nLab standard Courant algebroid

Idea

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.

Definition

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.

As a vector bundle with extra structure

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

  $$\langle X + \xi , Y +\eta \rangle = \eta(X) + \xi(Y)$$

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

• brackets

  $$[X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))$$

  where $\mathcal{L}_X \eta = \{d,\iota_X\} \eta$ denotes the Lie derivative of the 1-form $\eta$ by the vector field $X$.

As a dg-manifold

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

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

As a Lie 2-algebroid

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.

Properties

As the Atiyah Lie 2-algebroid of a $U(1)$-gerbe

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…)

Connections and generalized Riemannian metrics

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 $X$:

$$\pi : \mathfrak{c}(X) \to X \,.$$

A section

$$\sigma : T X \to \mathfrak{c}(X)$$

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 $X$.

Proposition

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?.

Proof

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

$$CE(\mathfrak{c}(X)) = \left( \wedge_{C^\infty(X)}^\bullet ( \langle \xi^i \rangle_{i=1}^n \oplus \langle \theta_i \rangle_{i=1}^n \oplus \langle p_i \rangle_{i=1}^n ) \,, d_{\mathfrak{c}(X)} \right)$$

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

$$d_{\mathfrak{c}(X)} : x^i = \xi^i$$

$$d_{\mathfrak{c}(X)} : \xi^i = 0$$

$$d_{\mathfrak{c}(X)} : \theta_i = p_i$$

$$d_{\mathfrak{c}(X)} : p_i = 0 \,$$

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

$$CE(T X ) = (\Omega^\bullet(X), d_{dR}) \,.$$

The morphism $\mathfrak{c}(X) \to T X$ is given by the dg-algebra morphism

$$(\Omega^\bullet(X),d_{dR}) \hookrightarrow CE(\mathfrak{c}(X))$$

that is the identity on $C^\infty(X)$ and identifies the $\xi^i$ with the deRham differentials of the standard coordinate functions

$$d x^i \mapsto \xi^i \,.$$

A section $\sigma : \mathfrak{c}(X) \to T X$ is accordingly a dg-algebra morphism

$$\sigma^* : CE(\mathfrak{c}(X)) \to (\Omega^\bullet(X), d_{dR}) \,.$$

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

$$\hat{t}_i \mapsto q_{i j} \d x^i \,.$$

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

$$\array{ \theta_i &\stackrel{d_{\mathfrak{c}(X)}}{\mapsto}&& p_i \\ \downarrow^{\mathrlap{\sigma^*}} &&& \downarrow^{\mathrlap{\sigma^*}} \\ q_{i j} d x^j &\stackrel{d_{dR}}{\mapsto}& (\partial_k q_{i j}) d x^k \wedge d x^j = & s^*(p_i) } \,.$$

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 $b \in \Gamma(\wedge^2 T X)$

$$q = g + b \,.$$

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.

Canonical $\infty$-Lie algebroid 3-cocycle

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

$$\mu|_U = \xi^i \wedge p_i \,.$$

Morphisms between standard Courant algebroids

Proposition

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 $\mathfrak{c}(X) \to T X$

  $$\array{ \mathfrak{c}(X) &&\stackrel{f}{\to}&& \mathfrak{c}(X) \\ & \searrow && \swarrow \\ && T X }$$

• 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, $d_{dR} b = 0$.

Proof

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

$$\array{ \theta_i &\stackrel{d_{\mathfrak{c}(X)}}{\mapsto}&& p_i \\ \downarrow^{\mathrlap{f^*}} &&& \downarrow^{\mathrlap{f^*}} \\ \theta_i + q_{i j} \theta^i & \stackrel{d_{\mathfrak{c}(X)}}{\mapsto}& p_i + \partial_k q_{i j} \theta^k \wedge \theta^j = & f^*(p_i) } \,.$$

For the 3-cocycle to be preserved, $f^*(\xi^i p_i) = \xi^i p_i$ we need that

$$0 = \partial_k q_{i j} \theta^i \wedge \theta^k \wedge \theta^j = \partial_k b_{i j} \theta^i \wedge \theta^k \wedge \theta^j = \pi^*(d_{dR} b) \,.$$

Related concepts

• higher Atiyah groupoid

  • Atiyah groupoid

  • Courant 2-groupoid

References

The description of the standard Courant algebroid in its incarnation as an dg-manifold is given for instance in section 5 of

• Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv:0203110)