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\section*{T-duality}

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

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

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

[[!include string theory - contents]]

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

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{PathIntegral}{Path integral heuristics deriving T-duality}\dotfill \pageref*{PathIntegral} \linebreak
\noindent\hyperlink{a_first_rough_look}{A first rough look}\dotfill \pageref*{a_first_rough_look} \linebreak
\noindent\hyperlink{the_path_integral}{The path integral}\dotfill \pageref*{the_path_integral} \linebreak
\noindent\hyperlink{topological_tduality}{Topological T-duality}\dotfill \pageref*{topological_tduality} \linebreak
\noindent\hyperlink{geometric_tduality}{Geometric T-duality}\dotfill \pageref*{geometric_tduality} \linebreak
\noindent\hyperlink{InDiffCohomology}{In terms of generalized differential cohomology}\dotfill \pageref*{InDiffCohomology} \linebreak
\noindent\hyperlink{TdualityIngcg}{In generalized complex geometry}\dotfill \pageref*{TdualityIngcg} \linebreak
\noindent\hyperlink{examples_of_tdual_pairs}{Examples of T-dual pairs}\dotfill \pageref*{examples_of_tdual_pairs} \linebreak
\noindent\hyperlink{in_mirror_symmetry}{In Mirror symmetry}\dotfill \pageref*{in_mirror_symmetry} \linebreak
\noindent\hyperlink{in_langlands_duality}{In Langlands duality}\dotfill \pageref*{in_langlands_duality} \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}

A $d$-dimensional [[sigma-model]] is a [[quantum field theory]] that is induced from certain [[differential geometry|differential geometric]] and [[differential cohomology|differential cohomological]] data, to be thought of as encoding the background geometry on which quantum objects of dimension $d$ propagate.

The operation of \textbf{T-duality} is a map that interchanges pairs of such geometric data for 2-dimensional [[conformal field theory]] [[sigma-model]]s, such that the induced QFTs are equivalent.

More specifically the space of [[differential geometry|differential geometric]] data consisting of

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


\item equipped with the structure of a $k$-torus [[bundle]] $X \simeq Y \times T^k$; -- the total space of this bundle modelling [[spacetime]];


\item and equipped with a [[Riemannian metric]] $g$ -- modelling the field of [[gravity]];


\item and a $U(1)$-[[gerbe]] with connection $G$; -- modelling the [[Kalb-Ramond field]];


\item possibly a cocycle in [[differential K-theory]] modelling the [[RR-field]];



\end{itemize}
admits a certain operation that, roughly, inverts the Riemannian circumference of the torus fibers and mixes the metric with the gerbe data, such that the induced 2-dimensional [[sigma-model]] QFTs for these backgrounds are equivalent. This is the operation called \textbf{T-duality}.

This was noticed originally in the study of [[conformal field theory|conformal field theories]] in the context of [[string theory]]: the conformal field theory [[sigma-model]]s with target space $X$ turn out to be equivalent as [[quantum field theory|quantum field theories]] for T-dual backgrounds $(X,g,G)$ and $(X',g',G')$ (at least to the approximate degree to which these are realized as full CFTs in the first place).

Further generalisations let $X$ be a nontrivial torus bundle, but the T-dual is then generically a bundle of [[noncommutative torus|non-commutative tori]]. (cite Mathai, Rosenberg and Hannabus)

\hypertarget{PathIntegral}{}\subsection*{{Path integral heuristics deriving T-duality}}\label{PathIntegral}

We indicate how one can see T-duality from formal manipulations of the [[path integral]] for the [[string theory|string]] [[sigma-model]]. We look at the simplest situation, where the torus bundle in question is a trivial circle bundle over a [[Cartesian space]] carrying the metric induced from the standard flat metric on $\mathbb{R}^n$ and where there are no other nontrivial background fields. In fact, for the purpose of the following computation we can entirely ignore the base of this bundle and consider target space to be nothing but a circle. Since the [[sigma-model]] for this is on the worldsheet just the theory of a single free field with values in $S^1$, this is often also called the ``free boson on the circle''.

This means that the only geometric datum determining the background geometry is the circumference $2 \pi R$ of the fiber of the circle bundle. The statement of T-duality in this situation is that the 2-dimensional [[sigma-model]] on this background yields the same 2-dimensional [[CFT]] as that for this kind of background with circumference of the circle being $2 \pi 1/R$.

