Generalized complex structure: Difference between revisions

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Revision as of 15:08, 3 December 2019

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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 \Complex.</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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==References==

*[[Nigel Hitchin|Hitchin, Nigel]] [https://dx.doi.org/10.1093/qmath/hag025 Generalized Calabi-Yau manifolds], ''[[Quarterly Journal of Mathematics]]'' '''54''' (2003), no. 3, 281&ndash;308.

*Gualtieri, Marco, [https://arxiv.org/abs/math.DG/0401221 Generalized complex geometry], PhD Thesis (2004).

*Gualtieri, Marco, [https://dx.doi.org/10.4007/annals.2011.174.1.3 Generalized complex geometry], ''[[Annals of Mathematics]]'' (2) '''174''' (2011), no. 1, 75&ndash;123.

*Graña, Mariana, [https://arxiv.org/abs/abs/hep-th/0509003 Flux compactifications in string theory: a comprehensive review], ''Phys. Rep.'' 423 (2006) 91-15891–158.

*[[Robbert Dijkgraaf]], [[Sergei Gukov]], Andrew Neitzke, and [[Cumrun Vafa]], [http://projecteuclid-org.ezproxy.neu.edu/euclid.atmp/1144070454 Topological M-theory as unification of form theories of gravity], ''[[Advances in Theoretical and Mathematical Physics]]'' '''9''' (2005), no. 4, 603&ndash;665.

{{String theory topics |state=collapsed}}