A standard Courant Lie algebroid of a manifold $X$ is a type of Courant algebroid constructed from the tangent bundle and cotangent bundle of $X$. This is the principal algebraic structure studied in generalized complex geometry.
Recall from the discussion at Courant algebroid that there are the following two equivalent definitions of Courant algebroids:
as a vector bundle equipped with a bracket and a bilinear form on its space of sections, satisfying various identities;
as a Lie 2-algebroid equivalently encoded in its Chevalley–Eilenberg algebra, equivalently the function algebra on a certain type of dg-manifolds.
In the first perspective a standard Courant algebroid of a manifold $X$ is the vector bundle $E = T X\oplus T^* X$ – the fiberwise direct sum of the tangent bundle and the cotangent bundle – with
bilinear form
for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$
brackets
where $\mathcal{L}_X \eta = \{d,\iota_X\} \eta$ denotes the Lie derivative of the 1-form $\eta$ by the vector field $X$.
As an dg-manifold a standard Courant algeebroid is is $\Pi T^* \Pi T X$, the shifted cotangent bundle of the shifted tangent bundle,
where the differential (homological vector field) 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
We may read the above dg-algebra as the Chevalley–Eilenberg algebra $CE(\mathfrak{c}(X))$ of the Lie 2-algebroid $\mathfrak{c}(X)$, the specification of which entirely specifies the Lie 2-algebroid itself.
More on this in the discussion below.
A standard Courant algebroid may be understood as being related to $\mathbf{B}U(1)$ principal 2-bundles ($U(1)$-gerbes) as an Atiyah Lie algebroid is related to a $U(1)$-principal bundle.
(…explain…)
Write $\mathfrak{c}(X)$ for the standard Courant algebroid of the manifold $X$. It comes canonically equipped with a projection down to the tangent Lie algebroid $T X$ of $X$:
A section
of this morphism of Lie ∞-algebroids is often called a connection on $\mathfrak{c}(X)$. One may regard it as being special flat ∞-Lie algebroid valued differential form data on $X$.
On base manifolds of the form $X = \mathbb{R}^n$ sections of $\mathfrak{c}(X) \to T X$ in the 1-category of Lie ∞-algebroids are in natural bijection with rank-2 tensor fields on $X$, i.e. with sections $q \in \Gamma(T X \oplus T X)$.
The proof is straightforward and easy, but spelling it out in detail also serves to establish concepts and notation for the treatment of the Courant algebroid in terms of its Chevalley–Eilenberg algebra.
The Chevalley–Eilenberg algebra of the Lie 2-algebroid $\mathfrak{c}(\mathbb{R}^n)$ is the semifree dga whose underlying algebra is the Grassmann algebra
where the generators $\xi_i$ and $\theta_i$ are in degree 1 and the $p_i$ in degree 2, equipped with the differential $d_{\mathfrak{c}(X)}$ that is defined on generators by
where $\{x^i\}_{i=1}^n$ are the canonical coordinate functions on $\mathbb{R}^n$.
The Chevalley–Eilenberg algebra of the tangent Lie algebroid $T X$ is the deRham complex
The morphism $\mathfrak{c}(X) \to T X$ is given by the dg-algebra morphism
that is the identity on $C^\infty(X)$ and identifies the $\xi^i$ with the deRham differentials of the standard coordinate functions
A section $\sigma : \mathfrak{c}(X) \to T X$ is accordingly a dg-algebra morphism
Being a section, it has to be the identity on $C^\infty(X)$ and send $\xi^i \mapsto d_{dR} x^i$.
The image of the generators $\theta_i$, being of degree 1, must be a linear combination over $C^\infty(X)$ of the degree-1 elements in $\Omega^\bullet(X)$, i.e. must be 1-forms on $X$. This defines the rank-2 tensor $q$ in question by
For this assignment to qualify as part of a morphism of dg-algebras, it has in addition to be compatible with the differential. The condition is that for all $i$ we have the equality in the bottom right corner of
This uniquely fixes the image under $\sigma^*$ of the generators $p_i$ and the differential is respected. So, indeed, the section $\sigma^*$ is specified by the tensor $q \in \Gamma(T X \otimes T X)$ and every such tensor gives rise to a section.
The rank-2 tensor $q$ appearing in the above may be uniquely writtes as sum of a symmetric and a skew-symmetric rank-2 tensor $g \Gamma(Sym^2(T X))$ and $b \in \Gamma(\wedge^2 T X)$
If the symmetric part happens to be non-degenerate, it may be regarded as a (possibly pseudo-)Riemannian metric. In this case the combination $q = g + b$ is called a generalized Riemannian metric in generalized complex geometry.
The standard Courant albebroid $\mathfrak{c}(X)$ is canonically equipped with the Lie ∞-algebroid 3-cocycle $\mu \in CE(\mathfrak{c}(X))$ that is on a local patch $\mathbb{R}^n \simeq U \to X$ given by
In the 1-category of Lie ∞-algebroids, automorphisms of the standard Courant algebroid of a cartesian space, $\mathfrak{c}(\mathbb{R}^n)$, that
respect the projection $\mathfrak{c}(X) \to T X$
fix the canonical 3-cocycle $\mu = \xi^i p_i$
come from (…say this more precisely…) rank-2 tensors $q = g + b$ such that the skew symmetric part $b$ is a closed 2-form, $d_{dR} b = 0$.
With the same kind of reasoning as above, we find that the action on the generators $\theta_i$ and $p_i$ is of the form
For the 3-cocycle to be preserved, $f^*(\xi^i p_i) = \xi^i p_i$ we need that
The description of the standard Courant algebroid in its incarnation as an dg-manifold is given for instance in section 5 of