Dirac and Generalized Complex Structures
The cotangent bundle, of M is the bundle of differential one-forms. In the case p=1 the Courant bracket maps two sections of, the direct sum of the tangent and cotangent bundles, to another section of . The fibers of admit inner products with signature (N,N) given by
A linear subspace of in which all pairs of vectors have zero inner product is said to be an isotropic subspace. The fibers of are 2N-dimensional and the maximal dimension of an isotropic subspace is N. An N-dimensional isotropic subspace is called a maximal isotropic subspace.
A Dirac structure is a maximally isotropic subbundle of whose sections are closed under the Courant bracket. Dirac structures include as special cases symplectic structures, Poisson structures and foliated geometries.
A generalized complex structure is defined identically, but one tensors 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 T*)C. Special cases of generalized complex structures include complex structure and a version of Kähler structure which includes the B-field.
Read more about this topic: Courant Bracket
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