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\section*{topological T-duality}

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

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Recall that \emph{geometric} [[T-duality]] is an operation acting on tuples roughly consisting of

\begin{itemize}%
\item a [[smooth manifold]] $X$ with the structure of a torus-[[principal bundle]] $T^n \to X \to X/T^n$ -- modelling [[spacetime]]


\item equipped with a [[circle n-bundle with connection|circle 2-bundle with connection]] -- modelling the [[Kalb-Ramond field]]


\item and in [[twisted K-theory]] refined to elements in [[differential K-theory|differential twisted K-theory]] -- modelling the [[RR-field]]


\item and notably equipped with a (pseudo)[[Riemannian metric]] -- modelling the field of [[gravity]].



\end{itemize}
The idea of \textbf{topological T-duality} (due to \hyperlink{BouwknegtEvslinMathai04}{Bouwknegt-Evslin-Mathai 04}, \hyperlink{BouwknegtHannabusMathai04}{Bouwknegt-Hannabus-Mathai 04}) is to disregard the [[Riemannian metric]] and the connection and study the remaining ``topological'' structure.

While the idea of [[T-duality]] originates in [[string theory]], topological T-duality has become a field of study in pure mathematics in its own right.

In the language of [[bi-brane]]s a topological T-duality transformation is a bi-brane of a special kind between the two gerbes involved. The induced [[integral transform]] (on sheaves of sections of, or K-classes of) (twisted) vector bundles is essentially the [[Fourier-Mukai transformation]]. More on the bi-brane interpretation of (topological and non-topological) T-duality is in (\hyperlink{SarkissianSchweigert08}{Sarkissian-Schweigert 08}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Two tuples $(X_i \to B, G_i)_{i = 1,2}$ consisting of a $T^n$-bundle $X_i$ over a manifold $B$ and a line [[bundle gerbe]] $G_i \to X_i$ over $X$ are \textbf{topological T-duals} if there exists an isomorphism $u$ of the two [[bundle gerbes]] pulled back to the [[fiber product]] [[correspondence]] space $X_1 \times_B X_2$:

\begin{displaymath}
\itexarray{
    && pr_1^* G_1 && \stackrel{u}{\leftarrow}
    && pr_2^* G_2
    \\
    & \swarrow && \searrow && \swarrow && \searrow
    \\
    G_1 &&&& X_1 \times_B X_2 &&&& G_2
    \\
    & \searrow && \swarrow && \searrow && \swarrow
    \\
    && X_1 &&&& X_2
    \\
    &&& \searrow && \swarrow
    \\
    &&&& B
  }
\end{displaymath}
of a certain prescribed [[integral transform]]-form (\hyperlink{BunkeRumpfSchick08}{Bunke-Rumpf-Schick 08, p. 9})).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[T-duality]]

\begin{itemize}%
\item \textbf{topological T-duality}


\item [[differential T-duality]]


\item [[T-duality 2-group]]



\end{itemize}

\item [[spherical T-duality]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept was introduced on the level of differential form data in

\begin{itemize}%
\item [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], \emph{T-Duality: Topology Change from H-flux}, Commun. Math. Phys. 249:383-415, 2004 (\href{http://arxiv.org/abs/hep-th/0306062}{hep-th/0306062})


\item [[Peter Bouwknegt]], [[Keith Hannabus]], [[Varghese Mathai]], \emph{T-duality for principal torus bundles}, JHEP 0403 (2004) 018 (\href{http://arxiv.org/abs/hep-th/0312284}{hep-th/0312284})



\end{itemize}
In these papers the $U(1)$-[[gerbe]] ([[circle 2-bundle with connection]]) does not appear, but an integral [[differential 3-form]], representing its [[Dixmier-Douady class]] does. Note that if the [[Eilenberg-MacLane spectrum|integral]] [[cohomology group]] $H^3(X,\mathbb{Z})$ of $X$ has [[torsion]] in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Schick]], \emph{On the topology of T-duality}, Rev.Math.Phys.17:77-112,2005, (\href{https://arxiv.org/abs/math/0405132}{arXiv:math/0405132})


\item  [[Ulrich Bunke|U. Bunke]], P. Rumpf, [[Thomas Schick]], \emph{The topology of $T$-duality for $T^n$-bundles}, Rev. Math. Phys. 18, 1103 (2006). (\href{http://arxiv.org/abs/math.GT/0501487}{arXiv:math.GT/0501487})


