_1$ is the first Chern class of the Hopf fibration, a generator of $\in {\textrm{H}}^2(S^2,{\mathbb {Z}})$, and $\theta \in {\textrm{H}}^1(S^1,{\mathbb {Z}})$ is a generator. Thus, $\textrm{DD}(\hat{\mathcal {G}})$ is a generator of ${\textrm{H}}^3(S^2 \times S^1,{\mathbb {Z}}) \cong {\mathbb {Z}}$. Moreover, the H-flux of $\hat{\mathcal {G}}$ is

$$\begin{aligned} \hat{H} = {\textrm{pr}}_{S^2}^{*}F \wedge {\textrm{pr}}_{S^1}^{*}\theta , \end{aligned}$$

where $F\in \Omega ^2(S^2)$ is the curvature of the Kaluza–Klein connection corresponding to the metric $g=g_3$.

As remarked above, the dual metric has been computed in [1]. The formula for the dual H-flux has been proved in the setting of T-duality with H-flux by Bouwknegt–Evslin–Mathai [5]. Our unifying setting of geometric T-duality correspondences implies both results.

7.3 The Hopf Fibration with the Basic Gerbe

The Hopf fibration $E:=S^3 \rightarrow S^2$ carries a canonical non-trivial bundle gerbe with connection, namely, the basic gerbe $\mathcal {G}_{bas}$ over , under the canonical diffeomorphism . In this section, we consider this bundle gerbe, while we keep E equipped with the round metric $g_3$ as in Sect. 7.2.

Proposition 7.4

The geometric T-background $(S^3,g_3,\mathcal {G}_{bas})$ is self-dual under geometric T-duality.

In the setting of T-duality with H-flux, the self-duality of $(S^3,g_3,H)$, where $H \in \Omega ^3(S^3)$ is the curvature of the basic gerbe, i.e., the canonical 3-form, was known before; Proposition 7.4 upgrades this to geometric T-duality.

In the remainder of this section, we prove Proposition 7.4. We recall that the diffeomorphism between and $S^3$ is

here, the resulting element $(a,b)\in {\mathbb {C}}^2$ is identified with $(\textrm{Re}(a),\textrm{Im}(a),\textrm{Re}(b), \textrm{Im}(b))\in S^3 \subseteq {\mathbb {R}}^4$. It is well-known that the round metric $g_3$ on $S^3$ corresponds to the Killing form $B(Y_1,Y_2) = 4\textrm{tr}(Y_1Y_2)$ on the Lie algebra $\mathfrak {su}(2)$. More precisely, we have under above diffeomorphism

$$\begin{aligned} g_3=-\frac{1}{8}B. \end{aligned}$$

Under the diffeomorphism with , the principal ${\mathbb {T}}$-action of the Hopf fibration is a map , and it is given by matrix multiplication along the group homomorphism

In other words, we have $\tau = m \circ \ ({\textrm{id}}\times \zeta )$, where m denotes the multiplication map of . We also remark that $\zeta ({\mathbb {T}})$ is a maximal torus. Now we consider the basic bundle gerbe $\mathcal {G}_{bas}$ over [26, 36]. Its canonical connection has the curvature $H=\frac{1}{6}\left\langle \theta \wedge [\theta \wedge \theta ] \right\rangle$, where $\left\langle -,- \right\rangle$ is the basic inner product, which, in case of , is $\left\langle -,- \right\rangle =-\textrm{tr}(-\cdot -)=-\frac{1}{4}B$. We recall that $\mathcal {G}_{bas}$ also has a canonical multiplicative structure [18, 49], consisting of a connection-preserving isomorphism

$$\begin{aligned} \mathcal {M}:{\textrm{pr}}_1^{*}\mathcal {G} \otimes {\textrm{pr}}_2^{*}\mathcal {G} \rightarrow m^{*}\mathcal {G} \otimes \mathcal {I}_{\rho } \end{aligned}$$

over , where ; here $\bar{\theta }$ is the right-invariant Maurer–Cartan form. Additionally, there is an “associator”, a connection-preserving 2-isomorphism

$$\begin{aligned} \alpha : \mathcal {M}_{1+2,3} \circ (\mathcal {M}_{1,2} \otimes {\textrm{id}}) \Rightarrow \mathcal {M}_{1,2+3} \circ ({\textrm{id}}\otimes \mathcal {M}_{2,3}) \end{aligned}$$

over , which in turn satisfies a pentagon axiom over .

We consider another multiplicative bundle gerbe, but over the Lie group ${\mathbb {T}}$. The underlying bundle gerbe with connection is the trivial one, $\mathcal {I}_0$. It is equipped with a multiplicative structure using the method of [49, Ex. 1.4 (b)]. Its multiplication isomorphism

$$\begin{aligned} \mathcal {P}:{\textrm{pr}}_1^{*}\mathcal {I}_0 \otimes {\textrm{pr}}_2^{*}\mathcal {I}_0 \rightarrow m^{*}\mathcal {I}_0 \otimes \mathcal {I}_{\Omega } \end{aligned}$$

over ${\mathbb {T}}^2$ is given by the Poincaré bundle $\mathcal {P}$ over ${\mathbb {T}}^2$, under the equivalence of Proposition 2.4. Its associator is

and one can easily check that the pentagon condition over ${\mathbb {T}}^4$ is satisfied. The bundle gerbe $\mathcal {I}_0$ together with the multiplicative structure will be denoted by $\mathcal {I}_0^{\mathcal {P}}$.

