Courant algebroid

In the area of differential geometry a topic of mathematics a Courant algebroid is the algebroid version of a quadratic Lie algebra. More precisely a Courant algebroid is a vector bundle E→M over a smooth manifold whose space of sections is endowed with a non-skew-symmetric R-bilinear bracket, a vector bundle map ρ: E → TM called the anchor, and a symmetric non-degenerate bilinear form $\langle .,.\rangle \colon E\otimes E\to M\times \mathbb {R}$ subject to the rules

$[\phi ,[\psi _{1},\psi _{2}]]=[[\phi ,\psi _{1}],\psi _{2}]+[\psi _{1},[\phi ,\psi _{2}]]$ (Jacobi identity)

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

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

$[\phi ,\phi ]=1/2\rho ^{*}\mathrm {d} \langle \phi ,\phi \rangle$ (violation of skew-symmetry)

for all φ, ψ_i ∈ Γ(E), and f ∈ C^∞(M) where d is the de Rham differential and ρ^*: T^*M → E^*=E the transpose of the anchor map.

Examples

0. Quadratic Lie algebras are Courant algebroids over a point, thus the anchor map vanishes. Remaining with vanishing anchor map also a bundle of quadratic Lie algebras is a Courant algebroid.

1. The Dorfman bracket with the Ševera twist is the following construction. Let E=TM⊕T^*M endowed with the projection ρ: E → TM and canonical symmetric pairing

$\langle X\oplus \alpha ,Y\oplus \beta \rangle =\alpha (Y)+\beta (X)$

where X, Y ∈ Γ(TM) and α, β ∈ Γ(T^*M). Let further H ∈ Ω³(M) closed under the de Rham differential, dH = 0. Then the Ševera bracket is

$[X\oplus \alpha ,Y\oplus \beta ]=[X,Y]\oplus {\mathcal {L}}_{X}\beta -i_{Y}\mathrm {d} \alpha +i_{X}i_{Y}H$

2. The Ševera bracket can be generalized to any Lie algebroid (A→M,ρ_A,[.,.]_A) straight-forwardly.

3. Given a Lie bialgebroid, i.e. a vector bundle A→M endowed with the structure of a Lie algebroid and also the dual bundle A^*→M endowed with a Lie algebroid structure that are compatible in the following sense

$\mathrm {d} _{*}[\phi ,\psi ]_{A}=[\mathrm {d} _{*}\phi ,\psi ]_{A}+[\phi ,\mathrm {d} _{*}\psi ]_{A}$

for all φ, ψ ∈ Γ(A) where d_* is the Lie algebroid differential induced by the structure on A^*. The double E = A⊕A^* is endowed with the usual

$\langle \phi \oplus \alpha ,\psi \oplus \beta \rangle =\alpha (\psi )+\beta (\phi )$,

$\rho (\phi \oplus \alpha )=\rho _{A}(\phi )+\rho _{*}(\alpha )$,

${\begin{aligned}{}[\phi \oplus \alpha ,\psi \oplus \beta ]=&[\phi ,\psi ]_{A}+{\mathcal {L}}_{\alpha }^{*}\psi -i_{\beta }\mathrm {d} _{*}\phi \\&\oplus [\alpha ,\beta ]_{*}+{\mathcal {L}}_{\phi }^{A}\beta -i_{\psi }\mathrm {d} _{A}\beta \end{aligned}}$

where in addition α, β ∈ Γ(A^*). Due to Xu et al this is a Courant algebroid.

Elementary properties

The reason to ask for a non-skew-symmetric bracket is that given a quadratic Lie algebroid, i.e. all the axioms but the bracket skew-symmetric, then the anchor map vanishes, i.e. we are left with a bundle of quadratic Lie algebras only. The original works of Courant describe an algebroid with a skew-symmetric bracket that fulfills a modified version of the Jacobi identity. Due to Roytenberg the two are equivalent, namely $[[\phi ,\psi ]]=1/2([\phi ,\psi ]-[\psi ,\phi ])$ and $[\phi ,\psi ]=[[\phi ,\psi ]]+1/2\rho ^{*}\mathrm {d} \langle \phi ,\psi \rangle$. It is also possible to replace the above axioms of a Courant algebroid with those for the skew-symmetric bracket, however these are more complicated so we omit them here.

It is worth noting that the Courant algebroid with the skew-symmetric bracket and the Jacobiator

$J(\phi _{1},\phi _{2},\phi _{3})=1/3(\langle [[\phi _{1},\phi _{2}]],\phi _{3}\rangle +\mathrm {cycl.} )$

form a two-term L_∞-algebra, i.e. V⁰ = Γ(E), V^-1 = C^∞(M), and

$l_{1}=\rho ^{*}\mathrm {d} \colon V^{-1}\to V^{0}:f\mapsto \rho ^{*}\mathrm {d} f$,

$l_{2}\colon V^{0}\times V^{\bullet }\to V^{\bullet }:(\phi ,\psi )\mapsto [[\phi ,\psi ]],(\phi ,f)\mapsto \rho (\phi )[f]$

$l_{3}=J\colon \wedge ^{3}V^{0}\to V^{-1}$

where φ, ψ ∈ Γ(E).

Another interesting observation (due to Uchino) is that the anchor map fulfills a morphism property, i.e. for φ, ψ ∈ Γ(E)

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

where the bracket on the right-hand side is the commutator bracket of vector fields.

