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#The exteriorExterior 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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:<math>\langle {\mathcal J}(X+\xi),{\mathcal J}(Y+\eta)\rangle=\langle X+\xi, Y+\eta \rangle.</math>

Like in the case of an ordinary [[almost complex structure]], a generalized almost complex structure is uniquely determined by its <math>\sqrt{-1}</math>-[[Vector bundle#Operations on vector bundles|eigenbundle]], i.e. a subbundle <math>L</math> of the complexified generalized tangent bundle <math>(\mathbf{T}\oplus\mathbf{T}^*)\otimes\CComplex </math>

given by

:<math>L=\{X+\xi\in (\mathbf{T}\oplus\mathbf{T}^*)\otimes\CComplex \ :\ {\mathcal J}(X+\xi)=\sqrt{-1}(X+\xi)\}</math>

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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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]].

===The definitionDefinition===

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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===Classification===

There is a one-to-one correspondence between maximal isotropic [[subbundle]] of <math>\mathbf{T} \oplus \mathbf{T}^*</math> and pairs <math>('''\mathbf{E'''}, ''ε''\varepsilon)</math> where '''E''' is a subbundle of '''T''' and ''ε''<math>\varepsilon</math> is a 2-form. This correspondence extends straightforwardly to the complex case.

Given a pair <math>('''\mathbf{E'''},''ε'' \varepsilon)</math> one can construct a maximally isotropic subbundle ''<math>L''('''\mathbf{E'''},''ε'' \varepsilon)</math> of <math>\mathbf{T} \oplus \mathbf{T}^*</math> as follows. The elements of the subbundle are the [[formal sum]]s ''<math>X''&nbsp;+&nbsp;''ξ''\xi</math> where the [[vector field]] ''X'' is a section of '''E''' and the one-form ''ξ'' restricted to the [[dual space]] '''E'''<supmath>\mathbf{E}^*</supmath> is equal to the one-form ε<math>\varepsilon(''X'').</math>

To see that ''<math>L''('''\mathbf{E'''},&nbsp;''ε'' \varepsilon)</math> is isotropic, notice that if ''Y'' is a section of '''E''' and ''ξ''<math>\xi</math> restricted to <math>\mathbf{E}^*</math> is ''ε''<math>\varepsilon(''X'')</math> then ''ξ''<math>\xi(''Y'') = ''ε''\varepsilon(''X'',&nbsp;''Y''),</math> as the part of ''ξ''<math>\xi</math> orthogonal to <math>\mathbf{E}^*</math> annihilates ''Y''. TheseforeTherefore if ''<math>X''&nbsp;+&nbsp;''ξ''\xi</math> and ''<math>Y''&nbsp;+&nbsp;''η''\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}(\epsilonvarepsilon(Y,X)+\epsilonvarepsilon(X,Y))=0</math>

and so ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> is isotropic. Furthermore, ''<math>L''('''\mathbf{E'''}, ''ε''\varepsilon)</math> is maximal because there are <math>\dim('''\mathbf{E'''})</math> (complex) dimensions of choices for '''<math>\mathbf{E'''},</math> and ''ε''<math>\varepsilon</math> is unrestricted on the [[complement (complexity)|complement]] of <math>\mathbf{E}^*,</math> which is of (complex) dimension ''<math>n''&nbsp;−&nbsp;-\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'''}, ''ε''\varepsilon)</math> for some '''<math>\mathbf{E'''}</math> and ''ε''<math>\varepsilon.</math>

===Type===

The '''type''' of a maximal isotropic subbundle ''<math>L''('''\mathbf{E'''}, ''ε''\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'''}, ''ε''\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 \CComplex.</math>

The type of a maximal isotropic subbundle is [[Invariant (mathematics)|invariant]] under [[diffeomorphisms]] and also under shifts of the [[Kalb-RamondKalb–Ramond field|B-field]], which are [[isometry|isometries]] of <math>\mathbf{T} \oplus \mathbf{T}^*</math> of the form

:<math>X+\xi\longrightarrow X+\xi+i_XB</math>

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===Real index===

The real index ''r'' of a maximal isotropic subspace ''L'' is the complex dimension of the [[intersection (set theory)|intersection]] of ''L'' with its complex conjugate. A maximal isotropic subspace of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex</math> is a generalized almost complex structure if and only if ''r'' = 0.

