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\section*{T-Duality and Differential K-Theory}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

This entry is about the article

\begin{itemize}%
\item [[Alexander Kahle]], [[Alessandro Valentino]], \emph{T-duality and Differential K-Theory}, Communications in Contemporary Mathematics, Volume 16, Issue 02, April 2014 (\href{http://arxiv.org/abs/0912.2516}{arXiv:0912.2516})

\end{itemize}
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

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], \emph{Differential K-theory. A survey} (\href{http://arxiv.org/abs/1011.6663}{arXiv:1011.6663}).

\end{itemize}
We try to indicate some of the content. There are two main ingredients:

\begin{itemize}%
\item a \hyperlink{FormalizationOfTheSetup}{formalization of the setup}

\end{itemize}
which proposes a refinement of the ingredients of [[topological T-duality]] to [[differential cohomology]]; and then the

\begin{itemize}%
\item \hyperlink{StatementOfDifferentialTduality}{statement of differential T-duality},

\end{itemize}
which is the assertion that this setup naturally induces the T-duality operation as an [[isomorphism]] on [[twisted differential K-theory]].

\hypertarget{FormalizationOfTheSetup}{}\subsection*{{Formalization of the setup}}\label{FormalizationOfTheSetup}

Let $\Lambda \subset \mathbb{R}^n$ be a [[lattice]] (an [[discrete group|discrete]] subgroup of the [[abelian group]] of [[real number]]s to the $n$th [[cartesian product|cartesian power]]).

Write

\begin{displaymath}
\hat \Lambda := Hom_{Grp}(\Lambda, \mathbb{Z})
\end{displaymath}
for the dual lattice and

\begin{displaymath}
(-,-) : \Lambda \times \hat \Lambda \to \mathbb{Z}
\end{displaymath}
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 infinity-groupoid|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 \emph{smooth} bundle morphisms between. Write $\mathbf{H}_{conn}(X,\mathbf{B} \mathbb{R}^n /\Lambda)$ for the corresponding differential refinement to bundles with [[connection on a bundle|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 complex]]es

\begin{displaymath}
\Lambda[2]\otimes\hat{\Lambda}[2]\cong (\Lambda\otimes\hat{\Lambda})[4]
\end{displaymath}
together with the map of complexes

\begin{displaymath}
(\Lambda \otimes \hat \Lambda)[4]\stackrel{(-,-)}{\to} \mathbb{Z}[4]
\end{displaymath}
induces the [[cup product]]

\begin{displaymath}
H^k(X, \Lambda) \times
  H^l(X, \hat \Lambda)
  \to 
  H^{k+l}(X, \Lambda \otimes \hat \Lambda)
   \stackrel{(-,-)}{\to}
  H^{k+l}(X, \mathbb{Z})
  \,.
\end{displaymath}
This can be refined to a pairing in [[differential cohomology]]

\begin{displaymath}
\bar{H}^k(X, \Lambda) \times
  \bar{H}^l(X, \hat \Lambda)
  \to 
  \bar{H}^{k+l}(X, \Lambda \otimes \hat \Lambda)
   \stackrel{(-,-)}{\to}
  \bar{H}^{k+l}(X, \mathbb{Z})
  \,.
\end{displaymath}
by considering the [[Deligne cohomology|Deligne complex]] of sheaves

\begin{displaymath}
\Lambda[2]^\infty_D:=(\Lambda\hookrightarrow C^\infty(-,\mathbb{R}^n)\xrightarrow{d_\Lambda} \Omega^1(-,\mathbb{R}^n)),
\end{displaymath}
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

\begin{displaymath}
d_\Lambda=(d\circ e^i)\otimes e_i
\end{displaymath}
(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

\begin{displaymath}
(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}
\end{displaymath}
induced by the composition of morphisms of complexes

\begin{displaymath}
\Lambda[2]^\infty_D\otimes \hat{\Lambda}[2]^\infty_D 
\stackrel{\cup}{\to}
(\Lambda\otimes\hat\Lambda)[4]^\infty_D \stackrel{}{\to}\mathbb{Z}[4]^\infty_D,
\end{displaymath}
where $\cup$ is the [[Beilinson-Deligne cup-product]].

