for the canonical pairing (the evaluation map). Notice that Λ^\hat\Lambda is canonically identified with the lattice of (ℝn)*(\mathbb{R}^n)^* consisting of those linear functionals φ:ℝn→ℝ\varphi:\mathbb{R}^n\to\mathbb{R} such that φ(Λ)⊆ℤ\varphi(\Lambda)\subseteq \mathbb{Z}. With this identification, the canonical pairing Λ×Λ^→ℤ\Lambda \times \hat \Lambda \to \mathbb{Z} can be seen as the restriction to Λ×Λ^\Lambda \times \hat \Lambda of the canonical pairing ℝn⊗(ℝn)*→ℝ\mathbb{R}^n\otimes(\mathbb{R}^n)^*\to \mathbb{R}.

Torus bundles on a smooth manifoldXX are classified by H2(X,Λ)H^2(X, \Lambda). Following the discussion at smooth ∞-groupoid we write here H(X,Bℝn/Λ)\mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda) for the groupoid of smooth torus bundles and smooth bundle morphisms between. Write Hconn(X,Bℝn/Λ)\mathbf{H}_{conn}(X,\mathbf{B} \mathbb{R}^n /\Lambda) for the corresponding differential refinement to bundles with connection.

For AA an abelian group, let A[n]A[n] be the chain complex consisting of AA concentrated in degree nn. Then the tensor product of chain complexes

where the differential dΛd_\Lambda is defined as follows: if e1,…ene_1,\dots e_n is a ℤ\mathbb{Z}-basis of Λ\Lambda and e1,…,ene^1,\dots,e^n are the corresponding projections ei:ℝn→ℝe^i:\mathbb{R}^n\to \mathbb{R}, then

dΛ=(d∘ei)⊗ei
d_\Lambda=(d\circ e^i)\otimes e_i

(this is independent of the chosen basis). The definition of dΛd_\Lambda is clearly chosen so to have an isomorphism of complexes (Z[2]D∞)⊗n≅Λ[2]D∞(\mathbf{Z}[2]^\infty_D)^{\otimes n}\cong \Lambda[2]^\infty_D induced by the choice of a ℤ\mathbb{Z}-basis of Λ\Lambda.

Write (Bℝn/Λ)conn(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn} for the smooth groupoid associated by the Dold-Kan correspondence to the Deligne complex Λ[2]D∞\Lambda[2]^\infty_D. Then we have the morphism of smooth groupoids to a morphism

The differential T-duality pairs of def. 1 are those elements (P,P^,σ)∈TDualityPairs(X)conn(P,\hat P, \sigma) \in TDualityPairs(X)_{conn} for which the twisttw(P,P^,σ)∈Hdiff4(X)tw(P,\hat P, \sigma) \in H^4_{diff}(X) in ordinary differential cohomology has an underlying trivial circle 3-bundle. We could restrict the homotopy pullback to these, but it seems natural to include the full collection of twists. (Notice that these “twist” here are not the twists in “twisted K-theory”, rather we are observing that already the notion of T-duality pairs itself is an example of cocycles in twisted cohomology in the general sense.)

Definition (roughly)

Write Hdiff,2(X,B3U(1))\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1)) for a 3-groupoid whose objects are cocycles in ordinary differential cohomology in degree 4, but whose morphisms need not preserve connections and are instead such that the automorphism2-groupoid of the 0-object is that of circle 2-bundles with connection Hdiff(X,B2U(1))\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1)).

A 1-groupoid truncation of this idea is the object denoted ℋp(X)\mathcal{H}^p(X) in KahleValentino, A.2.

Remark

(Notice the filt1filt_1 instead of filt0filt_0. ) By this proposition this has the right properties.

Lemma

The choice σ\sigma of the trivialization of the cup product of the two torus bundles induces canonically elments in degree 3 ordinary differential cohomology (two circle 2-bundles with connection) on PP and on P^\hat P, respectively, whose pullbacks to the fiber productP×XP^P \times_X \hat P are equivalent there.

Notice that in the top left we indeed have P×XP^P \times_X \hat P: the bottom left homotopy pullback of the product coefficients is equivalently given by the following pasting composite of homotopy pullbacks

Notice also that this is again directly analogous to the situation for string structures: as discussed there, a string structure on XX induces a BU(1)\mathbf{B}U(1)-2-bundle on the total space of a SpinSpin-principal bundle over XX.