Courant algebroid

In a field of mathematics known as differential geometry, a Courant algebroid is a combination of a Lie algebroid and a quadratic Lie algebra. It was originally introduced in 1990 by Theodore James Courant in his dissertation at UC Berkeley where he first called them Dirac Manifolds, and then were re-named after him in 1997 by Zhang-Ju Liu, Alan Weinstein and Ping Xu to describe the double of a Lie bialgebroid.

Definition

A Courant algebroid consists of the data a vector bundle $E\to M$ with a bracket $[.,.]:\Gamma E \times \Gamma E \to \Gamma E$, a non degenerate inner product $\langle.,.\rangle: E\times E\to \R$, and a bundle map $\rho:E\to TM$ subject to the following axioms,

[ϕ,[χ,ψ]] = [[ϕ,χ],ψ] + [χ,[ϕ,ψ]]

[ϕ,fψ] = ρ(ϕ)fψ + f[ϕ,ψ]

$[\phi,\phi]= \frac12 D\langle \phi,\phi\rangle$

$\rho(\phi)\langle \psi,\psi\rangle= 2\langle [\phi,\psi],\psi\rangle$

where φ,ψ are sections into E and f is a smooth function on the base manifold M. D is the combination β ^{− 1}ρ^Td with d the de Rham differential, ρ^T the dual map of ρ, and β the map from E to E ^* induced by the metric.

Properties

The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfils a certain Jacobi-identity (first axiom) and a Leibniz rule (second axiom). From these two axioms one can derive that the anchor map ρ is a morphism of brackets:

ρ[ϕ,ψ] = [ρ(ϕ),ρ(ψ)].

The fourth rule is an invariance of the inner product under the bracket. Polarization leads to

$\rho(\phi)\langle \chi,\psi\rangle= \langle [\phi,\chi],\psi\rangle +\langle \chi,[\phi,\psi]\rangle .$

Examples

An example of the Courant algebroid is the Dorfman bracket on the direct sum $TM\oplus T^*M$ with a twist introduced by Ševera, defined as:

$[X+\xi, Y+\eta] = [X,Y]+(\mathcal{L}_X\eta -i(Y) d\xi +i(X)i(Y)H)$

where X,Y are vector fields, ξ,η are 1-forms and H is a closed 3-form twisting the bracket. This bracket is used to describe the integrability of generalized complex structures.

A more general example arises from a Lie algebroid A whose induced differential on A ^* will be written as d again. Then use the same formula as for the Dorfman bracket with H an A-3-form closed under d.

Another example of a Courant algebroid is a quadratic Lie algebra, i.e. a Lie algebra with an invariant scalar product. Here the base manifold is just a point and thus the anchor map (and D) are trivial.

The example described in the paper by Weinstein et al. comes from a Lie bialgebroid, i.e. A a Lie algebroid (with anchor ρ_A and bracket [.,.]_A), also its dual A ^* a Lie algebroid (inducing the differential $d_{A^*}$ on Λ ^* A) and $d_{A^*}[X,Y]_A=[d_{A^*}X,Y]_A-[X,d_{A^*}Y]_A$ (where on the RHS you extend the A-bracket to Λ ^* A using graded Leibniz rule). This notion is symmetric in A and A ^* (see Roytenberg). Here $E=A\oplus A^*$ with anchor $\rho(X+\alpha)=\rho_A(X)+\rho_{A^*}(\alpha)$ and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

$[X+\alpha,Y+\beta]= ([X,Y]_A +\mathcal{L}^{A^*}_{\alpha}Y-i_\beta d_{A^*}X) +([\alpha,\beta]_{A^*} +\mathcal{L}^A_X\beta-i_Yd_{A}\alpha)$

Skew-symmetric bracket

Instead of the definition above one can introduce a skew-symmetric bracket as

$[[\phi,\psi]]= \frac12\big([\phi,\psi]-[\psi,\phi]\big.)$

This fulfils a homotopic Jacobi-identity.

$[[\phi,[[\psi,\chi]]\,]] +\text{cycl.} = DT(\phi,\psi,\chi)$

where T is

$T(\phi,\psi,\chi)=\frac13\langle [\phi,\psi],\chi\rangle +\text{cycl.}$

The Leibniz rule and the invariance of the scalar product become modified by the relation $[[\phi,\psi]] = [\phi,\psi] -\frac12 D\langle \phi,\psi\rangle$ and the violation of skew-symmetry gets replaced by the axiom

$\rho\circ D = 0$

The skew-symmetric bracket together with the derivation D and the Jacobiator T form a strongly homotopic Lie algebra.

References

• Liu, Weinstein, and Xu: Manin triples for Lie Bialgebroids, Journ. of Diff.geom. 45 pp.647–574 (1997)
• Dmitry Roytenberg: Courant algebroids, derived brackets, and even symplectic supermanifolds, PhD thesis Univ. of California Berkeley (1999)
• Pavol Ševera: Letters to A. Weinstein, unpublished (1998)