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Concept
Generalized complex structure
Summary
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 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 functionals of differential forms, a connection which formed the basis of Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke and Cumrun Vafa's 2004 proposal that 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 supersymmetric flux compactifications, which relate 10-dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.
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 vectors to M. A 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 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 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 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 signature (N, 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
A generalized almost complex structure is just an almost complex structure of the generalized tangent bundle which preserves the natural inner product:
such that and
Like in the case of an ordinary almost complex structure, a generalized almost complex structure is uniquely determined by its -eigenbundle, i.
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