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]].

These structures first arose in Hitchin's program of characterizing geometrical structures via [[functional (mathematics)|functional]]s of [[differential forms]], a connection which formed the basis of [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew NietzkeNeitzke]] and [[Cumrun Vafa]]'s 2004 proposal that [[topological string theory|topological string theories]] are special cases of a [[topological M-theory]]. Today generalized complex structures also play a leading role in physical [[string theory]], as [[supersymmetry|supersymmetric]] [[Compactification (physics)#Flux compactification|flux compactification]]s, which relate 10 -dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.

==Definition==

===The generalized tangent bundle===

Consider an ''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''.

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 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]] '''T'''&nbsp;<math>\oplus</math>

'''T'''^* of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.

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]] '''T'''&nbsp;<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>

A '''generalized almost complex structure''' is just an [[almost complex structure]] of the generalized tangent bundle which preserves the natural inner product:

:::<math>{\mathcal J}: \mathbf{T}\oplus\mathbf{T}^*\rightarrowto \mathbf{T}\oplus\mathbf{T}^*</math>

such that <math>{\mathcal J}^2=-{\rm Id},</math> and

such that

:::<math>{\mathcal J}^2=-{\rm Id}, \mbox{ and }\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\mathbb{C}Complex </math>

given by

:::<math>L=\{X+\xi\in (\mathbf{T}\oplus\mathbf{T}^*)\otimes\mathbb{C}Complex \ :\ {\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> }}

ViceversaVice 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.

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

(ii) ''L'' is '''maximal isotropic''', i.e. its complex [[rank]] equals ''N'' and <math>\langle\ell,\ell'\rangle=0</math> for all <math>\ell,\ell'\in L.</math>

Viceversa, any subbundle ''L'' satisfying (i), (ii) is the <math>\sqrt{-1}</math>-[[eigenbundle]] of a unique generalized complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.

===Courant bracket===

In ordinary complex geometry, an [[almost complex structure]] is [[Foliation#FoliationsIntegrable and integrabilitysystem|integrable]] to a [[linear complex structure|complex structure]] if and only if the [[Lie derivative|Lie bracket]] of two sections of the [[Holomorphic function|holomorphic]] subbundle is another section of the holomorphic subbundle.

In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called the [[Courant bracket]] which is defined by

:<math>[X+\xi,Y+\eta]=[X,Y] +\mathcal{L}_X\eta-\mathcal{L}_Y\xi -\frac{1}{2}d(i(X)\eta-i(Y)\xi)</math>

+\mathcal{L}_X\eta-\mathcal{L}_Y\xi

-\frac{1}{2}d(i(X)\eta-i(Y)\xi)</math>

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]].

===Definition===

Unlike the Lie bracket, the Courant bracket does not satisfy the [[Jacobi identity]].

A '''generalized complex structure''' is a generalized almost complex structure such that the space of smooth sections of the eigenbundle ''L'' is closed under the [[Courant bracket]].

===The definition===

A '''generalized complex structure''' is a generalized almost complex structure such that the space of smooth sections of the eigenbundle ''L'' is closed under the [[Courant bracket]].

==Maximal isotropic subbundles==

===Classification===

There is a one-to-one correspondence between maximal isotropic [[subbundle]] of '''T'''&nbsp;<math>\mathbf{T} \oplus</math>&nbsp;''' \mathbf{T'''<sup>}^*</supmath> 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 '''T'''&nbsp;<math>\mathbf{T} \oplus</math>&nbsp;''' \mathbf{T'''<sup>}^*</supmath> 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 '''E'''<supmath>\mathbf{E}^*</supmath> 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 '''E'''<supmath>\mathbf{E}^*</supmath> annihilates ''Y''. TheseforeTherefore if ''<math>X''&nbsp;+&nbsp;ξ\xi</math> and ''<math>Y''&nbsp;+&nbsp;η\eta</math> are sections of '''T'''&nbsp;<math>\mathbf{T} \oplus</math>&nbsp;''' \mathbf{T'''<sup>}^*</supmath> 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 '''E'''<supmath>\mathbf{E}^*,</supmath>, 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''2NN'' 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''2NN'', 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 ('''T'''<math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math>

'''T'''^*)<math>\otimes</math>'''C'''.

