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\section*{Courant algebroid}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{lie_algebras_of_compact_type}{Lie algebras of compact type}\dotfill \pageref*{lie_algebras_of_compact_type} \linebreak
\noindent\hyperlink{standard_courant_algebroid_and_gerbes}{Standard Courant algebroid and $U(1)$-gerbes}\dotfill \pageref*{standard_courant_algebroid_and_gerbes} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generalized_complex_geometry}{Generalized complex geometry}\dotfill \pageref*{generalized_complex_geometry} \linebreak
\noindent\hyperlink{chernsimons_element_and_courant_model}{Chern-Simons element and Courant $\sigma$-model}\dotfill \pageref*{chernsimons_element_and_courant_model} \linebreak
\noindent\hyperlink{lagrangian_submanifolds_and_dirac_structures}{Lagrangian submanifolds and Dirac structures}\dotfill \pageref*{lagrangian_submanifolds_and_dirac_structures} \linebreak
\noindent\hyperlink{RelationToAtiyahGroupoids}{Relation to Atiyah Lie 2-algebroid and quantomorphism 2-group}\dotfill \pageref*{RelationToAtiyahGroupoids} \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 \textbf{Courant algebroid} -- or better: \textbf{Courant Lie 2-algebroid} -- (named after [[Theodore Courant]]) is precisely a [[symplectic Lie n-algebroid|symplectic Lie 2-algebroid]] (\hyperlink{RoytenbergStructure}{Roytenberg}):

it is a [[L-infinity algebroid|Lie 2-algebroid]] $\mathfrak{P}$ whose [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P})$ is equipped with the structure of a [[Poisson n-algebra|Poisson 3-algebra]] whose Poisson bracket

\begin{displaymath}
\{-,-\} : CE(\mathfrak{P})\otimes CE(\mathfrak{P})
  \to 
  CE(\mathfrak{P})
\end{displaymath}
of degree -2 is non-degenerate.

Therefore the [[differential]] $d_{CE(\mathfrak{P})}$ on $CE(\mathfrak{P})$ has a [[Hamiltonian]] with respect to this bracket in that there is an element $\Theta \in CE(\mathfrak{P})$ such that

\begin{displaymath}
d_{CE(\mathfrak{P})} = \{\Theta, -\}
  \,.
\end{displaymath}
\hypertarget{history}{}\subsection*{{History}}\label{history}

The concept of Courant algebroids was originally introduced by [[Irene Dorfman]] and [[Ted Courant]] to study [[geometric quantization]] in the presence of constraints. Later it was considered by Liu, [[Alan Weinstein]] and [[Ping Xu]] in the study of [[double Lie algebroids]].

In these parts of the literature Courant algebroids are considered in the form of [[Lie algebroid]]s with relaxed axioms on the bracket. Even of this type there are two different definitions:

\begin{itemize}%
\item in one there is a skew-symmetric bracket which fails to satisfy a Jacobi identity by a coherent term -- this is the Courant bracket definition proper;


\item in the other there is a bracket which satisfies a Jacobi identity but is skew-symmetric only up to a correction term -- this is the Dorfman version.



\end{itemize}
So there are several different ways to present the structure encoded in a Courant algebroid. The picture that seems to be emerging is that the \emph{true} meaning of the notoin of Courant algebroids is given by the notion of [[n-symplectic manifold|2-symplectic manifold]]s.

Moreover, the way [[Lie algebroid]]s may be expressed in terms of [[Lie-Rinehart algebra]]s, Courant algebroids yield [[Courant-Dorfman algebra]]s.


\vspace{.5em} \hrule \vspace{.5em}
(\ldots{} need to say more about the way the Courant Lie algebroid is obtained from a Lie bialgebroid by derived brackets \ldots{})


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{lie_algebras_of_compact_type}{}\subsubsection*{{Lie algebras of compact type}}\label{lie_algebras_of_compact_type}

A Courant Lie 2-algebroid over the point is precisely an ordinary [[Lie algebra]] $\mathfrak{g}$ that is equipped with a quadratic and non-degenerate [[invariant polynomial]].

\hypertarget{standard_courant_algebroid_and_gerbes}{}\subsubsection*{{Standard Courant algebroid and $U(1)$-gerbes}}\label{standard_courant_algebroid_and_gerbes}

The [[standard Courant algebroid]] of a [[manifold]] $X$ is the one which

\begin{itemize}%
\item as a [[vector bundle]] with extra structure is $E = T X\oplus T^* X$, the fiberwise direct sum of the [[tangent bundle]] and the cotangent bundle; with

\begin{itemize}%
\item bilinear form

\begin{displaymath}
\langle X + \xi , Y +\eta \rangle = \eta(X) + \xi(Y)
\end{displaymath}


\end{itemize}
for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$

\begin{itemize}%
\item brackets

\begin{displaymath}
[X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta 
  - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))
\end{displaymath}
where $\mathcal{L}_X \eta = \{d,\iota_X\} \eta$ denotes the [[Lie derivative]] of the 1-form $\eta$ by the vector field $X$.



