Courant algebroid

In a field of mathematics known as differential geometry, a Courant algebroid is a structure which, in a certain sense, blends the concepts of Lie algebroid and of quadratic Lie algebra. This notion, which plays a fundamental role in the study of Hitchin's generalized complex structures, was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997.^[1] Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990^[2] the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on $TM\oplus T^*M$, called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.

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 fiber-wise inner product $\langle.,.\rangle: E\times E\to M\times\R$, and a bundle map $\rho:E\to TM$ subject to the following axioms,

$[\phi, [\chi, \psi]] = [[\phi, \chi], \psi] + [\chi, [\phi, \psi]]$

$[\phi, f\psi] = \rho(\phi)f\psi +f[\phi, \psi]$

$[\phi,\phi]= \tfrac12 D\langle \phi,\phi\rangle$

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

where φ,ψ,χ are sections of E and f is a smooth function on the base manifold M. D is the combination $\kappa^{-1}\rho^T d$ with d the de Rham differential, $\rho^T$ the dual map of $\rho$, and κ the map from E to $E^{*}$ induced by the inner product.

Skew-Symmetric Definition

An alternative definition can be given to make the bracket skew-symmetric as

$[[\phi,\psi]]= \tfrac12\big([\phi,\psi]-[\psi,\phi]\big.)$

This no longer satisfies the Jacobi-identity axiom above. It instead fulfills 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 $[[\phi,\psi]] = [\phi,\psi] -\tfrac12 D\langle \phi,\psi\rangle$ 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.

Properties

The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfills 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

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

Examples

An example of the Courant algebroid is the Dorfman bracket^[3] on the direct sum $TM\oplus T^*M$ with a twist introduced by Ševera,^[4] (1998) 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 $\rho _{A}$ and bracket $[.,.]_A$), also its dual $A^{*}$ a Lie algebroid (inducing the differential $d_{A^*}$ on $\wedge^* A$) 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 $\wedge^*A$ 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):

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

Dirac structures

Given a Courant algebroid with the inner product $\langle.,.\rangle$ of split signature (e.g. the standard one $TM\oplus T^*M$), then a Dirac structure is a maximally isotropic integrable vector subbundle L → M, i.e.

$\langle L,L\rangle \equiv 0$,

$\mathrm{rk}\,L=\tfrac12\mathrm{rk}\,E$,

$[\Gamma L,\Gamma L]\subset \Gamma L$.

Examples

As discovered by Courant and parallel by Dorfman, the graph of a 2-form ω ∈ Ω²(M) is maximally isotropic and moreover integrable iff dω = 0, i.e. the 2-form is closed under the de Rham differential, i.e. a presymplectic structure.

A second class of examples arises from bivectors $\Pi\in\Gamma(\wedge^2 TM)$ whose graph is maximally isotropic and integrable iff [Π,Π] = 0, i.e. Π is a Poisson bivector on M.

Generalized complex structures

(see also the main article generalized complex geometry)

Given a Courant algebroid with inner product of split signature. A generalized complex structure L → M is a Dirac structure in the complexified Courant algebroid with the additional property

$L \cap \bar{L} = 0$

where $\bar{\ }$ means complex conjugation with respect to the standard complex structure on the complexification.

As studied in detail by Gualtieri^[5] the generalized complex structures permit the study of geometry analogous to complex geometry.

Examples

Examples are beside presymplectic and Poisson structures also the graph of a complex structure J: TM → TM.

References

1. Z-J. Liu, A. Weinstein, and P. Xu: Manin triples for Lie Bialgebroids, Journ. of Diff.geom. 45 pp.647–574 (1997).
2. T.J. Courant: Dirac Manifolds, Transactions of the AMS, vol. 319, pp.631–661 (1990).
3. I.Y. Dorfman: Dirac structures of integrable evolution equations, Physics Letters A, vol.125, pp.240–246 (1987).
4. P. Ševera: Letters to A. Weinstein, unpublished.
5. M. Gualtieri: Generalized complex geometry, Ph.D. thesis, Oxford university, (2004)

Further reading

• Dmitry Roytenberg: Courant algebroids, derived brackets, and even symplectic supermanifolds, PhD thesis Univ. of California Berkeley (1999)