nLab
Courant algebroid

Idea

A Courant algebroid – or better: Courant Lie 2-algebroid – (named after Theodore Courant) is precisely a symplectic Lie 2-algebroid (Roytenberg):

it is a Lie 2-algebroid $𝔓$ whose Chevalley-Eilenberg algebra $\mathrm{CE}(𝔓)$ is equipped with the structure of a Poisson 3-algebra whose Poisson bracket

$$\{-,-\}:\mathrm{CE}(𝔓)\otimes \mathrm{CE}(𝔓)\to \mathrm{CE}(𝔓)$$\{-,-\} : CE(\mathfrak{P})\otimes CE(\mathfrak{P}) \to CE(\mathfrak{P})

of degree -2 in non-degenerate.

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

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

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 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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Examples

Lie algebras of compact type

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

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…).

Properties

Generalized complex geometry

The study of Courant algebroids is to a large extent known as generalized complex geometry.

Chern-Simons element and Courant $\sigma$-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 ∞-Chern-Simons theory induced by this element is the Courant sigma-model.

Related concepts

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant algebroid

References

The original references in order of appearance are

• Pavol Severa, Letters to A. Weinstein (web)

• Dmitry Roytenberg, Alan Weinstein, Courant algebroids and strongly homotopy Lie algebras Letters in Mathematical Physics, 46(1):81-93, 1998.

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis, University of California, Berkeley, 1999. (math.DG/9910078)

• Pavol Severa, Some title containing the words “homotopy” and “symplectic”, e.g. this one, In Travaux mathématiques. Fasc. XVI, chapter Trav. Math., XVI, pages 121-137. Univ. Luxemb., Luxembourg, 2005.

• Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids In Theodore Voronov, editor, Quantization, Poisson brackets and beyond (Manchester, 2001), volume 315 of Contemporary Mathematics: Quantization, Poisson brackets, and beyond, pages 169{185. American Mathematical Society, Providence, RI, 2002. (arXiv:math/0203110)

• Dmitry Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds , Letters in Mathematical Physics, 61(2):123-137 (2002) (arXiv:math/0112152)

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

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

• Marco Gualtieri, Generalized complex geometry (arXiv:math/0401221)

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 (hep-th/0212195)

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/0712.3461)

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:1001.0040)

• Chris Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization, (arxiv:1009.2975)

The relation to ∞-Chern-Simons theory is discussed in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher Chern-Weil Derivation of AKSZ Sigma-Models