Courant algebroid

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 ${\displaystyle TM\oplus T^{*}M}$, 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 ${\displaystyle E\to M}$ with a bracket ${\displaystyle [.,.]:\Gamma E\times \Gamma E\to \Gamma E}$, a non degenerate fiber-wise inner product ${\displaystyle \langle .,.\rangle :E\times E\to M\times \mathbb {R} }$, and a bundle map ${\displaystyle \rho :E\to TM}$ subject to the following axioms,

${\displaystyle [\phi ,[\chi ,\psi ]]=[[\phi ,\chi ],\psi ]+[\chi ,[\phi ,\psi ]]}$

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

${\displaystyle [\phi ,\phi ]={\tfrac {1}{2}}D\langle \phi ,\phi \rangle }$

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

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

Skew-Symmetric Definition

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

${\displaystyle [[\phi ,\psi ]]={\tfrac {1}{2}}{\big (}[\phi ,\psi ]-[\psi ,\phi ]{\big .})}$

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

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

where T is

${\displaystyle T(\phi ,\psi ,\chi )={\frac {1}{3}}\langle [\phi ,\psi ],\chi \rangle +{\text{cycl.}}}$

The Leibniz rule and the invariance of the scalar product become modified by the relation ${\displaystyle [[\phi ,\psi ]]=[\phi ,\psi ]-{\tfrac {1}{2}}D\langle \phi ,\psi \rangle }$ and the violation of skew-symmetry gets replaced by the axiom

${\displaystyle \rho \circ D=0}$

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:

${\displaystyle \rho [\phi ,\psi ]=[\rho (\phi ),\rho (\psi )].}$

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

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

Examples

An example of the Courant algebroid is the Dorfman bracket[3] on the direct sum ${\displaystyle TM\oplus T^{*}M}$ with a twist introduced by Ševera,[4] (1998) defined as:

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

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 ${\displaystyle A^{*}}$ 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 ${\displaystyle \rho _{A}}$ and bracket ${\displaystyle [.,.]_{A}}$), also its dual ${\displaystyle A^{*}}$ a Lie algebroid (inducing the differential ${\displaystyle d_{A^{*}}}$ on ${\displaystyle \wedge ^{*}A}$) and ${\displaystyle d_{A^{*}}[X,Y]_{A}=[d_{A^{*}}X,Y]_{A}+[X,d_{A^{*}}Y]_{A}}$ (where on the RHS you extend the A-bracket to ${\displaystyle \wedge ^{*}A}$ using graded Leibniz rule). This notion is symmetric in A and ${\displaystyle A^{*}}$ (see Roytenberg). Here ${\displaystyle E=A\oplus A^{*}}$ with anchor ${\displaystyle \rho (X+\alpha )=\rho _{A}(X)+\rho _{A^{*}}(\alpha )}$ and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

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

Dirac structures

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

${\displaystyle \langle L,L\rangle \equiv 0}$,

${\displaystyle \mathrm {rk} \,L={\tfrac {1}{2}}\mathrm {rk} \,E}$,

${\displaystyle [\Gamma L,\Gamma L]\subset \Gamma L}$.

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 ${\displaystyle \Pi \in \Gamma (\wedge ^{2}TM)}$ 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

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

where ${\displaystyle {\bar {\ }}}$ 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 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)