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\begin{document}

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\section*{type II geometry}

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

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

[[!include synthetic differential geometry - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher 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{FiberwiseMetric}{By fiberwise metric on the generalized tangent bundle}\dotfill \pageref*{FiberwiseMetric} \linebreak
\noindent\hyperlink{ReductionOfStructureGroup}{By reduction of the generalized tangent bundle}\dotfill \pageref*{ReductionOfStructureGroup} \linebreak
\noindent\hyperlink{GeometricAndNonGeometric}{Geometric and ``non-geometric'' type II geometries}\dotfill \pageref*{GeometricAndNonGeometric} \linebreak
\noindent\hyperlink{application_in_type_ii_supergravity}{Application in type II supergravity}\dotfill \pageref*{application_in_type_ii_supergravity} \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{ReferencesDoubledSupergeometry}{Doubled super-geometry}\dotfill \pageref*{ReferencesDoubledSupergeometry} \linebreak


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

\emph{Type II geometry} (often \emph{doubled geometry} as in ``[[double field theory]]'') is to [[Riemannian geometry]] as [[generalized complex geometry]] is to [[complex geometry]].

Where the latter is the [[geometry]] induced by [[reduction of the structure group]] of the [[generalized tangent bundle]] of an even dimensional [[manifold]] along the inclusion $U(d,d) \to O(2d,2d)$ of the indefinite [[unitary group]] into the [[orthogonal group]], type II geometry is the geometry induced by reduction along the inclusion of the product of [[orthogonal group|orthogonal groups]]

\begin{displaymath}
O(n) \times O(n) \to O(n,n)
  \,,
\end{displaymath}
which is the inclusion of the [[maximal compact subgroup]] into the [[Narain group]].

This notion takes its name from the fact that it describes a good bit of the geometry of [[type II supergravity]].

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

The definition of type II geometry proceeds in direct analogy with that of [[Riemannian geometry]] in terms of [[orthogonal structure]]/[[vielbein]] fields on the tangent bundle, generalized here to the [[generalized tangent bundle]]:

\begin{enumerate}%
\item \hyperlink{FiberwiseMetric}{As a fiberwise metric on the generalized tangent bundle}


\item \hyperlink{ReductionOfStructureGroup}{By reduction of the structure group of the generalized tangent bundle}



\end{enumerate}
\hypertarget{FiberwiseMetric}{}\subsubsection*{{By fiberwise metric on the generalized tangent bundle}}\label{FiberwiseMetric}

(\ldots{})

\hypertarget{ReductionOfStructureGroup}{}\subsubsection*{{By reduction of the generalized tangent bundle}}\label{ReductionOfStructureGroup}

We discuss how a type II geometry is the [[reduction of the structure group]] of the [[generalized tangent bundle]] along the inclusion $O(d) \times O(d) \to O(d,d)$.

\begin{defn}
\label{InclusionForTypeIIGeometry}\hypertarget{InclusionForTypeIIGeometry}{}
Consider the [[Lie group]] inclusion

\begin{displaymath}
\mathrm{O}(d) \times \mathrm{O}(d)
  \to 
  \mathrm{O}(d,d)
\end{displaymath}
of those [[orthogonal group|orthogonal transformations]], that preserve the positive definite part or the negative definite part of the [[bilinear form]] of signature $(d,d)$, respectively.