Naive cohomology

Due to Stiénon and Xu the formula that gives rise to the Lie algebroid differential

${\begin{aligned}\langle \mathrm {d} \alpha ,\psi _{0}\wedge \dots \psi _{n}\rangle =&\sum _{i}(-1)^{i}\rho (\psi _{i})\langle \alpha ,\psi _{0}\wedge \dots {\hat {\psi }}_{i}\dots \psi _{n}\rangle \\&+\sum _{i<j}(-1)^{i+j}\langle \alpha ,[\psi _{i},\psi _{j}]\wedge \psi _{0}\dots {\hat {\psi }}_{i}\dots \psi _{j}\dots \psi _{n}\rangle \end{aligned}}$

with ψ ∈ Γ(E) gives rise to a differential if we restrict to the naive cochains α ∈ Γ(Λⁿker ρ). The naive cohomology is now the usual quotient of closed chains modulo exact chains.

N-graded realization and standard cohomology

Given a pseudo Euclidian vector bundle (E,g) we can consider this as a graded Poisson manifold E[1] where the Poisson bracket induced by g has degree -2. A symplectic realization is now a surjective map from a (graded) symplectic manifold (S,ω) to this Poisson manifold. The map is required to be a Poisson morphism. Minimality means that the fibers are of minimal dimension. Due to Roytenberg the minimal symplectic realization of this Poisson manifold can be constructed as the commutative square

${\begin{aligned}{\mathcal {E}}\,&\to T^{*}[2]E[1]\\p\downarrow &\quad \quad \downarrow \pi \\E[1]\,&{\overset {i}{\to }}(E\oplus E^{*})[1]\end{aligned}}$

where $i\colon E\to E\oplus E^{*}:\psi \mapsto \psi \oplus 1/2g(\psi ,.)$. Given now a super function Θ ∈ O(Ε) of degree 3, then the derived bracket construction

$\rho (\phi )[f]=\{\{\phi ,\Theta \},f\}$,

$[\phi ,\psi ]=\{\{\phi ,\Theta \},\psi \}$,

$\langle \phi ,\psi \rangle =\{\phi ,\psi \}$

for φ, ψ ∈ Γ(E) and f ∈ C^∞(M) gives a Courant algebroid iff $\{\Theta ,\Theta \}=0$. Due to Roytenberg's theorem the converse is also true, i.e. given a Courant algebroid there is a unique cubic function Θ such that the Courant structure arises from the derived bracket construction.

The standard cohomology is now the cohomology of the super functions O(Ε) under the differential Q = {Θ,.}.

The meaning of the lowest cohomology groups is the following: H⁰ are the smooth functions on M that are constant along the integral leaves of the image of the anchor map. H¹ are the sections of E whose bracket with every other section vanishes modulo the exact sections ρ^*df for f any smooth function. H² are the infinitesimal automorphisms of the Courant algebroid modulo the inner automorphisms [φ,.]. H³ are the inequivalent infinitesimal deformations of the Courant structure. H⁴ governs the obstructions of extending an infinitesimal deformation to a formal deformation.

As an example Roytenberg (and Ševera) computed the standard cohomology of the standard Courant algebroid which turns out to be equivalent to the de Rham cohomology of the underlying smooth manifold and Ševera discovered that the third real cohomology classes of the manifold are the only deformations of the standard Courant algebroid (see the Ševera bracket above).

In the case of a transitive Courant algebroid (the anchor map is surjective) the naive cohomology is isomorphic to the standard cohomology. The advantage of the naive cohomology is that it is defined analogous to Lie algebroids and can thus be computed with similar methods.

Generalized complex geometry

Given a Courant algebroid we can look for isotropic integrable vector subspaces, i.e. D ⊂ E is called isotropic if g(D,D) = 0 and integrable if [Γ(D),Γ(D)] ⊂ Γ(D). These subspaces inherit the structure of a Lie algebroid, because the bracket restricts to them and its violation of skew-symmetry vanishes.

A Dirac structure in a Courant algebroid with a metric of split signature is a maximally isotropic integrable vector subbundle.

1. Given the standard Courant algebroid (with the Dorfman bracket, i.e. the Ševera bracket with H = 0) then the graph of a bilinear form ω ∈ Γ(T^*M⊗T^*M) is isotropic iff ω is skew-symmetric (i.e. a 2-form) and then integrable iff dω = 0, i.e. ω is presymplectic.
2. In the same Courant algebroid the graph of a bivector Π ∈ Γ(TM⊗TM) is isotropic iff Π is skew-symmetric. It is then integrable iff Π is Poisson, i.e. [Π,Π] = 0. Therefore Dirac structures generalize the notion of presymplectic and Poisson manifolds.
3. Given an exact Courant algebroid (i.e. a standard one with a Ševera bracket with H ∈ Ω³(M)), then the graph of a bivector Π is integrable iff $1/2[\Pi ,\Pi ]=(\wedge ^{3}\Pi ^{\#})(H)$, i.e. we have an H-twisted Poisson structure.

Given a complex Courant algebroid E of split signature, then a generalized complex structure is a complex vector subbundle D ⊂ E that is integrable, maximal isotropic, and regular, i.e. $\mathrm {rk} \,D\cap {\bar {D}}=\mathrm {const}$ where ¯ means complex conjugation.

Beside complex presymplectic and Poisson structures, also the graph of a complex structure J on E are examples of generalized complex structures on the complexified standard Courant algebroid E = TM⊕T^*M.

Also a generalized complex structure inherits the structure of a complex Lie algebroid. These can be used as generalizations of the tangent bundle and thus permit many constructions paralleling those of complex geometry.

Integration of the Courant brackets

Because a Courant algebroid gives rise to a 2-term L_∞-algebra it can be integrated by Kan complexes. The converse way is roughly via the 1-jet of the Kan complex. however it is not yet clear how to reconstruct the Q-structure of the Courant algebroid.