This entry is about the article
A review is in section 7.4 of
We try to indicate some of the content. There are two main ingredients:
for the dual lattice and
for the canonical pairing (the evaluation map). Notice that is canonically identified with the lattice of consisting of those linear functionals such that . With this identification, the canonical pairing can be seen as the restriction to of the canonical pairing .
Torus bundles on a smooth manifold are classified by . Following the discussion at smooth ∞-groupoid we write here for the groupoid of smooth torus bundles and smooth bundle morphisms between. Write for the corresponding differential refinement to bundles with connection.
For an abelian group, let be the chain complex consisting of concentrated in degree . Then the tensor product of chain complexes
together with the map of complexes
induces the cup product
This can be refined to a pairing in differential cohomology
by considering the Deligne complex of sheaves
where the differential is defined as follows: if is a -basis of and are the corresponding projections , then
(this is independent of the chosen basis). The definition of is clearly chosen so to have an isomorphism of complexes induced by the choice of a -basis of .
Write for the smooth groupoid associated by the Dold-Kan correspondence to the Deligne complex . Then we have the morphism of smooth groupoids to a morphism
induced by the composition of morphisms of complexes
where is the Beilinson-Deligne cup-product.
Notice that this is the one which defines abelian Chern-Simons theories. The higher holonomy of the circle 3-bundle with connection appearing here is the action functional of torus-Chern-Simons theory.
Differential T-duality pairs form the homotopy fiber of the morphism It relates differential cohomological structures on -principal bundles with that on certain dual -principal bundles.
A differential T-duality pair is
The differential T-duality pairs of def. 1 are those elements for which the twist 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.)
Write 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 automorphism 2-groupoid of the 0-object is that of circle 2-bundles with connection .
A 1-groupoid truncation of this idea is the object denoted in KahleValentino, A.2.
In terms of the notion of differential function complex we should simply set
(Notice the instead of . ) By this proposition this has the right properties.
The choice 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 and on , respectively, whose pullbacks to the fiber product are equivalent there.
This is (KahleValentino, 2.2, 2.3), where an explicit construction of the classes and their equivalence is given.
This is a special case of the general statement about extensions of higher bundles discussed here:
The characteristic map of a pair of torus bundles factors through precisely if these form a T-duality pair. Such a factorization induces a -principal 2-bundle on the fiber product . This follows from the following pasting diagram of homotopy pullbacks
The here is the class on the fiber product in question.
Notice that in the top left we indeed have : 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 induces a -2-bundle on the total space of a -principal bundle over .