==Canonical bundle==

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===Generalized almost complex structures===

The canonical bundle is a one complex dimensional subbundle of the bundle <math>\mathbf{\Lambda}^* \mathbf{T} \otimes \CComplex</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 \CComplex</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 \CComplex.</math> Therefore, a given pure spinor is annihilated by a half-dimensional subbundle '''E''' of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex.</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 \CComplex.</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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===Regular point===

Define the subbundle '''E''' of the complexified tangent bundle <math>\mathbf{T} \otimes \CComplex</math> to be the projection of the holomorphic subbundle '''L''' of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex</math> to <math>\mathbf{T} \otimes \CComplex.</math> In the definition of a generalized almost complex structure we have imposed that the intersection of '''L''' and its conjugate contains only the origin, otherwise they would be unable to span the entirety of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex.</math> However the intersection of their projections need not be trivial. In general this intersection is of the form

:<math>E\cap\overline{E}=\Delta\otimes\CComplex</math>

for some subbundle Δ. A point which has an [[open set|open]] [[neighborhood (mathematics)|neighborhood]] in which the dimension of the fibers of Δ is constant is said to be a '''regular point'''.

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===Darboux's theorem===

{{main|Darboux's theorem}}

Every regular point in a generalized complex manifold has an open neighborhood which, after a diffeomorphism and shift of the B-field, has the same generalized complex structure as the [[Cartesian product]] of the [[Linear complex structure|complex vector space]] <math>\CComplex^k</math> and the standard symplectic space <math>\R^{2n-2k}</math> with the standard symplectic form, which is the [[direct sum of matrices|direct sum]] of the two by two off-diagonal matrices with entries 1 and −1.

===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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===Complex manifolds===

The space of complex differential forms <math>\mathbf{\Lambda}^* \mathbf{T} \otimes \CComplex</math> has a complex conjugation operation given by complex conjugation in <math>\CComplex.</math> This allows one to define [[Holomorphic function|holomorphic]] and [[antiholomorphic]] one-forms and (''m'', ''n'')-forms, which are homogeneous polynomials in these one-forms with ''m'' holomorphic factors and ''n'' antiholomorphic factors. In particular, all (''n'', 0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.

(''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 <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex</math> to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex</math> defines an ordinary [[linear complex structure|complex structure]] on the tangent bundle.

As only half of a basis of vector fields are holomorphic, these complex structures are of type ''N''. In fact complex manifolds, and the manifolds obtained by multiplying the pure spinor bundle defining a complex manifold by a complex, <math>\partial</math>-closed (2,0)-form, are the only type ''N'' generalized complex manifolds.

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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 \CComplex</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>

A [[generalized Kähler structure|generalized almost Kähler structure]] is a pair of [[commutative operation|commuting]] generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \CComplex.</math> Generalized Kähler structures are reductions of the structure group to <math>U(n) \times U(n).</math> Generalized Kähler manifolds, and their twisted counterparts, are equivalent to the [[bihermitian manifolds]] discovered by [[Sylvester James Gates]], [[Chris Hull (physicist)|Chris Hull]] and [[Martin Rocek|Martin Roček]] in the context of 2-dimensional [[supersymmetry|supersymmetric]] [[quantum field theory|quantum field theories]] in 1984.

Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to <math>SU(n) \times SU(n).</math>

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==References==

*[[{{cite journal |authorlink=Nigel Hitchin |last=Hitchin, |first=Nigel]] [https://dx.|doi.org/=10.1093/qmath/hag025 |title=Generalized Calabi-Yau manifolds], |journal=[[Quarterly Journal of Mathematics]] '''|volume=54''' (|year=2003), no. |issue=3, |pages=281&ndash;308. }}

*{{cite thesis |last=Gualtieri, |first=Marco, [http://xxx.lanl.gov/abs/|arxiv=math.DG/0401221 |title=Generalized complex geometry], |type=PhD Thesis (|date=2004). }}

*{{cite journal |last=Gualtieri, |first=Marco, [https://dx.|doi.org/=10.4007/annals.2011.174.1.3 |title=Generalized complex geometry], |journal=[[Annals of Mathematics]] |series=(2) '''|volume=174''' (|year=2011), no. |issue=1, |pages=75&ndash;123 |doi-access=free |arxiv=0911.0993 }}

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

*[[{{cite journal |authorlink=Robbert Dijkgraaf]], [[|first1=Robbert |last1=Dijkgraaf |authorlink2=Sergei Gukov]], |first2=Sergei |last2=Gukov |first3=Andrew |last3=Neitzke, and|authorlink4=Cumrun [[Vafa |first4=Cumrun |last4=Vafa]], [http://projecteuclid-org.ezproxy.neu.edu/euclid.atmp/1144070454 |title=Topological M-theory as unification of form theories of gravity], |journal=[[Advances in Theoretical and Mathematical Physics]] '''|volume=9''' (|year=2005), no. |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}}