Courant algebroid explained

In a field of mathematics known as differential geometry, a Courant geometry 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 ⊕ 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

E\to M with a bracket [ ⋅ , ⋅ ]:\Gamma E x \Gamma E \to \Gamma E , a non degenerate fiber-wise inner product \langle ⋅ , ⋅ \rangle: E x E\to M x \R , and a bundle map \rho:E\to TM (called anchor) subject to the following axioms:

1. Jacobi identity:

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

1. Leibniz rule:

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

1. Obstruction to skew-symmetry:

[\phi,\psi] + [\psi,\phi] = \tfrac 12 D\langle \phi,\psi \rangle

1. Invariance of the inner product under the bracket:

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

where

\phi, \chi, \psi are sections of E and f is a smooth function on the base manifold M . The map D: l{C} ^{infty(M)} \to \Gamma E is the composition \kappa ^{-1} \rho ^{T} d: l{C} ^{infty(M)} \to \Gamma E , with d: l{C} ^{infty(M)} \to \Omega ^{1} (M) the de Rham differential, \rho ^{T} the dual map of \rho , and \kappa the isomorphism 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]] = \tfrac 12 ( [\phi,\psi] - [\psi,\phi] . )

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

[[\phi,[[\psi,\chi]]]] + cycl. = DT(\phi,\psi,\chi)

where

T isT(\phi,\psi,\chi) = (13 \langle)/([\phi,\psi],\chi\rangle) + cycl.

The Leibniz rule and the invariance of the scalar product become modified by the relation

[[\phi,\psi]] = [\phi,\psi] - \tfrac 12 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 \rho 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 given by the Dorfman bracket^[3] on the direct sum

TM ⊕ T ^{*M} with a twist introduced by Ševera,^[4] (1998) defined as:[X + \xi, Y +η ] = [X,Y] + ( l{L} _{Xη} - \iota _{Y} d\xi + \iota _{X} \iota _{Y} H) where X,Y are vector fields, \xi, 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 right-hand side 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 ⊕ 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 \alpha (equivalently in Y and \beta ):[X + \alpha,Y + \beta] = ([X,Y] _{A} + l{L} ^{A ^{*}} _{\alpha} Y - \iota _{\beta} d _{A ^{*}} X) + ([\alpha,\beta] _{A ^{*}} + l{L} ^{A} _{X\beta - \iota} _{Yd} _{A} \alpha)

Dirac structures

See also: Dirac structure. Given a Courant algebroid with the inner product

\langle ⋅ , ⋅ \rangle of split signature (e.g. the standard one TM ⊕ T ^{*M} ), then a Dirac structure is a maximally isotropic integrable vector subbundle L \to M , i.e.\langle L,L\rangle \equiv 0 ,rk L = \tfrac 12 rk E ,[\Gamma L,\Gamma L]\subset \Gamma L .

Examples

As discovered by Courant and parallel by Dorfman, the graph of a 2-form

\omega \in \Omega ^{2(M)} is maximally isotropic and moreover integrable if and only if d \omega = 0 , i.e. the 2-form is closed under the de Rham differential, i.e. is a presymplectic structure.

A second class of examples arises from bivectors

\Pi\in\Gamma(\wedge ^{2} TM) whose graph is maximally isotropic and integrable if and only if [\Pi,\Pi] = 0 , i.e. \rho is a Poisson bivector on M .

Generalized complex structures

Given a Courant algebroid with inner product of split signature, a generalized complex structure

L \to 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

J: TM \to 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 American Mathematical Society, 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)