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Enyokoyama (talk | contribs) m →Courant bracket: modify the destination of "integrable" |
clean up and general fixes, http: --> https:, typo(s) fixed: Furthermore → Furthermore, (5) using AWB |
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:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\epsilon(Y,X)+\epsilon(X,Y))=0</math>
and so ''L''('''E''', ''ε'') is isotropic. Furthermore, ''L''('''E''', ''ε'') is maximal because there are dim('''E''') (complex) dimensions of choices for '''E''', and ''ε'' is unrestricted on the [[complement (complexity)|complement]] of '''E'''<sup>*</sup>, which is of (complex) dimension ''n'' − dim('''E'''). Thus the total (complex) dimension in ''n''. Gualtieri has proven that all maximal isotropic subbundles are of the form ''L''('''E''',ε) for some '''E''' and ε.
===Type===
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A spinor is said to be a '''pure spinor''' if it is annihilated by half of a set of a set of generators of the Clifford algebra. Spinors are sections of our bundle '''Λ<sup>*</sup>T''', and generators of the Clifford algebra are the fibers of our other bundle ('''T''' <math>\oplus</math> '''T'''<sup>*</sup>) <math>\otimes</math> '''C'''.
Therefore, a given pure spinor is annihilated by a half-dimensional subbundle '''E''' of ('''T''' <math>\oplus</math> '''T'''<sup>*</sup>) <math>\otimes</math> '''C'''.
Such subbundles are always isotropic, so to define an almost complex structure one must only impose that the sum of '''E''' and its complex conjugate is all of ('''T''' <math>\oplus</math> '''T'''<sup>*</sup>) <math>\otimes</math> '''C'''. This is true whenever the [[wedge product]] of the pure spinor and its complex conjugate contains a top-dimensional component. Such pure spinors determine generalized almost complex structures.
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(''n,0'')-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from ('''T'''<math>\oplus</math>
'''T'''<sup>*</sup>)<math>\otimes</math>'''C''' to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on ('''T'''<math>\oplus</math>
'''T'''<sup>*</sup>)<math>\otimes</math>'''C''' defines an ordinary [[linear complex structure|complex structure]] on the tangent bundle.
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The above pure spinor is globally defined, and so the canonical bundle is trivial. This means that symplectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.
The pure spinor <math>\phi</math> is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the [[Kähler form]]. Therefore, these generalized complex structures are of the same type as those corresponding to a [[scalar (mathematics)|scalar]] pure spinor. A scalar is annihilated by the entire tangent space, and so these structures are of type ''0''.
Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called '''B-symplectic'''.
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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'''<sup>*</sup>) <math>\otimes</math> '''C''' with the above inner product is an O(2''n'', 2''n'') structure. A generalized almost complex structure is a reduction of this structure to a U(''n'', ''n'') structure. Therefore, the space of generalized complex structures is the coset
:::::<math>\frac{O(2n,2n)}{U(n,n)}.</math>
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