If $\mathrm{O}(d,d)$ is presented as the group of $2d \times 2d$-[[matrix|matrices]] that preserve the [[bilinear form]] given by the $2d \times 2d$-matrix

\begin{displaymath}
\eta 
  \coloneqq
  \left( 
     \itexarray{
       0 & \mathrm{id}_d
       \\
       \mathrm{id}_d & 0
     }
  \right)
\end{displaymath}
then this inclusion sends a pair $(A_+, A_-)$ of [[orthogonal group|orthogonal]] $n \times n$-matrices to the matrix

\begin{displaymath}
(A_+ , A_-) 
    \mapsto 
  \frac{1}{\sqrt{2}}
  \left(
    \itexarray{
	   A_+ + A_- & A_+ - A_-
	   \\
	   A_+ - A_- & A_+ + A_-
     }
  \right)
  \,.
\end{displaymath}
\end{defn}
This inclusion of [[Lie groups]] induces the corresponding morphism of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of [[principal bundles]]

\begin{displaymath}
\mathbf{TypeII}
  :
  \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d))
  \to 
  \mathbf{B} \mathrm{O}(d,d)  
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
There is a [[fiber sequence]] of [[smooth infinity-groupoid|smooth stacks]]

\begin{displaymath}
O(d) \backslash O(d,d) / O(d)
	 \to
  \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d))
     \stackrel{\mathbf{TypeII}}{\to}
    \mathbf{B} \mathrm{O}(d,d)  	
  \,,
\end{displaymath}
where the fiber on the left is the [[coset space]] of the [[action]] of $O(d) \times O(d)$ on $O(d,d)$.

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
There is a canonical embedding

\begin{displaymath}
\mathrm{GL}(d) \hookrightarrow \mathrm{O}(d,d)
\end{displaymath}
of the [[general linear group]].

In the above matrix presentation this is given by sending

\begin{displaymath}
a
  \mapsto 
  \left( 
     \itexarray{
	    a & 0
		\\
		0 & a^{-T}
      }
  \right)
  \,,
\end{displaymath}
where in the bottom right corner we have the [[transpose matrix|transpose]] of the inverse matrix of the invertble matrix $a$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Under inclusion of def. \ref{InclusionForTypeIIGeometry}, the [[tangent bundle]] of a $d$-[[dimension|dimensional]] [[manifold]] $X$ defines an $\mathrm{O}(d,d)$-[[cocycle]]

\begin{displaymath}
T X \oplus T^* X
  :
  
    X 
     \stackrel{T X}{\to}
    \mathbf{B}\mathrm{GL}(d)  
      \stackrel{}{\to}
    \mathbf{B} \mathrm{O}(d,d) 
  \,.
\end{displaymath}
The [[vector bundle]] canonically associated to this composite cocycles may canonically be identified with the [[direct sum]] vector bundle $T X \oplus T^* X$, and so we will refer to this cocycle by these symbols, as indicated. This is also called the \textbf{[[generalized tangent bundle]]} of $X$.

\end{defn}
Therefore we may canonically consider the groupoid of $T X \oplus T^* X$-twisted $\mathbf{TypeII}$-structures, according to the general notion of [[twisted differential c-structures]].

More generally, instead of $E = T X \oplus T^* X$ one considers bundle [[extensions]] $E$ of the form

\begin{displaymath}
T^* X \to E \to T X
  \,.
\end{displaymath}
These may have structure froups in $O(n,n)$ but not in the inclusion $GL(n) \hookrightarrow O(n,n)$. For more on this see the section \emph{\hyperlink{GeometricAndNonGeometric}{Geometric and non-geometric type II geometries}} below. Accordingly, in all of the following $T X \oplus T^* X$ could be replaced by a more general extension $E$.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{type II generalized vielbein} on a [[smooth manifold]] $X$ is a diagram

\begin{displaymath}
\itexarray{
    X &&\stackrel{\widetilde(T X \oplus T^* X)}{\to}&& \mathbf{B}(O(n) \times O(n))
    \\
    & {}_{\mathllap{T X \oplus T^* X}}\searrow &\swArrow_{E}& \swarrow_{\mathrlap{\mathbf{TypeII}}}
    \\
    && \mathbf{B} O(n,n)
  }
\end{displaymath}
in $\mathbf{H} =$ [[Smooth∞Grpd]], hence a cocycle in the smooth [[twisted cohomology]]

\begin{displaymath}
E 
   \in 
  \mathbf{TypeII}Struc(X)
  \coloneqq
  \mathbf{H}_{/\mathbf{B} O(n,n)}(T X \oplus T^* X, \mathbf{TypeII})
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The [[groupoid]] $\mathbf{TypeII}\mathrm{Struc}(X)$ is that of ``generalized vielbein fields'' on $X$, as considered for instance around equation (2.24) of (\hyperlink{GMPW}{GMPW}) (there only locally, but the globalization is evident).

