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| In a field of [[mathematics]] known as [[differential geometry]], the '''Courant bracket''' is a generalization of the [[Lie derivative|Lie bracket]] from an operation on the [[tangent bundle]] to an operation on the [[direct sum of vector bundles|direct sum]] of the tangent bundle and the [[vector bundle]] of [[differential form|''p''-forms]].
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| The case ''p'' = 1 was introduced by [[Theodore James Courant]] in his 1990 doctoral dissertation as a structure that bridges [[Poisson manifold|Poisson geometry]] and pre[[symplectic geometry]], based on work with his advisor [[Alan Weinstein]]. The twisted version of the Courant bracket was introduced in 2001 by [[Pavol Severa]], and studied in collaboration with Weinstein.
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| Today a [[complex number|complex]] version of the ''p''=1 Courant bracket plays a central role in the field of [[generalized complex geometry]], introduced by [[Nigel Hitchin]] in 2002. Closure under the Courant bracket is the [[integrability condition]] of a [[Almost_complex_manifold#Generalized_almost_complex_structure|generalized almost complex structure]].
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| ==Definition==
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| Let ''X'' and ''Y'' be [[vector field]]s on an N-dimensional real [[manifold (mathematics)|manifold]] ''M'' and let ''ξ'' and ''η'' be ''p''-forms. Then ''X+ξ'' and ''Y+η'' are [[fiber_bundle#Sections|sections]] of the direct sum of the tangent bundle and the bundle of ''p''-forms. The Courant bracket of ''X+ξ'' and ''Y+η'' is defined to be
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| :<math>[X+\xi,Y+\eta]=[X,Y]
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| +\mathcal{L}_X\eta-\mathcal{L}_Y\xi
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| -\frac{1}{2}d(i(X)\eta-i(Y)\xi)</math>
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| where <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field ''X'', ''d'' is the [[exterior derivative]] and ''i'' is the [[Exterior_algebra#The_interior_product_or_insertion_operator|interior product]].
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| ==Properties==
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| The Courant bracket is [[antisymmetric]] but it does not satisfy the [[Jacobi identity]] for ''p'' greater than zero.
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| ===The Jacobi identity===
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| However, at least in the case ''p=1'', the [[Jacobiator]], which measures a bracket's failure to satisfy the Jacobi identity, is an [[exact form]]. It is the exterior derivative of a form which plays the role of the [[Nijenhuis tensor]] in generalized complex geometry.
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| The Courant bracket is the antisymmetrization of the [[Courant_bracket#Dorfman_bracket|Dorfman bracket]], which does satisfy a kind of Jacobi identity.
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| ===Symmetries===
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| Like the Lie bracket, the Courant bracket is invariant under diffeomorphisms of the manifold ''M''. It also enjoys an additional symmetry under the vector bundle [[automorphism]]
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| :::<math>X+\xi\mapsto X+\xi+i(X)\alpha</math>
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| where ''α'' is a closed ''p+1''-form. In the ''p=1'' case, which is the relevant case for the geometry of [[Compactification (physics)#Flux compactification|flux compactification]]s in [[string theory]], this transformation is known in the physics literature as a shift in the [[Kalb-Ramond field|B field]].
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| ==Dirac and generalized complex structures==
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| The [[cotangent bundle]], <math>{\mathbf T}^*</math> of M is the bundle of differential one-forms. In the case ''p''=1 the Courant bracket maps two sections of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math>, the direct sum of the tangent and cotangent bundles, to another section of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math>. The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> admit [[inner product]]s with [[signature of a quadratic form|signature]] (N,N) given by
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| :::<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).</math>
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| A [[linear subspace]] of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> in which all pairs of vectors have zero inner product is said to be an [[isotropic subspace]]. The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> are ''2N''-dimensional and the maximal dimension of an isotropic subspace is ''N''. An ''N''-dimensional isotropic subspace is called a maximal isotropic subspace.
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| A [[Dirac structure]] is a maximally isotropic subbundle of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> whose sections are closed under the Courant bracket. Dirac structures include as special cases [[symplectic geometry|symplectic structures]], [[Poisson manifold|Poisson structures]] and [[foliation|foliated geometries]].
