Generalized complex structure: Difference between revisions

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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''. TheseforeTherefore if <math>X+\xi</math> and <math>Y+\eta</math> are sections of <math>\mathbf{T} \oplus \mathbf{T}^*</math> then

:<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 inis ''n''. Gualtieri has proven that all maximal isotropic subbundles are of the form <math>L(\mathbf{E}, \varepsilon)</math> for some <math>\mathbf{E}</math> and <math>\varepsilon.</math>

===Type===

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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 of a set of a set of generators of the Clifford algebra. Spinors are sections of our bundle <math>\mathbf{\Lambda}^* \mathbf{T},</math> and generators of the Clifford algebra are the fibers of our other bundle <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math> Therefore, a given pure spinor is annihilated by a half-dimensional subbundle '''E''' of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math> 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 <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math> 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.

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.