nLab
Courant algebroid

Idea

A Courant algebroid is a Lie-algebraic structure encoding a 2-symplectic manifold.

Various different incarnations of this data are considered in the literature. Following Roytenberg’s analysis (see below) we may identify a Courant algebroid as a Lie 2-algebroid that generalizes the notion of Poisson Lie algebroid to one degree higher:

on its defining Chevalley-Eilenberg algebra $\mathrm{CE}(\Theta )$ exists a graded Poisson bracket $\{-,-\}$ of degree $-2$ and in it a degree 3 element $\Theta$ – the higher analog of the Poisson bivector $\pi$ – such that the differential on $\mathrm{CE}(\theta )$ is

$$d_{\mathrm{CE}(\theta )}=\{\Theta ,-\}.$$d_{CE(\theta)} = \{\Theta, - \} \,.

In the literature, this Lie 2-algebroid perspective on Courant algebroids is not usually considered. Instead, by the red herring principle what is usually called a Courant algebroid is neither an algebroid nor in fact in general a Lie algebroid, but is thought of as a generalization of a Lie bialgebroid: every Lie bialgebroid induces a Courant algebroid.

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 studied by Liu, Alan Weinstein and Xu in the study of double Lie algebroid?s.

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

• 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;

• 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.

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 true meaning of the notoin of Courant algebroids is given by the notion of 2-symplectic manifolds.

Moreover, the way Lie algebroids may be expressed in terms of Lie-Rinehart algebras, Courant algebroids yield Courant-Dorfman algebras.

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(… need to say more about the way the Courant Lie algebroid is obtained from a Lie bialgebroid by derived brackets …)

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Standard Courant algebroid and $U(1)$-gerbes

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

• as a vector bundle with extra structure is $E=TX\oplus T^{*}X$, the fiberwise direct sum of the tangent bundle and the cotangent bundle; with

  • bilinear form

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

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

  • brackets

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

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

• as an NQ-supermanifold is $\Pi T^{*}\Pi TX$, the shifted cotangent bundle of the shifted tangent bundle,

  where the differential (homological vector field) is on each local coordinate patch ${ℝ}^n\simeq U\subset X$ with coordinates $\{x^i\}$ in degree 0, $\{d{x}^i\}$ and $\{{\theta }_i\}$ in degree 1 and $\{p_i\}$ in degree 2 given by

  $$\begin{array}{rl}{d}_C& =d_{\mathrm{dR}}+p_i\frac{\partial }{\partial {\theta }_i}\\ & ={\mathrm{dx}}^i\frac{\partial }{\partial x^i}+p_i\frac{\partial }{\partial {\theta }_i}\end{array}.$$\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} \,.

Such a standard Courant algebroid may be understood as the higher analog of the Atiyah Lie algebroid of a line bundle (…explain…).

References

A useful place to start reading about Courant algebroids with an emphasis on its Lie-2-algebroid nature (in NQ-supermanifold-language) is

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

This proceeds from the perspective of n-symplectic manifolds and derives the fact that a 2-symplectic manifold encodes and is encoded by a Courant algebroid. The last section is about the standard Courant algebroid.

Another useful summary of the theory of Courant algebroids is in section 3 of

• Marco Gualtieri, Generalized complex geometry (arXiv)

The identification of the Lie 3-algebra incarnation of the same date was given by

• Dmitry Roytenberg and Alan Weinstein, Courant algebroids and strongly homotopy Lie algebras (arXiv)

Equation (9) and theorem 4.3 there gives the Lie 3-algebra corresponding to a Courant algebroid (the way the tangent Lie algebroid gives the Lie algebra of vector fields). In this perspective the Courant algebroid with base space a point is the string Lie 2-algebra.

This is reviewed and further developed in

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv)

A discussion of Courant algebroids with an eye towards the relation of the standard Courant algebroid to bundle gerbes is

• Paul Bressler, Alexander Chervov?, Courant algebroids (arXiv)

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 Lie 2-algebras the treatment

• Dmitry Roytenberg, On weak Lie 2-algebras (arXiv)

Chris Rogers paper discusses 2-plectic manifolds, manifolds with nondegenerate closed 3 forms and shows that they are related to a special class of Courant algebroids, those that are exact.

• Chris Rogers Courant algebroids from categorified symplectic geometry, arXiv:1001.0040v1 [math-ph]