\end{itemize}

\item as a [[dg-manifold]] is $\Pi T^* \Pi T X$, the shifted cotangent bundle of the [[shifted tangent bundle]],

where the [[differential]] is on each local [[coordinate patch]] $\mathbb{R}^n \simeq U \subset X$ with coordinates $\{x^i\}$ in degree 0, $\{d x^i\}$ and $\{\theta_i\}$ in degree 1 and $\{p_i\}$ in degree 2 given by

\begin{displaymath}
\begin{aligned}
    d_C &= d_{dR} + p_i \frac{\partial}{\partial \theta_i}
    \\
    &= dx^i \frac{\partial}{\partial x^i }
     + p_i \frac{\partial}{\partial \theta_i}
  \end{aligned}
  \,.
\end{displaymath}


\end{itemize}
Such a standard Courant algebroid may be understood as the higher analog of the [[Atiyah Lie algebroid]] of a line bundle. See below in \emph{\hyperlink{RelationToAtiyahGroupoids}{Relation to Atiyah groupoids}}.

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

\hypertarget{generalized_complex_geometry}{}\subsubsection*{{Generalized complex geometry}}\label{generalized_complex_geometry}

The study of Courant algebroids is to a large extent known as [[generalized complex geometry]], where the Courant algebroid appears as the [[generalized tangent bundle]].

\hypertarget{chernsimons_element_and_courant_model}{}\subsubsection*{{Chern-Simons element and Courant $\sigma$-model}}\label{chernsimons_element_and_courant_model}

As every [[symplectic Lie n-algebroid]] the defining [[invariant polynomial]] on a Courant Lie 2-algebroid transgresses to a cocycle in [[∞-Lie algebroid cohomology]] and this transgression is witnessed by a [[Chern-Simons element]]. The [[schreiber:∞-Chern-Simons theory]] induced by this element is the [[Courant sigma-model]].

\hypertarget{lagrangian_submanifolds_and_dirac_structures}{}\subsubsection*{{Lagrangian submanifolds and Dirac structures}}\label{lagrangian_submanifolds_and_dirac_structures}

The [[Lagrangian dg-submanifolds]] of a Courant Lie 2-algebroid corespond to its [[Dirac structures]].

\hypertarget{RelationToAtiyahGroupoids}{}\subsubsection*{{Relation to Atiyah Lie 2-algebroid and quantomorphism 2-group}}\label{RelationToAtiyahGroupoids}

We discuss how the following tower of notions works

\begin{tabular}{l|l}
[[circle n-bundle]] with $(n-1)$-form connection&[[Lie n-algebra]] of [[group of bisections\\
\hline 
[[circle bundle]]&[[Lie algebra]] of [[sections]] of [[Atiyah Lie algebroid]]\\
[[circle 2-bundle]] with 1-form connection&[[Lie 2-algebra]] of [[sections]] of [[Courant Lie 2-algebroid]]\\
\end{tabular}

For $n,k \in \mathbb{N}$ and $k \leq n$ write

\begin{displaymath}
\mathbf{B}^n U(1)_{conn^k}
  \coloneqq
  DK\left[
    U(1) \stackrel{d log}{\to}
    \Omega^1
    \stackrel{d}{\to}
    \cdots
    \stackrel{d}{\to} 
     \Omega^{k-1}
    \stackrel{d}{\to} \Omega^k
     \to 
     \underbrace{
     0
     \to 0
     \to \cdots
     \to 0
     }_{n-k}
  \right]
\end{displaymath}
for the [[smooth ∞-groupoid]] which is presented under the [[Dold-Kan correspondence]] by the [[sheaf]] of [[chain complexes]], as indicated (see also at \emph{\href{differential+cohomology+diagram#DeligneCoefficients}{differential cohomology diagram -- Examples -- Deligne coefficients}}). This is such that for $k = n$ we have the [[Deligne complex]], representing the [[moduli ∞-stack]] of [[circle n-bundles with connection]]

\begin{displaymath}
\mathbf{B}^n U(1)_{conn^n} \simeq \mathbf{B}^n U(1)_{conn}
\end{displaymath}
and for $k = 0$ we have the moduli $\infty$-stack for plain [[circle n-group]] [[principal ∞-bundles]]

\begin{displaymath}
\mathbf{B}^n U(1)_{conn^0} \simeq \mathbf{B}^n U(1)
  \,.
\end{displaymath}
For $k_2 \lt k_1$ there are evident truncation maps

