Generalized complex structure: Difference between revisions

Revision as of 17:04, 22 May 2013

Revision as of 16:11, 1 June 2013

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==Definition==

===The generalized tangent bundle===

Consider an ''N''-manifold ''M''. The [[tangent bundle]] of ''M'', which will be denoted '''T''', is the [[vector bundle]] over ''M'' whose fibers consist of all [[tangent vector]]s to ''M''. A [[fiber bundle#Sections|section]] of '''T''' is a [[vector field]] on ''M''. The [[cotangent bundle]] of ''M'', denoted '''T'''^*, is the vector bundle over ''M'' whose sections are [[differential form|one-forms]] on ''M''.

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==Maximal isotropic subbundles==

===Classification===

There is a one-to-one correspondence between maximal isotropic [[subbundle]] of '''T'''&nbsp;<math>\oplus</math>&nbsp;'''T'''^* and pairs ('''E''',''ε'') where '''E''' is a subbundle of '''T''' and ''ε'' is a 2-form. This correspondence extends straightforwardly to the complex case.

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==Local classification==

===Canonical bundle===

Locally all pure spinors can be written in the same form, depending on an integer ''k'', the B-field 2-form ''B'', a nondegenerate symplectic form ω and a ''k''-form Ω. In a local neighborhood of any point a [[pure spinor]] Φ which generates the canonical bundle may always be put in the form

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==Examples==

===Complex manifolds===

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Some of the almost structures in generalized complex geometry may be rephrased in the language of [[G-structure]]s. The word "almost" is removed if the structure is integrable.

The bundle ('''T'''<math>\oplus</math>'''T'''^*)&nbsp;<math>\otimes</math>&nbsp;'''C''' with the above inner product is aan O(2''n'',&nbsp;2''n'') structure. A generalized almost complex structure is a reduction of this structure to a U(''n'',&nbsp;''n'') structure. Therefore the space of generalized complex structures is the coset

:::::<math>\frac{O(2n,2n)}{U(n,n)}.</math>