Courant algebroid

In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid.

It is named after Theodore Courant, who had implicitly devised in 1990 ^[1] 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. The general notion of Courant algebroid was introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. ^[2]

Definition

A Courant algebroid consists of the data a vector bundle ${\displaystyle E\to M}$ with a bracket ${\displaystyle [\cdot ,\cdot ]:\Gamma E\times \Gamma E\to \Gamma E}$, a non degenerate fiber-wise inner product ${\displaystyle \langle \cdot ,\cdot \rangle :E\times E\to M\times \mathbb {R} }$, and a bundle map ${\displaystyle \rho :E\to TM}$ (called anchor) subject to the following axioms:

1. Jacobi identity: ${\displaystyle [\phi ,[\chi ,\psi ]]=[[\phi ,\chi ],\psi ]+[\chi ,[\phi ,\psi ]]}$
2. Leibniz rule: ${\displaystyle [\phi ,f\psi ]=\rho (\phi )f\psi +f[\phi ,\psi ]}$
3. Obstruction to skew-symmetry: ${\displaystyle [\phi ,\psi ]+[\psi ,\phi ]={\tfrac {1}{2}}D\langle \phi ,\psi \rangle }$
4. Invariance of the inner product under the bracket: ${\displaystyle \rho (\phi )\langle \psi ,\chi \rangle =\langle [\phi ,\psi ],\chi \rangle +\langle \psi ,[\phi ,\chi ]\rangle }$

where ${\displaystyle \phi ,\chi ,\psi }$ are sections of ${\displaystyle E}$ and ${\displaystyle f}$ is a smooth function on the base manifold ${\displaystyle M}$. The map ${\displaystyle D:{\mathcal {C}}^{\infty }(M)\to \Gamma E}$ is the composition ${\displaystyle \kappa ^{-1}\rho ^{T}d:{\mathcal {C}}^{\infty }(M)\to \Gamma E}$, with ${\displaystyle d:{\mathcal {C}}^{\infty }(M)\to \Omega ^{1}(M)}$ the de Rham differential, ${\displaystyle \rho ^{T}}$ the dual map of ${\displaystyle \rho }$, and ${\displaystyle \kappa }$ the isomorphism ${\displaystyle E\to E^{*}}$ 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 ${\displaystyle T}$ 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 ${\displaystyle [[\cdot ,\cdot ]]}$ together with the derivation ${\displaystyle D}$ and the Jacobiator ${\displaystyle T}$ form a strongly homotopic Lie algebra.

Properties

The bracket ${\displaystyle [\cdot ,\cdot ]}$ 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 ${\displaystyle \rho }$ 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 given by the Dorfman bracket ^[3] on the direct sum ${\displaystyle TM\oplus T^{*}M}$ with a twist introduced by Ševera in 1988, ^[4] defined as:

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

where ${\displaystyle X,Y}$ are vector fields, ${\displaystyle \xi ,\eta }$ are 1-forms and ${\displaystyle H}$ 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 ${\displaystyle A}$ whose induced differential on ${\displaystyle A^{*}}$ will be written as ${\displaystyle d}$ again. Then use the same formula as for the Dorfman bracket with ${\displaystyle H}$ an A-3-form closed under ${\displaystyle d}$.

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 ${\displaystyle D}$) are trivial.

The example described in the paper by Weinstein et al. comes from a Lie bialgebroid: if ${\displaystyle A}$ is a Lie algebroid (with anchor ${\displaystyle \rho _{A}}$ and bracket ${\displaystyle [.,.]_{A}}$), also its dual ${\displaystyle A^{*}}$ is 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 right-hand side you extend the ${\displaystyle A}$-bracket to ${\displaystyle \wedge ^{*}A}$ using graded Leibniz rule). This notion is symmetric in ${\displaystyle A}$ 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 ${\displaystyle X}$ and ${\displaystyle \alpha }$ (equivalently in ${\displaystyle Y}$ and ${\displaystyle \beta }$):

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

Dirac structures

See also: Dirac structure

Given a Courant algebroid with the inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ of split signature (e.g. the standard one ${\displaystyle TM\oplus T^{*}M}$), a Dirac structure is a maximally isotropic integrable vector subbundle ${\displaystyle L\to M}$, 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 ${\displaystyle \omega \in \Omega ^{2}(M)}$ is maximally isotropic and moreover integrable if and only if ${\displaystyle d\omega =0}$, 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 ${\displaystyle \Pi \in \Gamma (\wedge ^{2}TM)}$ whose graph is maximally isotropic and integrable if and only if ${\displaystyle [\Pi ,\Pi ]=0}$, i.e. ${\displaystyle \rho }$ is a Poisson bivector on ${\displaystyle M}$.

Generalized complex structures

See also: Generalized complex geometry

Given a Courant algebroid with inner product of split signature, a generalized complex structure ${\displaystyle L\to M}$ 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, besides presymplectic and Poisson structures, also the graph of a complex structure ${\displaystyle J:TM\to TM}$.

References

1. Courant, Theodore James (1990). "Dirac manifolds". Transactions of the American Mathematical Society . 319 (2): 631–661. doi:10.1090/S0002-9947-1990-0998124-1. ISSN   0002-9947.
2. Liu, Zhang-Ju; Weinstein, Alan; Xu, Ping (1997-01-01). "Manin triples for Lie bialgebroids". Journal of Differential Geometry . 45 (3). arXiv: dg-ga/9508013 . doi:10.4310/jdg/1214459842. ISSN   0022-040X.
3. Dorfman, Irene Ya. (1987-11-16). "Dirac structures of integrable evolution equations". Physics Letters A . 125 (5): 240–246. Bibcode:1987PhLA..125..240D. doi:10.1016/0375-9601(87)90201-5. ISSN   0375-9601.
4. Ševera, Pavol (2017-07-05). "Letters to Alan Weinstein about Courant algebroids". arXiv: 1707.00265 [math.DG].
5. Gualtieri, Marco (2004-01-18). "Generalized complex geometry". arXiv: math/0401221 .

Further reading

• Roytenberg, Dmitry (1999). "Courant algebroids, derived brackets and even symplectic supermanifolds". arXiv: math.DG/9910078 .