Courant algebroid

Definition

A Courant algebroid consists of the data a vector bundle E → M with a bracket [ . , . ] : Γ E × Γ E → Γ E , a non degenerate fiber-wise inner product ⟨ . , . ⟩ : E × E → M × R , and a bundle map ρ : E → T M subject to the following axioms,

where φ,ψ,χ are sections of E and f is a smooth function on the base manifold M. D is the combination κ − 1 ρ T d with d the de Rham differential, ρ T the dual map of ρ , 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

This no longer satisfies the Jacobi-identity axiom above. It instead fulfills a homotopic Jacobi-identity.

where T is

The Leibniz rule and the invariance of the scalar product become modified by the relation [ [ ϕ , ψ ] ] = [ ϕ , ψ ] − 1 2 D ⟨ ϕ , ψ ⟩ and the violation of skew-symmetry gets replaced by the axiom

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:

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

Examples

An example of the Courant algebroid is the Dorfman bracket on the direct sum T M ⊕ T ∗ M with a twist introduced by Ševera, (1998) defined as:

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 ∧ ∗ 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 ∧ ∗ A using graded Leibniz rule). This notion is symmetric in A and A ∗ (see Roytenberg). Here E = A ⊕ A ∗ with anchor ρ ( X + α ) = ρ A ( X ) + ρ A ∗ ( α ) and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

Dirac structures

Given a Courant algebroid with the inner product ⟨ . , . ⟩ of split signature (e.g. the standard one T M ⊕ T ∗ M ), then a Dirac structure is a maximally isotropic integrable vector subbundle L → M, i.e.

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 Π ∈ Γ ( ∧ 2 T M ) 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

where   ¯ means complex conjugation with respect to the standard complex structure on the complexification.

As studied in detail by Gualtieri 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.