\begin{displaymath}
\mathbf{B}^n U(1)_{conn^{k_1}}
  \to 
  \mathbf{B}^n U(1)_{conn^{k_2}}
  \,.
\end{displaymath}
Now for $X \in$ [[SmthMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] a [[smooth manifold]], a map

\begin{displaymath}
\nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}
\end{displaymath}
modulates a [[circle n-bundle with connection]] ([[bundle gerbe|bundle (n-1)-gerbe]]), which we may think of as a \emph{[[prequantum circle n-bundle]]}. Regarding this as an [[object]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^n U(1)}$ this has an [[automorphism ∞-group]]. The [[concretification]] of this (\ldots{}) is the [[quantomorphism n-group]] $QuantMorph(\nabla)$.

\begin{displaymath}
\mathbf{QuantMorph}(\nabla)
  \coloneqq
  conc\mathbf{Aut}(\nabla)
  =
  \left\{
    \itexarray{
      X && \underoverset{\simeq}{\phi}{\to} && X
      \\
      & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\nabla}}
      \\
      && \mathbf{B}^n U(1)_{conn}
    }
  \right\}
  \,.
\end{displaymath}
But we can also first forget the $n$-form pieces of the prequantum $n$-bundle away and consider

\begin{displaymath}
\nabla_{n-1} 
    \;\colon\;
   X 
    \stackrel{\nabla}{\to}
   \mathbf{B}^n U(1)_{conn}
   \simeq 
   \mathbf{B}^n U(1)_{conn^n}
   \to
   \mathbf{B}^n U(1)_{conn^{n-1}} 
  \,.
\end{displaymath}
For $n = 2$ this is sometimes known in the literature as a ``bundle gerbe with connective structure but without curving''.

The concretified [[automorphism ∞-group]] of that truncated connection is the [[n-group]] of [[group of bisections|of bisection]] of the [[Atiyah n-groupoid]]

\begin{displaymath}
conc \mathbf{Aut}(\nabla_{n-1}) \in Grp(Smooth \infty Grpd)
  \,.
\end{displaymath}
For $n = 1$ this is the [[group of bisections]] of the [[Atiyah Lie groupoid]] of the underlying [[circle principal bundle]] $\nabla_0 \colon X \to \mathbf{B} U(1)$. Hence its [[Lie differentiation]] is the [[Lie algebra]] of sections of the corresponding [[Atiyah Lie algebroid]].

For $n = 2$ the [[Lie differentiation]] of this [[Lie 2-group]] is the [[Lie 2-algebra]] of sections of the corresponding Courant Lie 2-algebroid. With a little bit of translation, this is what is shown in (\hyperlink{Collier}{Collier}).

Finally notice that the forgetful map $\mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)_{conn^{n-1}}$ induces an [[homomorphism]] of [[∞-groups]]

\begin{displaymath}
conc \mathbf{Aut}(\nabla) \to conc \mathbf{Aut}(\nabla_{n-1})
\end{displaymath}
hence an embedding of the [[quantomorphism n-group]] into the $n$-[[group of bisections]] of the Atiyah n-groupoid. For $n = 2$ and after [[Lie differentiation]], this is an embedding of the [[Poisson Lie 2-algebra]] into the sections of the Courant Lie 2-algebroid. This embedding had been observed in (\hyperlink{Rogers}{Rogers}).

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

[[!include higher Atiyah groupoid - table]]

\begin{itemize}%
\item [[symplectic Lie n-algebroid]]

\begin{itemize}%
\item [[symplectic manifold]]


\item [[Poisson Lie algebroid]]


\item \textbf{Courant algebroid}



\end{itemize}

\item [[Dirac structure]]


\item [[Courant sigma-model]]



\end{itemize}
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

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

The original references in order of appearance are

\begin{itemize}%
\item [[Pavol Ševera]], \emph{Letters to A. Weinstein} (\href{http://sophia.dtp.fmph.uniba.sk/~severa/letters/}{web}, \href{https://arxiv.org/abs/1707.00265}{arXiv:1707.00265})


\item [[Dmitry Roytenberg]], [[Alan Weinstein]], \emph{Courant algebroids and strongly homotopy Lie algebras}, Lett. Math. Physics \textbf{46}(1):81-93, 1998.


\item [[Dmitry Roytenberg]], \emph{Courant algebroids, derived brackets and even symplectic supermanifolds} PhD thesis, University of California, Berkeley, 1999. (\href{http://arxiv.org/abs/math.DG/9910078}{math.DG/9910078})


\item [[Pavol Ševera]], \emph{Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one}, In Travaux math\'e{}matiques. Fasc. XVI, chapter Trav. Math., XVI, pp. 121-137. Univ. Luxembourg, 2005.