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| A [[generalized complex structure]] is defined identically, but one [[tensor product|tensors]] <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> by the complex numbers and uses the [[complex dimension]] in the above definitions and one imposes that the direct sum of the subbundle and its [[complex conjugate]] be the entire original bundle ('''T'''<math>\oplus</math>
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| '''T'''<sup>*</sup>)<math>\otimes</math>'''C'''. Special cases of generalized complex structures include [[complex structure]]{{dn|date=September 2012}} and a version of [[Kähler manifold|Kähler structure]] which includes the B-field.
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| ==Dorfman bracket==
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| In 1987 [[Irene Dorfman]] introduced the Dorfman bracket [,]<sub>D</sub>, which like the Courant bracket provides an integrability condition for Dirac structures. It is defined by
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| ::<math>[A,B]_D=[A,B]+d\langle A,B\rangle</math>.
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| The Dorfman bracket is not antisymmetric, but it is often easier to calculate with than the Courant bracket because it satisfies a [[Leibniz rule]] which resembles the Jacobi identity
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| ::<math>[A,[B,C]_D]_D=[[A,B]_D,C]_D+[B,[A,C]_D]_D.</math>
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| ==Courant algebroid==
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| The Courant bracket does not satisfy the [[Jacobi identity]] and so it does not define a [[Lie algebroid]], in addition it fails to satisfy the Lie algebroid condition on the [[anchor map]]. Instead it defines a more general structure introduced by [[Zhang-Ju Liu]], [[Alan Weinstein]] and [[Ping Xu]] known as a [[Courant algebroid]].
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| ==Twisted Courant bracket==
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| ===Definition and properties===
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| The Courant bracket may be twisted by a ''(p+2)''-form ''H'', by adding the interior product of the vector fields ''X'' and ''Y'' of ''H''. It remains antisymmetric and invariant under the addition of the interior product with a ''(p+1)''-form ''B''. When ''B'' is not closed then this invariance is still preserved if one adds ''dB'' to the final ''H''.
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| If ''H'' is closed then the Jacobiator is exact and so the twisted Courant bracket still defines a Courant algebroid. In [[string theory]], ''H'' is interpreted as the [[Kalb-Ramond field|Neveu-Schwarz 3-form]].
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| ===''p=0'': Circle-invariant vector fields===
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| When ''p''=0 the Courant bracket reduces to the Lie bracket on a [[principal bundle|principal]] [[circle bundle]] over ''M'' with [[Riemann curvature tensor|curvature]] given by the 2-form twist ''H''. The bundle of 0-forms is the trivial bundle, and a section of the direct sum of the tangent bundle and the trivial bundle defines a circle [[invariant vector field]] on this circle bundle.
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| Concretely, a section of the sum of the tangent and trivial bundles is given by a vector field ''X'' and a function ''f'' and the Courant bracket is
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| ::<math>[X+f,Y+g]=[X,Y]+Xg-Yf</math>
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| which is just the Lie bracket of the vector fields
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| :::<math>[X+f,Y+g]=[X+f\frac{\partial}{\partial\theta},Y+g\frac{\partial}{\partial\theta}]_{Lie}</math>
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| where ''θ'' is a coordinate on the circle fiber. Note in particular that the Courant bracket satisfies the Jacobi identity in the case ''p=0''.
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| ===Integral twists and gerbes===
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| The curvature of a circle bundle always represents an integral [[cohomology]] class, the [[Chern class]] of the circle bundle. Thus the above geometric interpretation of the twisted ''p=0'' Courant bracket only exists when ''H'' represents an integral class. Similarly at higher values of ''p'' the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by [[gerbe]]s when ''H'' is an integral cohomology class.
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| ==References==
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| *Courant, Theodore, ''Dirac manifolds'', Trans. Amer. Math. Soc., 319:631-661, (1990).
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| *Gualtieri, Marco, [http://xxx.lanl.gov/abs/math.DG/0401221 Generalized complex geometry], PhD Thesis (2004).
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| [[Category:Differential geometry]]
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| [[Category:Binary operations]]
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