Generalized complex structure: Difference between revisions

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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'''^*, 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#Exterior power|exterior powers]] of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the '''generalized tangent bundle''', which is the [[direct sum of vector bundles|direct sum]] <math>\mathbf{T} \oplus \mathbf{T}^*</math> of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.

The fibers are endowed with a natural [[inner product]] with [[Metric signature (topology)|signature]] (''N'',&nbsp;''N''). If ''X'' and ''Y'' are vector fields and ''ξ'' and ''η'' are one-forms then the inner product of ''X+ξ'' and ''Y+η'' is defined as

:<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

(i)|1= the intersection with its [[complex conjugate]] is the zero section: <math>L\cap\overline{L}=0</math>;

(ii)|2= ''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>}}

(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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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.

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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'',&nbsp; 2''n'')}} structure. A generalized almost complex structure is a reduction of this structure to a {{math|U(''n'',&nbsp; ''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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*{{cite journal |authorlink=Nigel Hitchin |last=Hitchin |first=Nigel |doi=10.1093/qmath/hag025 |title=Generalized Calabi-Yau manifolds |journal=[[Quarterly Journal of Mathematics]] |volume=54 |year=2003 |issue=3 |pages=281&ndash;308 }}

*{{cite thesis |last=Gualtieri |first=Marco |arxiv=math.DG/0401221 |title=Generalized complex geometry |type=PhD Thesis |date=2004 }}

*{{cite journal |last=Gualtieri |first=Marco |doi=10.4007/annals.2011.174.1.3 |title=Generalized complex geometry |journal=[[Annals of Mathematics]] |series=(2) |volume=174 |year=2011 |issue=1 |pages=75&ndash;123 |doi-access=free |arxiv=0911.0993 }}

*{{cite journal |last=Graña |first=Mariana |arxiv=hep-th/0509003 |title=Flux compactifications in string theory: a comprehensive review |journal=Phys. Rep. |volume=423 |year=2006 |issue=3 |pages=91–158 |doi=10.1016/j.physrep.2005.10.008 |s2cid=119508517 }}

*{{cite journal |authorlink=Robbert Dijkgraaf |firstfirst1=Robbert |lastlast1=Dijkgraaf |authorlink2=Sergei Gukov |first2=Sergei |last2=Gukov |first3=Andrew |last3=Neitzke |authorlink4=Cumrun Vafa |first4=Cumrun |last4=Vafa |title=Topological M-theory as unification of form theories of gravity |journal=[[Advances in Theoretical and Mathematical Physics]] |volume=9 |year=2005 |issue=4 |pages=603&ndash;665 |doi=10.4310/ATMP.2005.v9.n4.a5 |doi-access=free |arxiv=hep-th/0411073 }}

{{String theory topics |state=collapsed}}