\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{generalized complex geometry}

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

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - 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{on_a_vector_space}{On a vector space}\dotfill \pageref*{on_a_vector_space} \linebreak
\noindent\hyperlink{on_a_manifold}{On a manifold}\dotfill \pageref*{on_a_manifold} \linebreak
\noindent\hyperlink{in_terms_of_reduction_of_the_structure_group}{In terms of reduction of the structure group}\dotfill \pageref*{in_terms_of_reduction_of_the_structure_group} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{as_targets_for_models}{As targets for $\sigma$-models}\dotfill \pageref*{as_targets_for_models} \linebreak
\noindent\hyperlink{mirror_symmetry}{Mirror symmetry}\dotfill \pageref*{mirror_symmetry} \linebreak
\noindent\hyperlink{geometry_of_supergravity}{Geometry of supergravity}\dotfill \pageref*{geometry_of_supergravity} \linebreak


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

\emph{Generalized complex geometry} is the study of the geometry of [[symplectic Lie n-algebroid|symplectic Lie 2-algebroid]] called [[standard Courant algebroid]]s $\mathfrak{c}(X)$ (over a [[smooth manifold]] $X$).

This geometry of symplectic Lie 2-algebroids turns out to unify, among other things, [[complex geometry]] with [[symplectic geometry]]. This unification notably captures central aspects of [[T-duality]].

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

\hypertarget{on_a_vector_space}{}\subsubsection*{{On a vector space}}\label{on_a_vector_space}

Let $V$ be a [[dimension|finite dimensional]] [[vector space]] over the [[real numbers]].

Recall that a \emph{[[complex structure]]} on $V$ is a [[linear map]]

\begin{displaymath}
J : V \to V
\end{displaymath}
such that $J \circ J = - id_V$. And a [[symplectic structure]] on $V$ is equivalently a linear [[isomorphism]]

\begin{displaymath}
\omega : V \to V^*
\end{displaymath}
such that

\begin{displaymath}
\omega^* = - \omega
  \,,
\end{displaymath}
where $V^*$ denotes the [[dual vector space]] and $\omega^*$ the dual linear map.

The following definition may be thought of as combining these two concepts.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{generalized complex structure} on $V$ is a [[linear map]]

\begin{displaymath}
\mathcal{J} : V \oplus V^* \to V \oplus V^*
\end{displaymath}
(an [[endomorphism]] of the [[direct sum]] of $V$ with its [[dual vector space]])

such that it is both

\begin{enumerate}%
\item a [[complex structure]] on $V \oplus V^*$ in that $\mathcal{J}^2 = - id$;


\item a [[symplectic structure]] on $V \oplus V^*$ in that $\mathcal{J}^* = - \mathcal{J}$.



\end{enumerate}
\end{defn}
The following shows that this is indeed a joint generalization of complex and symplectic structures.

\begin{example}
\label{}\hypertarget{}{}
Let $J : V \to V$ be an ordinary complex structure on $V$. Then the linear endomorphism of $V \oplus V^*$ defined by [[matrix calculus]] as

\begin{displaymath}
\mathcal{J}_j := 
  \left(
    \itexarray{
       -J & 0
       \\
       0 & J^*
    }
  \right)
\end{displaymath}
is a generalized complex structure on $V$.

Similarly, let $\omega : V \to V^*$ be an ordinary symplectic structure on $V$. then the endomorphism

\begin{displaymath}
\mathcal{J}_\omega :=
  \left(
    \itexarray{
      0 & - \omega^{-1}
      \\
      \omega & 0
    }
  \right)
\end{displaymath}
is a generalized complex structure on $V$.

\end{example}
\hypertarget{on_a_manifold}{}\subsubsection*{{On a manifold}}\label{on_a_manifold}

A generalized complex structure on a manifold is a generalized complex structure on the fibers of the [[generalized tangent bundle]].

(\ldots{})

\hypertarget{in_terms_of_reduction_of_the_structure_group}{}\subsubsection*{{In terms of reduction of the structure group}}\label{in_terms_of_reduction_of_the_structure_group}

A generalized complex structure on $V \oplus V^*$ is equivalently a [[reduction of the structure group]] along the inclusion

\begin{displaymath}
U(n,n) \hookrightarrow O(2n,2n)
  \,,
\end{displaymath}
where the left hand is identified as $U(n,n) = O(2n,2n) \cap GL(2n, \mathbb{C})$ (\hyperlink{Gualtieri}{Gualtieri, prop. 4.6}).

The analog of this with the unitary group replaced by the orthogonal group yields \emph{[[type II geometry]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

One finds (as described at [[standard Courant algebroid]]) that

\begin{itemize}%
\item choices of sub-[[Lie algebroid]]s of $\mathfrak{c}(X)$ encode (almost) \textbf{[[Dirac structure]]s} and -- after [[complexification]] -- \textbf{generalized [[complex structure]]s};


\item choices of [[section]]s of the canonical morphism $\mathfrak{c}(X) \to T X$ to the [[tangent Lie algebroid]]s encode \textbf{generalized Riemannian metrics}: pairs consisting of a (possibly [[pseudo-Riemannian metric|pseudo]]-)[[Riemannian metric]] and a [[differential form|2-form]].

In applications in [[string theory]], this encodes the field of [[gravity]] and the [[Kalb–Ramond field]], respectively. (There are also proposals for how the [[dilaton]] field appears in this context.)