Notice that this is the one which defines [[higher dimensional Chern-Simons theory|abelian Chern-Simons theories]]. The higher [[holonomy]] of the [[circle n-bundle with connection |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.

\begin{defn}
\label{DifferentialTDualityPair}\hypertarget{DifferentialTDualityPair}{}
A \textbf{differential T-duality pair} is

\begin{itemize}%
\item a [[smooth manifold]] $X$;


\item 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;


\item a choice of trivialization

\begin{displaymath}
\sigma 
   : 
  (0,C) \stackrel{\simeq}{\to} (P, \theta) \cup (\hat P , \hat \theta)
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
This is (\hyperlink{KahleValentino}{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 (\hyperlink{BunkeSchick}{BunkeSchick}).

\begin{remark}
\label{}\hypertarget{}{}
We may refine this naturally to a \textbf{[[2-groupoid]] of twisted T-duality pairs} $TDualityPairs(X)_{conn}$, the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    TDualityPairs(X)_{conn} &\stackrel{tw}{\to}& H^4_{diff}(X)
    \\
    \downarrow &\swArrow_{\sigma}& \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda)
    &\stackrel{}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn}
  }
  \,.
\end{displaymath}
This is itself an example of [[twisted cohomology]] (as discussed there). (We use here the notation at \emph{[[schreiber:differential cohomology in a cohesive topos]]} .)

The differential T-duality pairs of def. \ref{DifferentialTDualityPair} are those elements $(P,\hat P, \sigma) \in TDualityPairs(X)_{conn}$ for which the [[twisted cohomology|twist]] $tw(P,\hat P, \sigma) \in H^4_{diff}(X)$ in [[ordinary differential cohomology]] has an underlying trivial [[circle n-bundle|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 \emph{[[differential string structure]]s in $StringBund(X)_{tw,conn}$ over $X$: as discussed in detail there, this is the [[homotopy pullback]]}

\begin{displaymath}
\itexarray{
    StringBund(X)_{tw,conn} &\to& H^4_{diff}(X)
    \\
    \downarrow &\swArrow_{\sigma}& \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}Spin \times \mathbf{B} SU)
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \frac{1}{46}\hat \mathbf{c}}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn}
  }
  \,.
\end{displaymath}
In particular the [[looping and delooping|looping]] $TString$ of the [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    \mathbf{B}TString &\stackrel{}{\to}& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda
      &\stackrel{\langle - \cup -\rangle}{\to}&
    \mathbf{B}^3 U(1))_{conn}
  }
\end{displaymath}
has the right to be called the [[T-duality 2-group]] or similar. The [[principal 2-bundles]] for this are [[T-folds]] (see there).

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
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 $\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1))$.

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

\begin{remark}
\label{}\hypertarget{}{}
In terms of the notion of [[differential function complex]] we should simply set

\begin{displaymath}
\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1))
  :=
  filt_1 ( H \mathbb{Z}_4)^X
  \,.
\end{displaymath}
(Notice the $filt_1$ instead of $filt_0$. ) By \href{http://ncatlab.org/nlab/show/differential%20function%20complex#homotopy_groups_24}{this proposition} this has the right properties.

\end{remark}
\begin{lemma}
\label{}\hypertarget{}{}
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 n-bundle with connection|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.

\end{lemma}
This is (\hyperlink{KahleValentino}{KahleValentino, 2.2, 2.3}), where an explicit construction of the classes and their equivalence is given.

\begin{remark}
\label{}\hypertarget{}{}
This is a special case of the general statement about extensions of higher bundles discussed :

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)

\begin{displaymath}
\cdots \to \mathbf{B}^2 U(1) \to A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)
\end{displaymath}
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 pullback]]s

\begin{displaymath}
\itexarray{
    P \times_X \hat P &\stackrel{\tilde \tau}{\to}& \mathbf{B}^2 U(1) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\to& A & \to & \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda 
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
     && && P \times_X \hat P
     \\
     && & \swarrow && \searrow
     \\
     && P &&&& \hat P
     \\
     & \swarrow && \searrow && \swarrow && \searrow
     \\
     * && && X && && *
     \\
     & \searrow && \swarrow && \searrow && \swarrow
     \\
     && \mathbf{B}\mathbb{R}^n / \Lambda 
     && && \mathbf{B}\mathbb{R}^n / \hat \Lambda
  }
  \,.
\end{displaymath}
Notice also that this is again directly analogous to the situation for [[string structure]]s: 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 $X$.

\end{remark}
\hypertarget{StatementOfDifferentialTDuality}{}\subsection*{{Statement of differential T-duality}}\label{StatementOfDifferentialTDuality}

(\ldots{})

category: reference



\end{document}