Generalized complex structure: Difference between revisions

Revision as of 21:41, 29 April 2025

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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.