In particular, its set of equivalence classes is the set of type-II generalized geometry structures on $X$.

\end{prop}
\begin{proof}
Over a local [[coordinate chart]] $\mathbb{R}^d \simeq U_i \hookrightarrow X$, the most general such generalized vielbein (hence the most general $\mathrm{O}(d,d)$-valued function) may be parameterized as

\begin{displaymath}
E = 
  \frac{1}{2}
  \left(
    \itexarray{
	  (e_+ + e_-) + (e_+^{-T} - e_-^{-T})B & (e_+^{-T} - e_-^{-T})
	  \\
	  (e_+ - e_-) - (e_+^{-T} + e_-^{-T})B & (e_+^{-T} + e_-^{-T})
     }
  \right)
  \,,
\end{displaymath}
where $e_+, e_- \in C^\infty(U_i, \mathrm{O}(d))$ are thought of as two [[vielbein|ordinary vielbein]] fields, and where $B$ is any smooth skew-symmetric $n \times n$-matrix valued function on $\mathbb{R}^d \simeq U_i$.

By an $\mathrm{O}(d) \times \mathrm{O}(d)$-[[gauge transformation]] this can always be brought into a form where $e_+ = e_- =: \tfrac{1}{2}e$ such that

\begin{displaymath}
E = 
  \left(
    \itexarray{
	  e & 0
	  \\
	  - e^{-T}B  & e^{-T}
	}
  \right)
  \,.
\end{displaymath}
The corresponding ``generalized metric'' over $U_i$ is

\begin{displaymath}
E^T 
  E
  = 
  \left(
    \itexarray{
	  e^T & B e^{-1}
	  \\
	  0  & e^{-1}
    }
  \right)
  \left(
    \itexarray{
	  e & 0
	  \\
	  - e^{-T}B  & e^{-T}
    }
  \right)
  = 
  \left(
    \itexarray{
	  g - B g^{-1} B & B g^{-1}
	  \\
	  - g^{-1} B  & g^{-1}
     }
  \right)
  \,,
\end{displaymath}
where

\begin{displaymath}
g \coloneqq e^T e
\end{displaymath}
is the [[metric]] (over $\mathbb{R}^q \simeq U_i$ a smooth function with values in symmetric $n \times n$-matrices) given by the [[vielbein|ordinary vielbein]] $e$.

\end{proof}
\hypertarget{GeometricAndNonGeometric}{}\subsubsection*{{Geometric and ``non-geometric'' type II geometries}}\label{GeometricAndNonGeometric}

\begin{defn}
\label{}\hypertarget{}{}
An element in $O(d,d)$ which in the canonical matrix presentation is of the block form

\begin{displaymath}
e^\omega \coloneqq \left(
    \itexarray{
      1_d & 0 
      \\
      \omega &  1_d
    }
  \right)
\end{displaymath}
is called a \textbf{$B$-transform}. An element of the block form

\begin{displaymath}
e^\beta \coloneqq \left(
    \itexarray{
      1_d & \beta 
      \\
      0 &  1_d
    }
  \right)
\end{displaymath}
is called a \textbf{$\beta$-transform}. The [[subgroup]]

\begin{displaymath}
G_{geom}(d) \hookrightarrow O(d,d)
\end{displaymath}
generated by $Gl(d) \hookrightarrow O(d,d)$ and the B-transforms, hence that of matrices with vaishing top right block is called the \textbf{geometric subgroup} (e.g. \hyperlink{GMPW}{GMPW, p.5}).

