under construction
This entry is about the article
on the identification of the correct description in differential cohomology (ordinary differential cohomology and differential K-theory) of T-duality or topological T-duality: differential T-duality.
A review is in section 7.4 of
We try to indicate some of the content. There are two main ingredients:
which proposes a refinement of the ingredients of topological T-duality to differential cohomology; and then the
which is the assertion that this setup naturally induces the T-duality operation as an isomorphism on twisted differential K-theory.
Let $\Lambda \subset \mathbb{R}^n$ be a lattice (an discrete subgroup of the abelian group of real numbers to the $n$th cartesian power).
Write
for the dual lattice and
for the canonical pairing (the evaluation map). Notice that $\hat\Lambda$ is canonically identified with the lattice of $(\mathbb{R}^n)^*$ consisting of those linear functionals $\varphi:\mathbb{R}^n\to\mathbb{R}$ such that $\varphi(\Lambda)\subseteq \mathbb{Z}$. With this identification, the canonical pairing $\Lambda \times \hat \Lambda \to \mathbb{Z}$ can be seen as the restriction to $\Lambda \times \hat \Lambda$ of the canonical pairing $\mathbb{R}^n\otimes(\mathbb{R}^n)^*\to \mathbb{R}$.
The quotient $\mathbb{R}^n / \Lambda$ is a torus. A $\mathbb{R}^n/\Lambda$-principal bundle is a torus-bundle. Write $\mathcal{B}\mathbb{R}^n/\Lambda \in$ Top for the classifying space and $\mathbf{B} \mathbb{R}^n / \Lambda$ for its moduli space: the smooth groupoid delooping the Lie group $\mathbb{R}/\Lambda$.
Torus bundles on a smooth manifold $X$ are classified by $H^2(X, \Lambda)$. Following the discussion at smooth ∞-groupoid we write here $\mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda)$ for the groupoid of smooth torus bundles and smooth bundle morphisms between. Write $\mathbf{H}_{conn}(X,\mathbf{B} \mathbb{R}^n /\Lambda)$ for the corresponding differential refinement to bundles with connection.
For $A$ an abelian group, let $A[n]$ be the chain complex consisting of $A$ concentrated in degree $n$. 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 $d_\Lambda$ is defined as follows: if $e_1,\dots e_n$ is a $\mathbb{Z}$-basis of $\Lambda$ and $e^1,\dots,e^n$ are the corresponding projections $e^i:\mathbb{R}^n\to \mathbb{R}$, then
(this is independent of the chosen basis). The definition of $d_\Lambda$ is clearly chosen so to have an isomorphism of complexes $(\mathbf{Z}[2]^\infty_D)^{\otimes n}\cong \Lambda[2]^\infty_D$ induced by the choice of a $\mathbb{Z}$-basis of $\Lambda$.
Write $(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}$ for the smooth groupoid associated by the Dold-Kan correspondence to the Deligne complex $\Lambda[2]^\infty_D$. Then we have the morphism of smooth groupoids to a morphism
induced by the composition of morphisms of complexes
where $\cup$ 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 $(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}$ It relates differential cohomological structures on $\mathbb{R}^n / \Lambda$-principal bundles with that on certain dual $(\mathbb{R}^n)^* /\hat \Lambda$-principal bundles.
A differential T-duality pair is
a smooth manifold $X$;
a $\mathbb{R}^n/\Lambda$-principal bundle $P \to X$ with connection $\theta$ and a $\mathbb{R}^n / \hat \Lambda$-principal bundle $\hat P \to X$ with connection $\hat \theta$;
such that the underlying topological class of the cup product $(P, \theta) \cup (\hat P, \hat \theta )$ is trivial;
a choice of trivialization
This is (KahleValentino, def. 2.1). It is the evident differential generalization of the description in topological T-duality that appears for instance around (7.11) of (BunkeSchick).
We may refine this naturally to a 2-groupoid of twisted T-duality pairs $TDualityPairs(X)_{conn}$, the homotopy pullback
This is itself an example of twisted cohomology (as discussed there). (We use here the notation at differential cohomology in a cohesive topos .)
The differential T-duality pairs of def. 1 are those elements $(P,\hat P, \sigma) \in TDualityPairs(X)_{conn}$ for which the twist $tw(P,\hat P, \sigma) \in H^4_{diff}(X)$ 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.)
Notice that the above analogous to the notion of differential string structures in $StringBund(X)_{tw,conn}$ over $X$: as discussed in detail there, this is the homotopy pullback
In particular the looping $TString$ of the homotopy fiber
has the right to be called the T-duality 2-group or similar. The principal 2-bundles for this are T-folds (see there).
Write $\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1))$ 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 automorphism 2-groupoid of the 0-object is that of circle 2-bundles with connection $\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1))$.
A 1-groupoid truncation of this idea is the object denoted $\mathcal{H}^p(X)$ in KahleValentino, A.2.
In terms of the notion of differential function complex we should simply set
(Notice the $filt_1$ instead of $filt_0$. ) By this proposition this has the right properties.
The choice $\sigma$ 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 $P$ and on $\hat P$, respectively, whose pullbacks to the fiber product $P \times_X \hat P$ 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:
Let $A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ be the homotopy fiber of the pairing class $\mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)$. This leads to the long fiber sequence (as discussed there)
The characteristic map $X \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ of a pair of torus bundles $P , \hat P \to X$ factors through $A$ precisely if these form a T-duality pair. Such a factorization induces a $\mathbf{B} U(1)$-principal 2-bundle on the fiber product $P \times_X \hat P$. This follows from the following pasting diagram of homotopy pullbacks
The $\tilde \tau$ here is the class on the fiber product in question.
Notice that in the top left we indeed have $P \times_X \hat P$: 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 $X$ induces a $\mathbf{B}U(1)$-2-bundle on the total space of a $Spin$-principal bundle over $X$.
(…)