In differential geometry, a field of mathematics, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of p-forms.
was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges 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.
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 generalized almost complex structure.
is the Lie derivative along the vector field
The Courant bracket is antisymmetric but it does not satisfy the Jacobi identity for
, 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.
The Courant bracket is the antisymmetrization of the Dorfman bracket, which does satisfy a kind of Jacobi identity.
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 where
case, which is the relevant case for the geometry of flux compactifications in string theory, this transformation is known in the physics literature as a shift in the B field.
the Courant bracket maps two sections of
, the direct sum of the tangent and cotangent bundles, to another section of
in which all pairs of vectors have zero inner product is said to be an isotropic subspace.
-dimensional and the maximal dimension of an isotropic subspace is
A Dirac structure is a maximally isotropic subbundle of
whose sections are closed under the Courant bracket.
A generalized complex structure is defined identically, but one tensors
, which like the Courant bracket provides an integrability condition for Dirac structures.
It is defined by 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 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.
, by adding the interior product of the vector fields
It remains antisymmetric and invariant under the addition of the interior product with a
is not closed then this invariance is still preserved if one adds
Concretely, a section of the sum of the tangent and trivial bundles is given by a vector field
Note in particular that the Courant bracket satisfies the Jacobi identity in the case
The curvature of a circle bundle always represents an integral cohomology class, the Chern class of the circle bundle.
Courant bracket only exists when H represents an integral class.
the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by gerbes when H is an integral cohomology class.