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 '''T'''<math>\mathbf{T} \oplus</math>''' \mathbf{T'''<sup>}^*</supmath> of the form

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

The type of a maximal isotropic subbundle is [[Invariant (mathematics)|invariant]] under [[diffeomorphisms]] and also under shifts of the [[Kalb-Ramond field|B-field]], which are [[isometry|isometries]] of '''T'''<math>\oplus</math>'''T'''^* of the form

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

where ''B'' is an arbitrary closed 2-form called the B-field in the [[string theory]] literature.

The type of a generalized almost complex structure is in general not constant, it can jump by any even [[integer]]. However it is upper [[semi-continuous]], which means that each point has an open neighborhood in which the type does not increase. In practice this means that subsets of greater type than the ambient type occur on submanifolds with positive [[codimension]].

===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 ('''T'''<math>(\mathbf{T} \oplus</math> '''\mathbf{T'''^{}^*})&nbsp;<math> \otimes \Complex</math>&nbsp;'''C''' is a generalized almost complex structure if and only if ''r'' = 0.

==Canonical bundle==

As in the case of ordinary complex geometry, there is a correspondence between generalized almost complex structures and [[complex line bundle]]s. The complex line bundle corresponding to a particular generalized almost complex structure is often referred to as the '''canonical bundle''', as it generalizes the [[canonical bundle]] in the ordinary case. It is sometimes also called the [[pure spinor bundle]], as its sections are [[pure spinor]]s.

===Generalized almost complex structures===

The canonical bundle is a one complex dimensional subbundle of the bundle '''Λ<supmath>\mathbf{\Lambda}^*</sup> \mathbf{T'''<math>} \otimes \Complex</math>'''C''' 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 ('''T'''&nbsp;<math>(\mathbf{T} \oplus</math>&nbsp;''' \mathbf{T'''^{}^*})&nbsp;<math> \otimes \Complex</math>&nbsp;'''C''' 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 '''Λ<supmath>\mathbf{\Lambda}^* \mathbf{T},</supmath>T''', 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 '''TE'''&nbsp; of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \Complex.</math>&nbsp; Such subbundles are always isotropic, so to define an almost complex structure one must only impose that the sum of '''TE''' and its complex conjugate is all of <supmath>(\mathbf{T} \oplus \mathbf{T}^*</sup>)&nbsp;<math> \otimes \Complex.</math>&nbsp;'''C''' 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.

Therefore a given pure spinor is annihilated by a half-dimensional subbundle '''E''' of ('''T'''&nbsp;<math>\oplus</math>&nbsp;'''T'''^*)&nbsp;<math>\otimes</math>&nbsp;'''C'''.

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 ('''T'''&nbsp;<math>\oplus</math>&nbsp;'''T'''^*)&nbsp;<math>\otimes</math>&nbsp;'''C'''. 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.

===Integrability and other structures===

Line 93 ⟶ 91:

If a pure spinor that determines a particular complex structure is [[Closed and exact differential forms|closed]], or more generally if its exterior derivative is equal to the action of a gamma matrix on itself, then the almost complex structure is integrable and so such pure spinors correspond to generalized complex structures.

If one further imposes that the canonical bundle is holomorphically trivial, meaning that it is global sections which are closed forms, then it defines a [[generalized Calabi-Yau structure]] and ''M'' is said to be a '''generalized Calabi-Yau manifold'''.

==Local classification==

===Canonical bundle===

Locally all pure spinors can be written in the same form, depending on an integer ''k'', the B-field 2-form ''B'', a nondegenerate symplectic form ω and a ''k''-form Ω. In a local neighborhood of any point a [[pure spinor]] Φ which generates the canonical bundle may always be put in the form

::::<math>\Phi=e^{B+i\omega}\Omega</math>

where Ω is decomposable as the [[wedge product]] of one-forms.

===Regular point===

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

<math>\otimes</math>'''C''' to '''T'''<math>\otimes</math>'''C'''. 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 ('''T'''<math>\oplus</math>'''T'''^*)

<math>\otimes</math>'''C'''. 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\mathbf{C}</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'''.

::::<math>E\cap\overline{E}=\Delta\otimes\mathbf{C}Complex</math>

===Darboux 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]] '''C'''^k and the standard symplectic space '''R'''^2n-2k 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.