\hypertarget{a_first_rough_look}{}\subsubsection*{{A first rough look}}\label{a_first_rough_look}

A quick way to get an indication for this is to consider the center-of-mass energy of the string in such a circle-bundle background. In the simplified setup we mentioned before, a string on a circle of radius $R$ has quantized momentum $p = \frac{\ell \in \mathbb{Z}}{R}$. In a state in which the string winds around the circle $m$ times and has $\ell$ quanta of kinetic momentum for propagation around the circle, its energy is

\begin{displaymath}
E_0 = \sqrt{p^2 + M^2} = \sqrt{(\ell/R)^2 + (R m)^2 }
  \,,
\end{displaymath}
This energy is clearly invariant under exchanging

\begin{displaymath}
(R, (\ell, m)) \mapsto (\frac{1}{R}, (m,\ell))
  \,.
\end{displaymath}
This is of course far from being a proof that the corresponding two [[QFT]]s are equivalent, but it does already capture a good deal of the essence of what T-duality does and why it works.

In slightly more detail, but still at a very rough level, if we denote by

\begin{displaymath}
X : \Sigma \to S^1_R
\end{displaymath}
the $\sigma$-model field on the worldsheet $\Sigma = \mathbb{R}^2$ with values in target space $S^1_R$ then T-duality with respect to this circle may be thought of as exchanging worldsheet momentum $\partial_t X$ with worldsheet winding $\partial_\sigma X$.

This then also means that for the open string it exchanges von Neumann boundary conditions $\partial_\sigma X|_{\sigma = 0} = 0$ with Dirichlet boundary conditions $\partial_t X|_{\sigma = 0} = 0$. The first boundary condition is that describing an open string whose endpoints are free to propagate in worldsheet time, whereas the second boundary condition describes a situation where the endpoint of the string is fixed at some point in target space. In terms of the language of geometric target space data, a sigma-model with such a constraint is said to describe a [[D-brane]] in target space: the locus where the endpoints of the string are fixed. This is a first indication that the T-duality operation on geometric background also involves the [[RR-field]].

\hypertarget{the_path_integral}{}\subsubsection*{{The path integral}}\label{the_path_integral}

We follow [[Kentaro Hori]]`s [[path integral]] discussion of T-duality. Here the strategy is to consider a path integral over a certain space of auxiliary fields and show or argue that by ``algebraically integrating out'' some of these in two different ways, the path integral is equivalent to that over two different [[action functional]]s, which describe two T-dual geometric backgrounds.

Let the boundary components of the worldsheet $\Sigma$ be labeled by $\partial \Sigma_{(1)}$.

We consider the following fields on the worldsheet:

\begin{itemize}%
\item $\tilde X : \Sigma \to \mathbb{R}/(2\pi/R)\mathbb{Z} = S^1_R$ -- a circle-valued function; this is the standard $\sigma$-model field describing propagation of the string on the circle;


\item $X_{i} : \partial \Sigma_{(i)} \to S^1_R$ -- the boundary values of this field;


\item $b \in \Omega^1(\Sigma, \mathbb{R})$ -- a 1-form; this is the auxiliary field that will not contribute to the dynamics but serves to make the T-duality manifest.



\end{itemize}
Consider then the following [[action functional]] on this collection of fields given by the assignment

\begin{displaymath}
S'_E(\tilde X,b) = \frac{1}{2 \pi}
  \int_\Sigma
  \left(
     \frac{1}{2} b \wedge \star b  - b \wedge d \tilde X
  \right)
  -
  \frac{i}{2 \pi} \sum_{i = 1}^s
  \int_{\partial \Sigma_{(i)}}
  (\tilde X -a_i) d X_i
  \,,
\end{displaymath}
where the $(a_i)$ are a collection of real numbers.

We now want to formally perform the [[path integral]] over the fields in two different orders, which should give the same quantum field theories but in terms of different effective action functionals.

If we do first the path integral over the field $b$ then by the general formal rule of ``algebraically integrating out a non-dynamical field'' which says that we can evaluate this path integral that formally looks like a Gaussian integral by the usual formulas for Gaussian integrals, we obtain the action functional

\begin{displaymath}
\tilde S_E =
  \frac{1}{4 \pi} 
  \int_\Sigma d \tilde X \wedge \star d \tilde X
  +
  \frac{i}{2\pi}
  \int_{\partial \Sigma_{(i)}}
  (\tilde X - a_i)
  d X_i
\end{displaymath}
then doing the integral over the boundary values $X_i$ yields

\begin{displaymath}
\tilde X|_{\partial \Sigma_i} = a_i
\end{displaymath}
This is the action functional for a $\sigma$-model on $S^1_{1/R}$ with a D-brane at $\tilde X = a_i$.

