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\section*{double field theory}

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

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

[[!include string theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{tdualitycovariant_formalism}{T-duality-covariant formalism}\dotfill \pageref*{tdualitycovariant_formalism} \linebreak
\noindent\hyperlink{parahermitian_geometry}{Para-hermitian geometry}\dotfill \pageref*{parahermitian_geometry} \linebreak
\noindent\hyperlink{foliated_courant_algebroid_and_spacetime}{Foliated Courant algebroid and spacetime}\dotfill \pageref*{foliated_courant_algebroid_and_spacetime} \linebreak
\noindent\hyperlink{cbracket}{C-bracket}\dotfill \pageref*{cbracket} \linebreak
\noindent\hyperlink{analogy_with_geometric_quantization}{Analogy with geometric quantization}\dotfill \pageref*{analogy_with_geometric_quantization} \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}

The term \emph{double field theory} has come to be used for [[field theory]] ([[prequantum field theory]]/[[quantum field theory]]) on [[spacetimes]] which are [[T-folds]] ([[doubled geometries]]) hence for ``[[T-duality]]-[[equivariance|equivariant]] field theory''.

\hypertarget{tdualitycovariant_formalism}{}\subsection*{{T-duality-covariant formalism}}\label{tdualitycovariant_formalism}

\hypertarget{parahermitian_geometry}{}\subsubsection*{{Para-hermitian geometry}}\label{parahermitian_geometry}

The use of para-hermitian geometry in Double Field Theory was introduced by Izu Vaisman in (\hyperlink{Vaisman}{I.Vaisman 2012}), then by David Svoboda (\hyperlink{Svoboda}{D.Svoboda 2018}).

$\backslash$begin\{defn\} An \emph{almost para-complex manifold} is a [[manifold]] $M$ equipped with a vector bundle endomorphism $F\in\mathrm{End}(T M)$ such that $F^2=1$ and its $\pm 1$ eigenbundles $T^\pm M$ have same rank. $\backslash$end\{defn\}

$\backslash$begin\{defn\} The \emph{para-complex projectors} are the canonical projectors onto $T^\pm M$ defined by $P_\pm = \frac{1}{2}(1\pm F)$ $\backslash$end\{defn\}

$\backslash$begin\{defn\} A $\pm$\emph{-para-complex manifold} is an almost para-complex manifold $(M,F)$ such that $T^\pm M$ of $F$ is Frobenius integrable as a distribution. A \emph{para-complex} manifold is a manifold that is both $+$-para-complex and $-$-para-complex. $\backslash$end\{defn\}

$\backslash$begin\{rmk\} A doubled manifold $M$ equipped with the $O(d,d)$-structure $\eta$ carries a natural almost para-hermitian structure. On patches $U$ with coordinates $(x^\mu,\tilde{x}_\mu)$ we have the canonical para-complex structure

\begin{displaymath}
F\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial x^\mu},\,\, F\frac{\partial}{\partial \tilde{x}_\mu} = - \frac{\partial}{\partial \tilde{x}_\mu}
\end{displaymath}
with eigenbundles

\begin{itemize}%
\item $T^+U=\big\langle\frac{\partial}{\partial x^\mu}\big\rangle$,


\item $T^-U=\big\langle\frac{\partial}{\partial \tilde{x}_\mu}\big\rangle$,



\end{itemize}
with associated foliations

\begin{itemize}%
\item $\mathcal{F}_+=\{x^\mu = \mathrm{const}\}$,


\item $\mathcal{F}_-=\{\tilde{x}_\mu = \mathrm{const}\}$.



\end{itemize}
In analogy with [[complex geometry]], if we define $\Omega^{r,s}(M)$ as the space of sections of $\Lambda^r(T^+M)\wedge\Lambda^s(T^-M)$, we have the decomposition

\begin{displaymath}
\Omega^k(M)=\bigoplus_{k=r+s}\Omega^{r,s}(M)
\end{displaymath}
Let us call $\pi^{r,s}:\Omega^{r+s}(M)\rightarrow\Omega^{r,s}(M)$ the canonical projector induced by $P_\pm$.

In analogy with complex geometry, there are \emph{para-Dolbeault operators}:

\begin{itemize}%
\item $\mathrm{d}^+=\pi^{r+1,s}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r+1,s}(M)$,


\item $\mathrm{d}^-=\pi^{r,s+1}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r,s+1}(M)$.



\end{itemize}
with properties:

\begin{itemize}%
\item $\mathrm{d} = \mathrm{d}^+ + \mathrm{d}^-$,


\item $(\mathrm{d}^\pm)^2= 0$,


\item $\mathrm{d}^+\mathrm{d}^-+\mathrm{d}^-\mathrm{d}^+ =0$.