A type II background where the structure group of the [[generalized tangent bundle]] is not in the inclusion of the geometric subgroup is often called a \textbf{non-geometric background} (e.g. \hyperlink{GMPW}{GMPW, section 5}).

\end{defn}
\hypertarget{application_in_type_ii_supergravity}{}\subsection*{{Application in type II supergravity}}\label{application_in_type_ii_supergravity}

The [[target space]] geometry for [[type II superstrings]] in the NS-NS sector , [[type II supergravity]], is naturally encoded by type II geometry.

\ldots{}

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

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


\item [[non-geometric string theory vacua]]


\item [[T-folds]]


\item [[twisted smooth cohomology in string theory]]



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

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

The appearance of type II geometry in [[type II supergravity]]/[[type II string theory]] is discussed for instance in

\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 [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Daniel Waldram]], \emph{T-duality, Generalized Geometry and Non-Geometric Backgrounds} (\href{http://arxiv.org/abs/0807.4527}{arXiv:0807.4527})



\end{itemize}
The genuine reformulation of type II supergravity as a $(O(d)\times O(d) \hookrightarrow O(d,d))$-gauge/gravity theory is in

\begin{itemize}%
\item Andr\'e{} Coimbra, Charles Strickland-Constable, [[Daniel Waldram]], \emph{Supergravity as Generalised Geometry I: Type II Theories} (\href{http://arxiv.org/abs/1107.1733}{arXiv:1107.1733})

\end{itemize}
In

\begin{itemize}%
\item [[Chris Hull]], \emph{Generalised Geometry for M-Theory}, JHEP 0707:079,2007 (\href{http://arxiv.org/abs/hep-th/0701203}{arXiv:hep-th/0701203})

\end{itemize}
the geometry of the reduction $O(d) \times O(d) \to O(d,d)$ was referred to as ``type I geometry'', with ``type II geometry'' instead referring to further [[U-duality]] group extensions, discussed at \emph{[[exceptional generalized geometry]]}.

\hypertarget{ReferencesDoubledSupergeometry}{}\subsubsection*{{Doubled super-geometry}}\label{ReferencesDoubledSupergeometry}

Much of the discussion of type II geometry has in fact been purely bosonic, ignoring the [[super-geometry]] of [[supergravity]].

In contrast, discussion combining [[doubled geometry]] with [[super-geometry]] includes the following:

\begin{itemize}%
\item Machiko Hatsuda, Kiyoshi Kamimura, [[Warren Siegel]], \emph{Superspace with manifest T-duality from type II superstring}, J. High Energ. Phys. (2014) 2014: 39 (\href{https://arxiv.org/abs/1403.3887}{arXiv:1403.3887})


\item [[Igor Bandos]], \emph{Superstring in doubled superspace}, Physics Letters B Volume 751, 17 December 2015, Pages 408-412 (\href{https://arxiv.org/abs/1507.07779}{arXiv:1507.07779})


\item [[Martin Cederwall]], \emph{Double supergeometry}, J. High Energ. Phys. (2016) 155 (\href{https://arxiv.org/abs/1603.04684}{arXiv:1603.04684})


\item Bojan Nikolić, Branislav Sazdović, \emph{T-dualization of type II superstring theory in double space}, Eur. Phys. J. C77 (2017) no.3, 197 (\href{https://arxiv.org/abs/1505.06044}{arXiv:1505.06044})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 6 of: \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})


\item Bojan Nikolić, Branislav Sazdović, \emph{Advantage of the second-order formalism in double space T-dualization of type II superstring} (\href{https://arxiv.org/abs/1907.03571}{arXiv:1907.03571})



\end{itemize}
See also the references on the corresponding [[super-geometry]]-enhancement of [[exceptional generalized geometry]]: \emph{\href{exceptional+generalized+geometry#SuperExceptionalGeometryReferences}{Super-exceptional geometry -- References}}

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