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{{Short description|Property of a differential manifold that includes complex structures}}
{{No footnotes|date=June 2020}}
In the field of [[mathematics]] known as [[differential geometry]], a '''generalized complex structure''' is a property of a [[differential manifold]] that includes as special cases a [[linear complex structure|complex structure]] and a [[symplectic structure]]. Generalized complex structures were introduced by [[Nigel Hitchin]] in 2002 and further developed by his students [[Marco Gualtieri]] and [[Gil Cavalcanti]].
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===The generalized tangent bundle===
Consider an [[Manifold|''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'''<sup>*</sup>, is the vector bundle over ''M'' whose sections are [[differential form|one-forms]] on ''M''.
In [[complex geometry]] one considers structures on the tangent bundles of manifolds. In [[symplectic geometry]] one is instead interested in [[Exterior algebra#
The fibers are endowed with a natural [[inner product]] with [[Metric signature
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).</math>
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Such subbundle ''L'' satisfies the following properties:
{{ordered list | list-style-type=lower-roman
▲(ii) ''L'' is '''maximal isotropic''', i.e. its complex [[rank (linear algebra)|rank]] equals ''N'' and <math>\langle\ell, \ell' \rangle =0</math> for all <math>\ell,\ell'\in L.</math>
Vice versa, any subbundle ''L'' satisfying (i), (ii) is the <math>\sqrt{-1}</math>-eigenbundle of a unique generalized almost complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.
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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]].
===
A '''generalized complex structure''' is a generalized almost complex structure such that the space of smooth sections of ''L'' is closed under the Courant bracket.
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Given a pair <math>(\mathbf{E}, \varepsilon)</math> one can construct a maximally isotropic subbundle <math>L(\mathbf{E}, \varepsilon)</math> of <math>\mathbf{T} \oplus \mathbf{T}^*</math> as follows. The elements of the subbundle are the [[formal sum]]s <math>X+\xi</math> where the [[vector field]] ''X'' is a section of '''E''' and the one-form ''ξ'' restricted to the [[dual space]] <math>\mathbf{E}^*</math> is equal to the one-form <math>\varepsilon(X).</math>
To see that <math>L(\mathbf{E}, \varepsilon)</math> is isotropic, notice that if ''Y'' is a section of '''E''' and <math>\xi</math> restricted to <math>\mathbf{E}^*</math> is <math>\varepsilon(X)</math> then <math>\xi(Y) =\varepsilon(X,Y),</math> as the part of <math>\xi</math> orthogonal to <math>\mathbf{E}^*</math> annihilates ''Y''.
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\varepsilon(Y,X)+\varepsilon(X,Y))=0</math>
and so <math>L(\mathbf{E}, \varepsilon)</math> is isotropic. Furthermore, <math>L(\mathbf{E}, \varepsilon)</math> is maximal because there are <math>\dim(\mathbf{E})</math> (complex) dimensions of choices for <math>\mathbf{E},</math> and <math>\varepsilon</math> is unrestricted on the [[complement (complexity)|complement]] of <math>\mathbf{E}^*,</math> which is of (complex) dimension <math>n-\dim(\mathbf{E}).</math> Thus the total (complex) dimension
===Type===
The '''type''' of a maximal isotropic subbundle <math>L(\mathbf{E}, \varepsilon)</math> is the real dimension of the subbundle that annihilates '''E'''. Equivalently it is 2''N'' minus the real dimension of the [[projection (mathematics)|projection]] of <math>L(\mathbf{E}, \varepsilon)</math> onto the tangent bundle '''T'''. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the '''complex type'''. While the type of a subbundle can in principle be any integer between 0 and 2''N'', generalized almost complex structures cannot have a type greater than ''N'' because the sum of the subbundle and its complex conjugate must be all of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math>
The type of a maximal isotropic subbundle is [[Invariant (mathematics)|invariant]] under [[diffeomorphisms]] and also under shifts of the [[
:<math>X+\xi\longrightarrow X+\xi+i_XB</math>
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The canonical bundle is a one complex dimensional subbundle of the bundle <math>\mathbf{\Lambda}^* \mathbf{T} \otimes \Complex</math> of complex differential forms on ''M''. Recall that the [[gamma matrices]] define an [[isomorphism]] between differential forms and spinors. In particular even and odd forms map to the two chiralities of [[Spinor#Weyl spinors|Weyl spinors]]. Vectors have an action on differential forms given by the interior product. One-forms have an action on forms given by the wedge product. Thus sections of the bundle <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex</math> act on differential forms. This action is a [[group representation|representation]] of the action of the [[Clifford algebra]] on spinors.
A spinor is said to be a '''pure spinor''' if it is annihilated by half
Given a generalized almost complex structure, one can also determine a pure spinor up to multiplication by an arbitrary [[complex function]]. These choices of pure spinors are defined to be the sections of the canonical bundle.
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===Local holomorphicity===
Near non-regular points, the above [[classification theorem]] does not apply. However, about any point, a generalized complex manifold is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like Weinstein's theorem for the local structure of [[Poisson manifold]]s. The remaining question of the local structure is: what does a generalized complex structure look like near a point of complex type? In fact, it will be induced by a holomorphic [[Poisson manifold|Poisson structure]].
==Examples==
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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 <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex</math> with the above inner product is an {{math|O(2''n'',
:<math>\frac{O(2n,2n)}{U(n,n)}.</math>
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==References==
*
*{{cite thesis |last=Gualtieri
*{{cite journal |last=Gualtieri
*{{cite journal |last=Graña
*
{{String theory topics |state=collapsed}}
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