Now we evaluate the original path integral in a different way, this way first integrating over components of $\tilde X$. To do so, we imagine that we may re-encode the field $\tilde X$ in terms of its de Rham differential

\begin{displaymath}
d \tilde X = d f + \frac{2\pi}{R}\sum_A \eta_A \omega_A
\end{displaymath}
where $\eta_A$ are integers and

\begin{displaymath}
\{\omega_A\} \subset 
   H^1(\Sigma, \mathbb{Z})
   \simeq
   \mathbb{Z}^{\oplus 2g + s - 1}
 \,.
\end{displaymath}
Then formally performing the path integral over $f$ yields $d b = 0$ and $b|_{\partial \Sigma} = d X_i$. It follows that $b = d X$ for some other field $X : \Sigma \to S^1_R$.

So we get the action

\begin{displaymath}
S_E = \frac{1}{4\pi} \int_\Sigma d X \wedge \star d X
  +
  \frac{i}{2 \pi}
   \sum_{i = 1}^s
   \int_{\partial \Sigma_i}
   a_i d X
\end{displaymath}
in terms of the field $X$. This is the $\sigma$-model for string propagation on $S^1_R$. with D-brane wrapped on $S^1_R$ that carries on its worldvolume a [[gauge field]] given by a constant connection 1-form $a_i$.

\hypertarget{topological_tduality}{}\subsection*{{Topological T-duality}}\label{topological_tduality}

It turns out to be possible and useful to discuss just the \emph{topological} aspects of T-duality, meaning all the aspects that depend on the $X$ as a [[topological space]], on the topological class of the [[gerbe]] and of its 3-form curvature, but not on the [[Riemannian metric]] and not on the precise connection on the gerbe (there may be several inequivalent ones for a given curvature)!

This sub-phenomenon is discussed in more detail at [[topological T-duality]].

\hypertarget{geometric_tduality}{}\subsection*{{Geometric T-duality}}\label{geometric_tduality}

\hypertarget{InDiffCohomology}{}\subsubsection*{{In terms of generalized differential cohomology}}\label{InDiffCohomology}

[[gauge field|Gauge field]]s are [[cocycle]]s in [[differential cohomology]]. The [[Kalb-Ramond field]] is given by degree-3 [[ordinary differential cohomology]], the differential refinement on degree-3 [[integral cohomology]]. The [[RR-field]] is given by [[differential K-theory]].

Induced by the morphisms $\mathbf{c}(n)$ in the [[fiber sequence]]s

\begin{displaymath}
\mathbf{B}U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) 
  \stackrel{\mathbf{c}_n}{\to} \mathbf{B}^2 U(1)
\end{displaymath}
is induced a notion of [[twisted cohomology]] which makes the Kalb-Ramond field act as a twist for [[twisted K-theory]].

In these terms, the setup of T-duality is a [[correspondence]] of [[Kalb-Ramond field]]s over [[spacetime]] [[torus]]-bundles $P \to X$ and $\hat P \to X$ that induces an integral transform

\begin{displaymath}
K_{diff}^{\bullet + \tau}(P)
  \to 
  K_{diff}^{\bullet + \tau -1}(\hat P)
\end{displaymath}
of twisted differential K-theory classes.

This is an [[isomorphism]] -- the action of the T-duality isomorphism on the Kalb-Ramond field and the RR-field.

See (\hyperlink{KahleValentino}{KahleValentino}).

\hypertarget{TdualityIngcg}{}\subsubsection*{{In generalized complex geometry}}\label{TdualityIngcg}

Another approach to the study of T-duality takes a somewhat complementary point of view and ignores the [[Eilenberg-MacLane spectrum|integral cohomology]] class in $H^3(X,\mathbb{Z})$ of the [[gerbe]] but does consider the [[Riemannian metric]].

In this context T-duality is described as an isomorphism of [[standard Courant algebroid]]s. This picture emerged in the study of [[generalized complex geometry]].

