Generalized complex structure: Difference between revisions

Revision as of 08:59, 9 January 2019

Revision as of 13:33, 11 May 2019

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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:

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===Type===

The '''type''' of a maximal isotropic subbundle <math>L(\mathbf{E}, \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}, \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-Ramond field|B-field]], which are [[isometry|isometries]] of <math>\mathbf{T} \oplus \mathbf{T}^*</math> of the form

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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===

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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 O(2''n'',&nbsp;2''n'') structure. A generalized almost complex structure is a reduction of this structure to a 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>