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'''.

===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]] '''C'''<supmath>\Complex^k</supmath> and the standard symplectic space '''R'''<supmath>\R^{2n-2k}</supmath> 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 -1−1.

===Local holomorphicity===

Near non-regular points, the above [[classification theorem]] does not apply. However, about any point, a generalized complex manifoldismanifold 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==

===Complex manifolds===

The space of complex differential forms '''Λ<supmath>\mathbf{\Lambda}^*</sup> \mathbf{T'''<math>} \otimes \Complex</math>'''C''' has a complex conjugation operation given by complex conjugation in '''C'''<math>\Complex. </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'', 0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.

(''n,0'', 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 \Complex</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 \Complex</math> defines an ordinary [[linear complex structure|complex structure]] on the tangent bundle.

'''T'''^*)<math>\otimes</math>'''C''' to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore this generalized complex structure on ('''T'''<math>\oplus</math>

'''T'''^*)<math>\otimes</math>'''C''' 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.

===Symplectic manifolds===

The pure spinor bundle generated by

:::<math>\phi=e^{i\omega}</math>

:::<math>\phi=e^{i\omega}</math>

for a nondegenerate two-form ''ω'' defines a symplectic structure on the tangent space. Thus symplectic manifolds are also generalized complex manifolds.

Thefor abovea purenondegenerate spinortwo-form is''ω'' globallydefines defined,a andsymplectic sostructure on the canonicaltangent bundle is trivialspace. This means thatThus symplectic manifolds are not onlyalso generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.

The above pure spinor is globally defined, and so the canonical bundle is trivial. This means that symplectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.

The pure spinor <math>\phi</math> is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the [[Kähler form]]. Therefore these generalized complex structures are of the same type as those corresponding to a [[scalar (mathematics)|scalar]] pure spinor. A scalar is annihilated by the entire tangent space, and so these structures are of type ''0''.

The pure spinor <math>\phi</math> is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the [[Kähler form]]. Therefore, these generalized complex structures are of the same type as those corresponding to a [[scalar (mathematics)|scalar]] pure spinor. A scalar is annihilated by the entire tangent space, and so these structures are of type ''0''.

Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called '''B-symplectic'''.

Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called '''B-symplectic'''.

==Relation to G-structures==

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 ('''T'''<math>(\mathbf{T} \oplus</math>''' \mathbf{T'''^{}^*})&nbsp;<math> \otimes \Complex</math>&nbsp;'''C''' with the above inner product is aan {{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 \Complex.</math> Generalized Kähler structures are reductions of the structure group to <math>U(n) \oplustimes 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.

'''T'''^*)<math>\otimes</math>'''C'''.

Generalized Kähler structures are reductions of the structure group to U(''n'')<math>\times</math>U(''n''). 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'')<math> \times</math> SU(''n'').</math>

===Calabi-Yau versus Calabi-&ndash;Yau metric===

Notice that a generalized Calabi metric structure, which was introduced by Marco Gualtieri, is a stronger condition than a generalized Calabi-&ndash;Yau structure, which was introduced by [[Nigel Hitchin]]. In particular a generalized Calabi-&ndash;Yau metric structure implies the existence of two commuting generalized almost complex structures.

==References==

*[[{{cite journal |authorlink=Nigel Hitchin |last=Hitchin, |first=Nigel]] [http://xxx.lanl|doi=10.gov1093/absqmath/math.DG/0209099hag025 |title=Generalized Calabi-Yau manifolds], Quart.|journal=[[Quarterly J.Math.Journal Oxfordof Ser.Mathematics]] |volume=54 (|year=2003) |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, [http:|doi=10.4007//xxxannals.lanl2011.gov/abs/math174.DG/07032981.3 |title=Generalized complex geometry |journal=[[Annals of Mathematics]], |series=(20072) |volume=174 |year=2011 |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. ReptRep. |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 |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 }}

*Robert Dijikgraaf; Sergei Gukov; Andrew Neitzke & Cumrun Vafa [http://arxiv.org/pdf/hep-th/0411073 Topological M-theory as Unification of Form Theories of Gravity]

{{String theory topics |state=collapsed}}

[[Category:Differential geometry]]