\hypertarget{examples_of_tdual_pairs}{}\subsection*{{Examples of T-dual pairs}}\label{examples_of_tdual_pairs}

\hypertarget{in_mirror_symmetry}{}\subsubsection*{{In Mirror symmetry}}\label{in_mirror_symmetry}

One special case of T-duality is [[mirror symmetry]].

\begin{itemize}%
\item [[Andrew Strominger]], [[Shing-Tung Yau]], [[Eric Zaslow]], \emph{Mirror Symmetry is T-Duality}, Nucl. Phys. B 479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) \href{http://arxiv.org/abs/hep-th/9606040}{hep-th/9606040}

\end{itemize}
\hypertarget{in_langlands_duality}{}\subsubsection*{{In Langlands duality}}\label{in_langlands_duality}

In some cases the passage to the [[Langlands dual group]] in the [[geometric Langlands correspondence]] can be understood as T-duality. (\hyperlink{DaenzerErp}{Daenzer-vanErp})

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

[[duality in physics]], [[duality in string theory]]

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

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

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

\end{itemize}

\item [[Poisson-Lie T-duality]]


\item [[differential T-duality]]


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



\end{itemize}

\item [[T-fold]], [[double field theory]]


\item [[duality in string theory]]

\begin{itemize}%
\item [[U-duality]], [[S-duality]]


\item [[AdS/CFT correspondence]]



\end{itemize}


\end{itemize}
The geometry of the fiber product of two torus fiber bundles with a [[circle 2-bundle]] on it is sometimes referred to as \emph{[[Bn-geometry]]}.

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

The observation of T-duality is attributed to

\begin{itemize}%
\item [[Thomas Buscher]], \emph{A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 ()}


\item [[Thomas Buscher]], \emph{Path integral derivation of quantum duality in nonlinear sigma models}, Phys. Lett. B 201 (1988) 466 ()



\end{itemize}
Precursors include (according to \hyperlink{Schwarz96}{Schwarz 96, first paragraph}):

\begin{itemize}%
\item Keiji Kikkawa, Masami Yamasaki, \emph{Casimir effects in superstring theories}, Physics Letters B Volume 149, Issues 4–5, 20 December 1984, Pages 357-360 ()

\end{itemize}
Discussion for the [[superstring]] is in

\begin{itemize}%
\item M. Abou-Zeid, [[Chris Hull]], [[Ulf Lindström]], [[Martin Roček]], \emph{T-Duality in $(2,1)$ Superspace} (\href{https://arxiv.org/abs/1901.00662}{arXiv:1901.00662})

\end{itemize}
Review includes

\begin{itemize}%
\item Amit Giveona, Massimo Porrati, Eliezer Rabinovicia, \emph{Target space duality in string theory}, Physics Reports Volume 244, Issues 2–3, August 1994, Pages 77-202 ()


\item E. Alvarez, [[Luis Alvarez-Gaume]], Y. Lozano, \emph{An Introduction to T-Duality in String Theory}, Nucl.Phys.Proc.Suppl.41:1-20,1995 (\href{https://arxiv.org/abs/hep-th/9410237}{arXiv:hep-th/9410237})


\item [[Peter West]], section 17.1 of \emph{[[Introduction to Strings and Branes]]}, Cambridge University Press 2012


\item [[Mathai Varghese]], \emph{T-duality: A basic introduction}, 10th Geometry and Physics Conference \emph{Quantum Geometry}, Anogia 2012 ([[Mathai12.pdf:file]])


\item Jnanadeva Maharana, \emph{The Worldsheet Perspective of T-duality Symmetry in String Theory} (\href{http://arxiv.org/abs/1302.1719}{arXiv:1302.1719})


\item Mark Bugden, \emph{A Tour of T-duality: Geometric and Topological Aspects of T-dualities}, (\href{https://arxiv.org/abs/1904.03583}{arXiv:1904.03583})



\end{itemize}
Discussion at strong coupling (``[[F-theory]]'') includes

\begin{itemize}%
\item [[John Schwarz]], \emph{M Theory Extensions of T Duality} (\href{https://arxiv.org/abs/hep-th/9601077}{arXiv:hep-th/9601077})


\item [[Jorge Russo]], \emph{T-duality in M-theory and supermembranes}, Phys.Lett. B400 (1997) 37-42 (\href{https://arxiv.org/abs/hep-th/9701188}{arXiv:hep-th/9701188})