\end{itemize}
We can also define [[Lie derivatives]] on the eigenbundles $T^\pm M$ by

\begin{displaymath}
\mathcal{L}^\pm_{X_\pm}\xi = (\mathrm{d}^\pm\iota_{X_\pm} + \iota_{X_\pm}\mathrm{d}^\pm)\xi
\end{displaymath}
for any vector $X_\pm\in\Gamma(T^\pm M)$ and $\xi\in\Omega^{r,s}(M)$. $\backslash$end\{rmk\}

$\backslash$begin\{defn\} An \emph{almost para-hermitian manifold} is an almost para-complex manifold $M$ equipped with a compatible metric, i.e a symmetric tensor $\eta\in\mathrm{Sym}^2(T M)$ such that $\eta(F\cdot\,,F\cdot\,)=-\eta(\,\cdot\,,\,\cdot\,)$. $\backslash$end\{defn\}

$\backslash$begin\{rmk\} The contraction with the metric $\eta$ defines two isomorphisms

\begin{displaymath}
\phi^\pm : T^\pm M \rightarrow (T^\mp M)^\ast
\end{displaymath}
by

\begin{itemize}%
\item $\phi^\pm(X_\pm) =  X_\pm^\flat$ and


\item $(\phi^\pm)^{-1}(\xi_\pm)=\xi^\sharp$



\end{itemize}
that map a vector in $T^\pm M$ to a $1$-form in $(T^\mp M)^\ast$ and vice-versa. We used the notation $\flat,\sharp$ for the musical isomorphisms induced by the metric $\eta$ between $T M$ and $T^\ast M$.

This can be used to define a new couple of isomorphisms

\begin{displaymath}
\Phi^\pm : T M \longrightarrow T^\pm M\oplus(T^\pm M)^\ast
\end{displaymath}
that maps a vector a vector $X=X_++X_-$ with $X_\pm\in T^\pm M$ into $X_+ + X_-^\flat$. $\backslash$end\{rmk\}

$\backslash$begin\{defn\} A \emph{para-hermitian manifold} is an almost para-hermitian manifold $(M,\eta,F)$ such that $(M,F)$ is a para-complex manifold. $\backslash$end\{defn\}

\hypertarget{foliated_courant_algebroid_and_spacetime}{}\subsubsection*{{Foliated Courant algebroid and spacetime}}\label{foliated_courant_algebroid_and_spacetime}

Given a $+$-para-hermitian manifold $(M,\eta,F)$, consider the triple $\big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+M}\big)$ where

\begin{itemize}%
\item the skew-symmetric bracket $[\,\cdot\,,\,\cdot\,]_+:=[\,\cdot\,,\,\cdot\,]|_{T^+ M}$


\item the anchor $1_{T^\pm M}:=1_{T M}|_{T^\pm M}$ are the restrictions of Lie bracket and identity of $T M$.



\end{itemize}
Since $M$ is $+$-para-hermitian we have that $T^+M$ is integrable and therefore it is the tangent bundle $T^+M = T\mathcal{F}_\pm$ of a foliation $\mathcal{F}_+$. This means that this triple is just the tangent Lie algebroid of the foliation $\mathcal{F}_+$:

\begin{displaymath}
\big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+ M}\big) = \big(T\mathcal{F}_+,[\,\cdot\,,\,\cdot\,]_{T\mathcal{F}_+},1_{T\mathcal{F}_+}\big).
\end{displaymath}
There is a natural Courant algebroid structure on the bundle $T^+ M\oplus(T^+ M)^\ast$ with skew-symmetric pairing

\begin{displaymath}
[X+\alpha,Y+\beta]_{+} = [X,Y]_+ + \mathcal{L}_X^+\beta -\mathcal{L}_Y^+\alpha +\mathrm{d}_+(\iota_Y\alpha)
\end{displaymath}
and symmetric pairing

\begin{displaymath}
\langle X+\alpha,Y+\beta\rangle_+ = \iota_X\beta + \iota_Y\alpha.
\end{displaymath}
The isomorphism $\Phi^+:T^+ M\oplus (T^+ M)^\ast\rightarrow T^+ M\oplus T^- M = T M$ previously defined induces a Courant algebroid isomorphism and hence a Courant algebroid structure on $T M$. This induces a metric $\eta$ on $M$ and a skew-symmetric pairing on $T M$ by

\begin{displaymath}
[[X_++X_-,Y_++Y_-]]_+ := [X_+,Y_+] + \Big[\mathcal{L}_{X_+}^+ Y_-^\flat -\mathcal{L}_{Y_+}^+ X_-^\flat +\mathrm{d}_+\big(\eta(X_-,Y_+)\big)\Big]^\sharp,
\end{displaymath}
where $X\in T M$ is split in $X_\pm=P_\pm X$.

$\backslash$begin\{rmk\} Since $M$ is assumed $+$-para-hermitian, $T^+ M\oplus (T^+ M)^\ast$ can be written as $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$. Therefore we constructed an isomorphism between the Courant algebroid on the whole $T M$ and the generalized tangent bundle $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$ of the foliation. In other terms para-hermitian geometry of the doubled manifold $M$ reduces to Generalized Geometry of physical spacetime. $\backslash$end\{rmk\}

$\backslash$begin\{rmk\} The same argument can be clearly applied to $T^-M$ too. $\backslash$end\{rmk\}

\hypertarget{cbracket}{}\subsubsection*{{C-bracket}}\label{cbracket}

In previous section we assumed that the $+1$-eigenbundle $T^+M$ is integrable. This is equivalent to assuming that there exists a well defined foliation that can be interpreted as the physical spacetime. However it is possible to construct a more general bracket that does not require such an assumption, but only an almost para-complex structure. Therefore it works even when a global physical spacetime foliation is not defined. This is achieved by Vaisman with the definition of \emph{C-bracket} by using a generalization of the notion of Levi-Civita connection (look (\hyperlink{Vaisman}{I.Vaisman 2012})).