\end{itemize}
In components these are structures found on the [[vector bundle]]

\begin{displaymath}
T X \oplus T^* X
  \,,
\end{displaymath}
the [[direct sum]] of the [[tangent bundle]] with the cotangent bundle of $X$.

Generalized complex geometry thus generalizes and unifies

\begin{itemize}%
\item [[complex geometry]],
\item [[symplectic geometry]], and
\item [[Riemannian geometry]].

\end{itemize}
It was in particular motivated by the observation that this provides a natural formalism for describing [[T-duality]].

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\begin{itemize}%
\item [[generalized Calabi-Yau manifold]]

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

\begin{itemize}%
\item [[generalized tangent bundle]], [[generalized vielbein]]


\item [[type II geometry]]


\item [[exceptional generalized geometry]]


\item [[Bn-geometry]]


\item [[generalized contact geometry]]



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

\hypertarget{general}{}\paragraph*{{General}}\label{general}

Generalized complex geometry was proposed by [[Nigel Hitchin]] as a formalism in [[differential geometry]] that would be suited to capture the phenomena that physicists encountered in the study of [[T-duality]]. It was later and is still developed by his students, notably Gualtieri and Cavalcanti.

A standard reference is the PhD thesis

\begin{itemize}%
\item [[Marco Gualtieri]], \emph{Generalized complex geometry} (\href{http://arxiv.org/abs/math/0401221}{arXiv:math/0401221})

\end{itemize}
A survey set of slides with an eye towards the description of the [[Kalb-Ramond field]] and [[bundle gerbe]]s is

\begin{itemize}%
\item [[Nigel Hitchin]], \emph{B-Fields, gerbes and generalized geometry} Oxford Durham Symposium 2005 (\href{http://www.maths.dur.ac.uk/events/Meetings/LMS/2005/GCFTST/Talks/hitchin1.pdf}{pdf})


\item [[Nigel Hitchin]], \emph{Lectures on generalized geometry} (\href{http://arxiv.org/abs/1008.0973}{arxiv/1008.0973})



\end{itemize}
\hypertarget{as_targets_for_models}{}\paragraph*{{As targets for $\sigma$-models}}\label{as_targets_for_models}

Generalized complex structures may serve as [[target spaces]] for [[sigma-models]]. Relations to the [[Poisson sigma-model]] and the [[Courant sigma-model]] are discussed in

\begin{itemize}%
\item [[Alberto Cattaneo]], Jian Qiub, [[Maxim Zabzine]], \emph{2D and 3D topological field theories for generalized complex geometry}, \href{http://projecteuclid.org/getRecord?id=euclid.atmp/1288619156}{euclid} \href{http://www.ams.org/mathscinet-getitem?mr=2721659}{MR2012a:81255} \href{http://www.math.uzh.ch/fileadmin/math/preprints/GC_BV.pdf}{pdf}

\end{itemize}
\hypertarget{mirror_symmetry}{}\paragraph*{{Mirror symmetry}}\label{mirror_symmetry}

\begin{itemize}%
\item Oren Ben-Bassat, \emph{Mirror symmetry and generalized complex manifolds. I. The transform on vector bundles, spinors, and branes}, J. Geom. Phys. \textbf{56} (2006), no. 4, 533--558 \href{http://arxiv.org/abs/math/0405303}{math.AG/0405303} \href{http://www.ams.org/mathscinet-getitem?mr=2199280}{MR2006k:53067} \href{http://dx.doi.org/10.1016/j.geomphys.2005.03.004}{doi}

\end{itemize}
\hypertarget{geometry_of_supergravity}{}\paragraph*{{Geometry of supergravity}}\label{geometry_of_supergravity}

Generalized complex geometry and variants of [[exceptional generalized complex geometry]] are natural for describing [[supergravity]] background compactifications in [[string theory]] with their [[T-duality]] and [[U-duality]] symmetries ([[non-geometric vacua]]).

\begin{itemize}%
\item Ian Ellwood, \emph{NS-NS fluxes in Hitchin's generalized geometry} (\href{http://arxiv.org/abs/hep-th/0612100}{arXiv:hep-th/0612100})


\item Paulo Pires Pacheco, [[Daniel Waldram]], \emph{M-theory, exceptional generalised geometry and superpotentials} (\href{http://arxiv.org/abs/0804.1362}{arXiv:0804.1362})


\item [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Daniel Waldram]], \emph{T-duality, generalized geometry and non-geometric backgrounds}, J. High Energy Phys. 2009, no. 4, 075, 39 pp. \href{http://xxx.lanl.gov/abs/0807.4527}{arXiv:0807.4527} \href{http://www.ams.org/mathscinet-getitem?mr=2505954}{MR2010i:81323} \href{http://dx.doi.org/10.1088/1126-6708/2009/04/075}{doi}


\item David Andriot, [[Ruben Minasian]], Michela Petrini, \emph{Flux backgrounds from twists}, J. High Energy Phys. 2009, no. 12, 028 \href{http://arxiv.org/abs/0903.0633}{arXiv:0903.0633} \href{http://www.ams.org/mathscinet-getitem?mr=2593014}{MR2011c:81201} \href{http://dx.doi.org/10.1088/1126-6708/2009/12/028}{doi}



\end{itemize}
[[!redirects generalized complex structure]] [[!redirects generalized complex structures]]



\end{document}