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.

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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,

[\phi, [\chi, \psi]] = [[\phi, \chi], \psi] %2B [\chi, [\phi, \psi]]
[\phi, f\psi] = \rho(\phi)f\psi %2Bf[\phi, \psi]
[\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 \beta^{-1}\rho^T d with d the de Rham differential, \rho^T the dual map of \rho, 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:

\rho[\phi,\psi] = [\rho(\phi),\rho(\psi)] .

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 %2B\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%2B\xi, Y%2B\eta] = [X,Y]%2B(\mathcal{L}_X\eta -i(Y) d\xi %2Bi(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 \rho_A and bracket [.,.]_A), also its dual A^* a Lie algebroid (inducing the differential d_{A^*} on \Lambda^* 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 \Lambda^*A using graded Leibniz rule). This notion is symmetric in A and A^* (see Roytenberg). Here E=A\oplus A^* with anchor \rho(X%2B\alpha)=\rho_A(X)%2B\rho_{A^*}(\alpha) and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

[X%2B\alpha,Y%2B\beta]= ([X,Y]_A %2B\mathcal{L}^{A^*}_{\alpha}Y-i_\beta d_{A^*}X) %2B([\alpha,\beta]_{A^*} %2B\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]]\,]] %2B\text{cycl.} = DT(\phi,\psi,\chi)

where T is

T(\phi,\psi,\chi)=\frac13\langle [\phi,\psi],\chi\rangle %2B\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