\item [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], \emph{Periodic twisted cohomology and T-duality}, Ast\'e{}risque No. 337 (2011), vi+134 pp. ISBN: 978-2-85629-307-2



\end{itemize}
A refined version of this using [[smooth stacks]] is due to

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], \emph{T-Duality via Gerby Geometry and Reductions} (\href{http://arxiv.org/abs/1305.6050}{arXiv:1305.6050})


\item [[Thomas Nikolaus]], \emph{[[T-Duality in K-theory and Elliptic Cohomology]]}, talk at \emph{String Geometry Network Meeting}, Feb 2014, ESI Vienna (\href{http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html}{website})



\end{itemize}
There is also [[C\emph{-algebra|C}-algebraic]] version of toplogical T-duality, .e. in [[noncommutative topology]], which sees also topological T-duals in [[non-commutative geometry]]:

\begin{itemize}%
\item [[Varghese Mathai]], [[Jonathan Rosenberg]], \emph{T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology} (\href{http://arxiv.org/abs/hep-th/0401168}{arXiv:hep-th/0401168})

\end{itemize}
The equivalence of the [[C\emph{-algebra|C}-algebraic]] to the Bunke-Schick version, when the latter exists, is discussed in

\begin{itemize}%
\item Ansgar Schneider, \emph{Die lokale Struktur von T-Dualitn\"a{}tstripeln} (\href{https://arxiv.org/abs/0712.0260}{arXiv:0712.0260})

\end{itemize}
Jonathan Rosenberg has also written a little introductory book for mathematicians:

\begin{itemize}%
\item [[Jonathan Rosenberg]], \emph{Topology, $C^*$-algebras, and string duality} (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:05632075&type=pdf&format=complete}{ZMATH})

\end{itemize}
Another discussion that instead of [[noncommutative geometry]] uses [[topological groupoids]] is in

\begin{itemize}%
\item Calder Daenzer, \emph{A groupoid approach to noncommutative T-duality} (\href{http://arxiv.org/abs/0704.2592}{arXiv:0704.2592})

\end{itemize}
The [[bi-brane]] perspective on T-duality is amplified in

\begin{itemize}%
\item Gor Sarkissian, [[Christoph Schweigert]], \emph{Some remarks on defects and duality} (\href{http://arxiv.org
/abs/0810.3159}{arXiv:0810.3159})

\end{itemize}
Discussion for non-free torus actions (physically: [[KK-monopoles]]) is in

\begin{itemize}%
\item Ashwin S. Pande, \emph{Topological T-duality and Kaluza-Klein Monopoles}, Adv. Theor. Math. Phys., 12, (2007), pp 185-215 (\href{https://arxiv.org/abs/math-ph/0612034}{arXiv:math-ph/0612034})

\end{itemize}
Discussion in [[rational homotopy theory]]/[[dg-geometry]] is in

\begin{itemize}%
\item [[Ernesto Lupercio]], Camilo Rengifo, [[Bernardo Uribe]], \emph{T-duality and exceptional generalized geometry through symmetries of dg-manifolds} (\href{https://arxiv.org/abs/1208.6048}{arXiv:1208.6048})

\end{itemize}
and a derivation of the rules of topological T-duality from analysis of the [[super p-brane]] super-cocycles in super rational homotopy theory (with a [[doubled supergeometry]]) is given in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]}, \href{http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/}{ATMP Volume 22 (2018) Number 5} (\href{https://arxiv.org/abs/1611.06536}{arXiv:1611.06536}, \href{https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3}{doi:10.4310/ATMP.2018.v22.n5.a3})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:T-duality in rational homotopy theory via strong homomotopy Lie algebras]]}, \href{http://geo-top.org/GTMP/index.php/volume-1-2018/}{Geometry, Topology and Mathematical Physics Journal, Volume 1 (2018)} (\href{https://arxiv.org/abs/1712.00758}{arXiv:1712.00758})


\item [[Domenico Fiorenza]], \emph{T-duality in rational homotopy theory}, talk at \emph{\href{http://conference.math.muni.cz/srni/files/archiv/2018/}{38th Srni Winter School on Geometry and Physics}}, 2018 ([[FiorenzaSrni2018.pdf:file]])



\end{itemize}
The refinement of topological T-duality to [[differential cohomology]], hence to an operation on the [[differential K-theory]] classes that model the [[RR-field]] is in

\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}


\end{document}