Generalized complex structure: Difference between revisions

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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==

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

===Calabi 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==

*[[Nigel Hitchin|Hitchin, Nigel]] [http://xxxdx.lanldoi.govorg/abs/math10.DG1093/qmath/0209099hag025 Generalized Calabi-Yau manifolds], Quart.[[Quarterly J.Math.Journal Oxfordof Ser.Mathematics]] ''54'' (2003), no. 3, 281-&ndash;308.

*Gualtieri, Marco, [http://xxx.lanl.gov/abs/math.DG/0401221 Generalized complex geometry], PhD Thesis (2004).

*Gualtieri, Marco, [http://xxx.lanl.gov/abs/math.DG/0703298 Generalized complex geometry], (2007).