Generalized complex manifold

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 Nietzke 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.