nLab
generalized complex geometry

Idea

Generalized complex geometry is effectively the study of the Lie 2-algebroids called the standard Courant algebroids $𝔠(X)$ of manifolds $X$.

One finds (as described at standard Courant algebroid) that

• choices of sub-Lie algebroids of $𝔠(X)$ encode Dirac structure?s and – after complexification? – generalized complex structure?s;

• choices of sections of the canonical morphism $𝔠(X)\to TX$ to the tangent Lie algebroids encode generalized Riemannian metrics: pairs consisting of a (possibly pseudo-)Riemannian metric and a 2-form.

  In applications in string theory, this encodes the field of gravity and the Kalb–Ramond field, respectively. (There are also proposals for how the dilaton field appears in this context.)

In components these are structures found on the vector bundle

$$TX\oplus T^{*}X,$$T X \oplus T^* X \,,

the direct sum of the tangent bundle with the cotangent bundle of $X$.

Generalized complex geometry thus generalizes and unifies

• complex geometry?,
• symplectic geometry, and
• Riemannian geometry.

It was in particular motivated by the observation that this provides a natural formalism for describing T-duality.

References

Generalized complex geometry was propsoed by Nigel Hitchin as a formalism in differential geometry that would be suited to capture the phenomena that physicists encountered in the study of T-duality. It was later and is still developed by his students, notably Gualtieri and Cavalcanti.

A standard reference is the PhD thesis

• Marco Gualtieri, Generalized complex geometry (arXiv)

A survey set of slides with an eye towards the description of the Kalb-Ramond field and bundle gerbes is

• Nigel Hitchin, B-Fields, gerbes and generalized geometry (pdf)