\item [[Dmitry Roytenberg]], \emph{On the structure of graded symplectic supermanifolds and Courant algebroids}, pp. 169--185. in Contemporary Mathematics \textbf{315}, \emph{Quantization, Poisson brackets and beyond} (Manchester, 2001), Theodore Voronov, editor, Amer. Math. Soc. 2002. (\href{http://arxiv.org/abs/math/0203110}{arXiv:math/0203110})



\end{itemize}
\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{Quasi-Lie bialgebroids and twisted Poisson manifolds} , Letters in Mathematical Physics, 61(2):123-137 (2002) (\href{http://arxiv.org/abs/math/0112152}{arXiv:math/0112152})

\end{itemize}
Another useful summary of the theory of Courant algebroids is in \href{http://arxiv.org/PS_cache/math/pdf/0401/0401221v1.pdf#page=19}{section 3} of

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

\end{itemize}
A discussion of Courant algebroids with an eye towards the relation of the [[standard Courant algebroid]] to [[bundle gerbes]] is

\begin{itemize}%
\item [[Paul Bressler]], Alexander Chervov, \emph{Courant algebroids} (\href{http://arxiv.org/abs/hep-th/0212195}{hep-th/0212195})

\end{itemize}
The identification of the Lie 2-algebra of sctions of a Courant Lie 2-algebroid associated with a [[circle 2-bundle with connection]] as its Lie algebra of automorphisms after forgetting the ``curving'' is in

\begin{itemize}%
\item [[Braxton Collier]], \emph{Infinitesimal Symmetries of Dixmier-Douady Gerbes} (\href{http://arxiv.org/abs/1108.1525}{arXiv:1108.1525})

\end{itemize}
The embedding of the [[Poisson Lie 2-algebra]] of a given [[2-plectic geometry]] into the [[Lie 2-algebra]] of sections of the Courant Lie 2-algebroid of the corresponding [[prequantum 2-bundle]] is observed in

\begin{itemize}%
\item [[Chris Rogers]], \emph{Courant algebroids from categorified symplectic geometry}, (\href{http://arxiv.org/abs/1001.0040}{arXiv:1001.0040})

\end{itemize}
This is developed further in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]} (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Courant_algebroid}{Courant algebroid}}

\end{itemize}
The relation between the two different Lie-alebroid-like definition of Courant algebroids, one with skew, the other with non-skew brackets inspired on the level of [[L-infinity algebra|Lie 2-algebras]] the treatment

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{On weak Lie 2-algebras} (\href{http://arxiv.org/abs/0712.3461}{arXiv/0712.3461})


\item [[Chris Rogers]], \emph{2-plectic geometry, Courant algebroids, and categorified prequantization}, (\href{http://arxiv.org/abs/1009.2975}{arxiv:1009.2975})



\end{itemize}
A proposal for a higher analog of the [[standard Courant algebroid]] with the [[generalized tangent bundle]] $T X \oplus T^* X$ replaced by $T X \oplus \wedge^n T^* X$ -- for a notion of standard [[higher Courant Lie algebroid]] -- is discussed in

\begin{itemize}%
\item [[Marco Zambon]], \emph{$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids}, J. Symplectic Geometry, (\href{http://arxiv.org/abs/1003.1004}{arXiv:1003.1004})

\end{itemize}
The relation to [[schreiber:∞-Chern-Simons theory]] is discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]}

\end{itemize}
Discussion of [[Riemannian geometry]] on Courant algebroids and relation to [[supergravity]] [[equations of motion]] is in

\begin{itemize}%
\item [[Branislav Jurco]], [[Jan Vysoky]], \emph{Courant Algebroid Connections and String Effective Actions}, Proceedings of Tohoku Forum for Creativity, Special volume: Noncommutative Geometry and Physics IV (\href{https://arxiv.org/abs/1612.01540}{arXiv:1612.01540})

\end{itemize}
See also

\begin{itemize}%
\item Xu Xiaomeng, \emph{Twisted Courant algebroids and coisotropic Cartan geometries} (\href{http://arxiv.org/abs/1206.2282}{arXiv:1206.2282})


\item [[Jan Vysoky]], \emph{Hitchhiker's Guide to Courant Algebroid Relations} (\href{https://arxiv.org/abs/1910.05347}{arXiv:1910.05347})



\end{itemize}
[[!redirects Courant Lie algebroid]] [[!redirects Courant Lie algebroids]]

[[!redirects Courant algebroids]]

[[!redirects Courant Lie 2-algebroid]] [[!redirects Courant Lie 2-algebroids]]

[[!redirects Courant bracket]] [[!redirects Courant brackets]]



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