Courant algebroid

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Short description: Concept in differential geometry

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 TMT*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 EM with a bracket [,]:ΓE×ΓEΓE, a non degenerate fiber-wise inner product ,:E×EM×, and a bundle map ρ:ETM (called anchor) subject to the following axioms:

  1. Jacobi identity: [ϕ,[χ,ψ]]=ϕ,χ],ψ]+[χ,[ϕ,ψ
  2. Leibniz rule: [ϕ,fψ]=ρ(ϕ)fψ+f[ϕ,ψ]
  3. Obstruction to skew-symmetry: [ϕ,ψ]+[ψ,ϕ]=12Dϕ,ψ
  4. Invariance of the inner product under the bracket: ρ(ϕ)ψ,χ=[ϕ,ψ],χ+ψ,[ϕ,χ]

where ϕ,χ,ψ are sections of E and f is a smooth function on the base manifold M. The map D:𝒞(M)ΓE is the composition κ1ρTd:𝒞(M)ΓE, with d:𝒞(M)Ω1(M) the de Rham differential, ρT the dual map of ρ, and κ the isomorphism EE* induced by the inner product.

Skew-symmetric definition

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

ϕ,ψ=12([ϕ,ψ][ψ,ϕ])

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

ϕ,[[ψ,χ]]+cycl.=DT(ϕ,ψ,χ)

where T is

T(ϕ,ψ,χ)=13[ϕ,ψ],χ+cycl.

The Leibniz rule and the invariance of the scalar product become modified by the relation ϕ,ψ=[ϕ,ψ]12Dϕ,ψ and the violation of skew-symmetry gets replaced by the axiom

ρD=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:

ρ[ϕ,ψ]=[ρ(ϕ),ρ(ψ)].

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

ρ(ϕ)χ,ψ=[ϕ,χ],ψ+χ,[ϕ,ψ].

Examples

An example of the Courant algebroid is given by the Dorfman bracket[3] on the direct sum TMT*M with a twist introduced by Ševera in 1988,[4] defined as:

[X+ξ,Y+η]:=[X,Y]+(XηιYdξ+ιXιYH)

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: if A is a Lie algebroid (with anchor ρA and bracket [.,.]A), also its dual A* is a Lie algebroid (inducing the differential dA* on *A) and dA*[X,Y]A=[dA*X,Y]A+[X,dA*Y]A (where on the right-hand side you extend the A-bracket to *A using graded Leibniz rule). This notion is symmetric in A and A* (see Roytenberg). Here E=AA* with anchor ρ(X+α)=ρA(X)+ρA*(α) and the bracket is the skew-symmetrization of the above in X and α (equivalently in Y and β):

[X+α,Y+β]=([X,Y]A+αA*YιβdA*X)+([α,β]A*+XAβιYdAα).

Dirac structures

See also: Physics:Dirac structure

Given a Courant algebroid with the inner product , of split signature (e.g. the standard one TMT*M), a Dirac structure is a maximally isotropic integrable vector subbundle LM, i.e.

L,L0,
rkL=12rkE,
[ΓL,ΓL]ΓL.

Examples

As discovered by Courant and parallel by Dorfman, the graph of a 2-form ωΩ2(M) is maximally isotropic and moreover integrable if and only if dω=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 ΠΓ(2TM) whose graph is maximally isotropic and integrable if and only if [Π,Π]=0, i.e. ρ is a Poisson bivector on M.

Generalized complex structures

Given a Courant algebroid with inner product of split signature, a generalized complex structure LM is a Dirac structure in the complexified Courant algebroid with the additional property

LL¯=0

where  ¯ 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 J:TMTM.

References

  1. Courant, Theodore James (1990). "Dirac manifolds" (in en). Transactions of the American Mathematical Society 319 (2): 631–661. doi:10.1090/S0002-9947-1990-0998124-1. ISSN 0002-9947. https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/. 
  2. Liu, Zhang-Ju; Weinstein, Alan; Xu, Ping (1997-01-01). "Manin triples for Lie bialgebroids". Journal of Differential Geometry 45 (3). doi:10.4310/jdg/1214459842. ISSN 0022-040X. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-45/issue-3/Manin-triples-for-Lie-bialgebroids/10.4310/jdg/1214459842.full. 
  3. Dorfman, Irene Ya. (1987-11-16). "Dirac structures of integrable evolution equations". Physics Letters A 125 (5): 240–246. doi:10.1016/0375-9601(87)90201-5. ISSN 0375-9601. Bibcode1987PhLA..125..240D. https://linkinghub.elsevier.com/retrieve/pii/0375960187902015. 
  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.