\end{itemize}
Geometric T-duality in terms of [[differential cohomology]] as an operation on [[twisted K-theory|twisted]] [[differential K-theory]] is discussed in

\begin{itemize}%
\item [[Alexander Kahle]], [[Alessandro Valentino]], \emph{[[T-Duality and Differential K-Theory]]}

\end{itemize}
More physically oriented discussion of this is in

\begin{itemize}%
\item [[Katrin Becker]], Aaron Bergman, \emph{Geometric Aspects of D-branes and T-duality} (\href{http://arxiv.org/abs/0908.2249}{arXiv:0908.2249})

\end{itemize}
Geometric T-duality is identified as an [[isomorphism]] of [[standard Courant algebroid]]s ([[generalized complex geometry]]) in section 4 of

\begin{itemize}%
\item [[Gil Cavalcanti]], [[Marco Gualtieri]], \emph{Generalized complex geometry and T-duality} (\href{http://arxiv.org/abs/1106.1747}{arXiv:1106.1747})

\end{itemize}
Discussion of the [[sigma-model]] description of T-duality in this context includes

\begin{itemize}%
\item [[Ulf Lindstöm]], [[Martin Rocek]], Itai Ryb, [[Rikard von Unge]], [[Maxim Zabzine]], \emph{T-duality and Generalized K\"a{}hler Geometry}, JHEP 0802:056,2008 (\href{http://arxiv.org/abs/0707.1696}{arXiv:0707.1696})


\item Jonas Persson, \emph{T-duality and Generalized Complex Geometry} (\href{http://arxiv.org/abs/hep-th/0612034}{arXiv:hep-th/0612034})



\end{itemize}
Further references are

\begin{itemize}%
\item Willie Carl Merrell, \emph{Application of superspace techniques to effective actions, complex geometry and T-duality in String theory} (\href{http://www.lib.umd.edu/drum/bitstream/1903/6865/1/umi-umd-4355.pdf}{pdf})


\item Peggy Kao, \emph{T-duality and Poisson-Lie T-duality in generalized geometry} (\href{http://tpsrv.anu.edu.au/Members/bouwknegt/Kao.pdf}{pdf})



\end{itemize}
Discussion of the infinitesimal T-duality geometry, replacing [[gerbes]] on [[torus]]-[[fiber bundles]] with the corresponding [[dg-manifolds]] is in

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

\end{itemize}
For references on [[topological T-duality]] see there.

The relation to [[Langlands dual groups]] is discussed in

\begin{itemize}%
\item [[Calder Daenzer]], [[Erik Van Erp]], \emph{T-Duality for Langlands Dual Groups} (\href{http://arxiv.org/abs/1211.0763}{arXiv:1211.0763})

\end{itemize}
For [[RR-fields]]:

\begin{itemize}%
\item [[Kentaro Hori]], \emph{D-branes, T-duality and Index theory}, Adv. Theor. Math. Phys. 3 (1999) 281 (\href{https://arxiv.org/abs/hep-th/9902102}{arXiv:hep-th/9902102})

\end{itemize}
Discussion of T-duality that takes into account the [[super p-brane]] charges (i.e. the fermionic components of the [[RR-fields]]) on [[super spacetime]], hence also of [[Green-Schwarz action functionals]], includes the following:

\begin{itemize}%
\item [[Mirjam Cvetic]], H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], \emph{T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map}, Nucl.Phys.B573:149-176,2000 (\href{https://arxiv.org/abs/hep-th/9907202}{arXiv:hep-th/9907202})


\item Bogdan Kulik, Radu Roiban, \emph{T-duality of the Green-Schwarz superstring}, JHEP 0209 (2002) 007 (\href{https://arxiv.org/abs/hep-th/0012010}{arXiv:hep-th/0012010})


\item [[Igor Bandos]], [[Bernard Julia]], \emph{Superfield T-duality rules}, JHEP 0308 (2003) 032 (\href{https://arxiv.org/abs/hep-th/0303075}{arXiv:hep-th/0303075})

reviewed in \emph{Superfield T-duality rules in ten dimensions with one isometry} (\href{https://arxiv.org/abs/hep-th/0312299}{arXiv:hep-th/0312299})



\end{itemize}
[[!redirects Buscher rule]] [[!redirects Buscher rules]]



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