Lemma 7.5

We have $\zeta ^{*}\mathcal {G}_{bas} \cong \mathcal {I}_0^{\mathcal {P}}$ as multiplicative bundle gerbes with connection.

Proof

One considers for multiplicative bundle gerbes with connection the pair $(H,\rho )$ consisting of the curvature H of the bundle gerbe and the 2-form $\rho$ of their multiplicative structure. One can check that $\zeta ^{*}(H,\rho ) = (0,\Omega )$. By [49, Prop. 2.4], the pair $(H,\rho )$ characterizes the multiplicative bundle gerbe uniquely up to isomorphism provided that ${\textrm{H}}^4(BG,{\mathbb {Z}})$ is torsion-free. This is the case when $G={\mathbb {T}}$, as the cohomology of ${\mathbb {T}}$ is a polynomial ring.

In the following, we choose an isomorphism $\mathcal {T}:\zeta ^{*}\mathcal {G}_{bas} \rightarrow \mathcal {I}_0^{\mathcal {P}}$ of multiplicative bundle gerbes with connection (it is unique up to unique 2- isomorphism). The multiplicative structure $\mathcal {M}$ of $\mathcal {G}_{bas}$ then induces an isomorphism $\mathcal {M}'$

$$\begin{aligned} \tau ^{*}\mathcal {G}_{bas}= & {} ({\textrm{id}}\times \zeta )^{*}m^{*}\mathcal {G}_{bas} {\mathop {\cong }\limits ^{\mathcal {M}}} ({\textrm{id}}\times \zeta )^{*}({\textrm{pr}}_1^{*}\mathcal {G}_{bas} \otimes {\textrm{pr}}_2^{*}\mathcal {G}_{bas} \otimes \mathcal {I}_{-\rho }) \nonumber \\= & {} {\textrm{pr}}_1^{*}\mathcal {G}_{bas} \otimes {\textrm{pr}}_2^{*}\zeta ^{*}\mathcal {G}_{bas} \otimes \mathcal {I}_{-({\textrm{id}}\times \zeta )^{*}\rho }{\mathop {\cong }\limits ^{\mathcal {T}}} {\textrm{pr}}_1^{*}\mathcal {G}_{bas}\otimes \mathcal {I}_{-({\textrm{id}}\times \zeta )^{*}\rho }. \end{aligned}$$

over . Next we infer that is canonically diffeomorphic to the correspondence space for the self-dual situation: the diffeomorphism is

Note that ${\textrm{pr}}\circ \Psi = {\textrm{pr}}_1$ and $\hat{{\textrm{pr}}} \circ \Psi = \tau$. Thus, pulling back the isomorphism $\mathcal {M}'$ along $\Psi ^{-1}$, we obtain a candidate for the isomorphism $\mathcal {D}$. We first verify that the 2-form is correct, i.e.,

$$\begin{aligned} \Psi ^{*}\rho _{g_3,g_3} = ({\textrm{id}}\times \zeta )^{*}\rho . \end{aligned}$$

This can be checked explicitly using the given definitions. By this, we have a geometric correspondence.

Conditions (T1) and (T2) of Definition 4.9 are obviously satisfied, since we have the same bundle and metric on both sides. It remains to verify condition (T3). Consider an open set $U \subseteq S^2$ with a local trivialization $\varphi : U \times S^1 \rightarrow S^3|_U$. We denote by the corresponding section. For dimensional reasons, there exists a trivialization $\mathcal {S}: s^{*}\mathcal {G}_{bas} \rightarrow \mathcal {I}_\lambda$, where $\lambda \in \Omega ^2(S^2)$. Note that $\varphi (x,z)=s(x)\zeta (z)$, or, $\varphi =m \circ (s \times \zeta )$. Hence, we may produce a trivialization

$$\begin{aligned} \mathcal {U}: \varphi ^{*}\mathcal {G}_{bas} = (s \times \zeta )^{*}m^{*}\mathcal {G}_{bas} \cong {\textrm{pr}}_1^{*}s^{*}\mathcal {G}_{bas} \otimes {\textrm{pr}}_2^{*}\zeta ^{*}\mathcal {G}_{bas} \otimes \mathcal {I}_{-(s \times \zeta )^{*}\rho }\cong \mathcal {I}_{{\textrm{pr}}_1^{*}\lambda -(s\times \zeta )^{*}\rho } \end{aligned}$$

with $B:= {\textrm{pr}}_1^{*}\lambda -(s\times \zeta )^{*}\rho$. We choose the same trivializations $\varphi$ and $\mathcal {U}$ on both sides.

We observe that there is a commutative diagram

where $\psi (x,z_1,z_2):= (s(x)\zeta (z_1),z_2-z_1)$, and $\Phi =(\varphi ,\varphi )$. Over $U \times {\mathbb {T}}^2$ we then have to consider the isomorphism

$$\begin{aligned} (\mathcal {U}_{1,3} \otimes {\textrm{id}})\circ \psi ^{*}\Psi ^{*}\mathcal {D} \,\circ \, \mathcal {U}^{-1}_{1,2} \end{aligned}$$

Substituting the definitions of $\mathcal {U}$ and $\mathcal {D}$, it turns out that all occurrences of $\mathcal {M}$, and both occurrences of $\mathcal {S}$ cancel. Remaining are the contributions of $\mathcal {T}$, which are $\mathcal {T}_1$, $\mathcal {T}_2^{-1}$ and $\mathcal {T}_{2-1}$. By Lemma 7.5, this gives the Poincaré bundle. This proves (T3) and completes the proof of Proposition 7.4.