In the special case of an (inetgrable) para-hermitian manifold C-bracket is given by

\begin{displaymath}
[[ X+\alpha,Y+\beta ]]_{\mathrm{C}} = [X,Y] + \mathcal{L}_X\beta -\mathcal{L}_Y\alpha + \mathrm{d}\big(\eta(X+\alpha,Y+\beta)\big) +
    [\alpha,\beta]^\ast + \mathcal{L}^\ast_\alpha Y -\mathcal{L}^\ast_\beta X - \mathrm{d}^\ast\big(\eta(X+\alpha,Y+\beta)\big)
\end{displaymath}
where $[\,\cdot\,,\,\cdot\,]$, $\mathcal{L}_X$ and $\mathrm{d}$ are usual Lie bracket, Lie derivative and differential on $T\mathcal{F}_+$. On the other hand $[\,\cdot\,,\,\cdot\,]^\ast$, $\mathcal{L}^\ast_\alpha$ and $\mathrm{d}^\ast$ are Lie bracket, Lie derivative and differential on $T\mathcal{F}_+^\ast$ induced by the former ones.

\hypertarget{analogy_with_geometric_quantization}{}\subsubsection*{{Analogy with geometric quantization}}\label{analogy_with_geometric_quantization}

The analogy between [[geometric quantization]] and DFT was firstly noticed by [[David Berman]].

Given a symplectic manifold $(M,\omega)$ there exist couples of lagrangian foliations $\mathcal{F}_+,\mathcal{F}_-$ of $M$ defined by

\begin{displaymath}
\forall X,Y\in T\mathcal{F}_\pm, \: \omega(X,Y)=0.
\end{displaymath}
For example for a symplectic space $(\mathbb{R}^{2d},\omega=\mathrm{d}x^\mu\wedge\mathrm{p}_\mu)$ we can have $\mathcal{F}_+ = \{x^\mu = \mathrm{const} \}$ and $\mathcal{F}_-= \{p_\mu = \mathrm{const} \}$. But notice that any symplectic rotation of this choice is a couple of lagrangian foliations that works fine.

Heuristically, in geometrical quantization we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to ``select'' a physical spacetime $\mathcal{F}_+$ from the whole symplectic-covariant theory on $M$.

Similarly in DFT, when $M$ is an (integrable) para-hermitian manifold we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to ``select'' a physical spacetime $\mathcal{F}_+$ from the whole T-duality-covariant theory on $M$.

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

\begin{itemize}%
\item [[doubled geometry]], [[generalized geometry]], [[exceptional generalized geometry]]


\item [[exceptional field theory]]


\item [[T-fold]]



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

The idea of ``doubled spacetime geometry'' is a variant of the idea of [[T-folds]], due to

\begin{itemize}%
\item [[Chris Hull]], \emph{Doubled geometry and T-folds} JHEP0707:080,2007 (\href{http://arxiv.org/abs/hep-th/0605149}{arXiv:hep-th/0605149})

\end{itemize}
The coinage of the term ``double field theory'' for field theory on such doubled geometry goes back to

\begin{itemize}%
\item [[Chris Hull]], [[Barton Zwiebach]], \emph{Double Field Theory}, JHEP 0909:099,2009 (\href{http://arxiv.org/abs/0904.4664}{arXiv:0904.4664})

\end{itemize}
Discussion in the context of [[L-infinity algebra]] includes

\begin{itemize}%
\item [[Andreas Deser]], [[Jim Stasheff]], \emph{Even symplectic supermanifolds and double field theory}, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 (\href{http://arxiv.org/abs/1406.3601}{arXiv:1406.3601})

\end{itemize}
Discussion of an extended version of Riemannian geometry suitable for the description of double field theory

\begin{itemize}%
\item [[Andreas Deser]], [[Christian Saemann]], \emph{Extended Riemannian Geometry I: Local Double Field Theory}, (\href{https://arxiv.org/abs/1611.02772}{arXiv:1611.02772})

\end{itemize}
Discussion about para-Hermitian formalism started in

\begin{itemize}%
\item Izu Vaisman, \emph{On the geometry of double field theory} Journal of Mathematical Physics 53, 033509 (2012) (\href{https://arxiv.org/abs/1203.0836v1}{arXiv:1203.0836})

\end{itemize}
Para-Hermitian formalism further developed and generalized in

\begin{itemize}%
\item David Svoboda, \emph{Algebroid structures on para-Hermitian manifolds} Journal of Mathematical Physics 59, 122302 (2018) (\href{https://arxiv.org/abs/1802.08180}{arxiv:1802.08180})

\end{itemize}


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