geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Generalized complex geometry is the study of the geometry of symplectic Lie 2-algebroid called standard Courant algebroids $\mathfrak{c}(X)$ (over a smooth manifold $X$).
This geometry of symplectic Lie 2-algebroids turns out to unify, among other things, complex geometry with symplectic geometry. This unification notably captures central aspects of T-duality.
Let $V$ be a finite dimensional vector space over the real numbers.
Recall that a complex structure on $V$ is a linear map
such that $J \circ J = - id_V$. And a symplectic structure on $V$ is equivalently a linear isomorphism
such that
where $V^*$ denotes the dual vector space and $\omega^*$ the dual linear map.
The following definition may be thought of as combining these two concepts.
A generalized complex structure on $V$ is a linear map
(an endomorphism of the direct sum of $V$ with its dual vector space)
such that it is both
a complex structure on $V \oplus V^*$ in that $\mathcal{J}^2 = - id$;
a symplectic structure on $V \oplus V^*$ in that $\mathcal{J}^* = - \mathcal{J}$.
The following shows that this is indeed a joint generalization of complex and symplectic structures.
Let $J : V \to V$ be an ordinary complex structure on $V$. Then the linear endomorphism of $V \oplus V^*$ defined by matrix calculus as
is a generalized complex structure on $V$.
Similarly, let $\omega : V \to V^*$ be an ordinary symplectic structure on $V$. then the endomorphism
is a generalized complex structure on $V$.
A generalized complex structure on a manifold is a generalized complex structure on the fibers of the generalized tangent bundle.
(…)
A generalized complex structure on $V \oplus V^*$ is equivalently a reduction of the structure group along the inclusion
where the left hand is identified as $U(n,n) = O(2n,2n) \cap GL(2n, \mathbb{C})$ (Gualtieri, prop. 4.6).
The analog of this with the unitary group replaced by the orthogonal group yields type II geometry.
One finds (as described at standard Courant algebroid) that
choices of sub-Lie algebroids of $\mathfrak{c}(X)$ encode (almost) Dirac structures and – after complexification – generalized complex structures;
choices of sections of the canonical morphism $\mathfrak{c}(X) \to T X$ 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
the direct sum of the tangent bundle with the cotangent bundle of $X$.
Generalized complex geometry thus generalizes and unifies
It was in particular motivated by the observation that this provides a natural formalism for describing T-duality.
Generalized complex geometry was proposed 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
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 Oxford Durham Symposium 2005 (pdf)
Nigel Hitchin, Lectures on generalized geometry (arxiv/1008.0973)
Generalized complex structures may serve as target spaces for sigma-models. Relations to the Poisson sigma-model and the Courant sigma-model are discussed in
Generalized complex geometry and variant of exceptional generalized complex geometry are natural for describing supergravity background compactifications in string theory with their T-duality and U-duality symmetries.
Ian Ellwood, NS-NS fluxes in Hitchin’s generalized geometry (arXiv:hep-th/0612100)
Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)
Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, generalized geometry and non-geometric backgrounds, J. High Energy Phys. 2009, no. 4, 075, 39 pp. arXiv:0807.4527 MR2010i:81323 doi
David Andriot, Ruben Minasian, Michela Petrini, Flux backgrounds from twists, J. High Energy Phys. 2009, no. 12, 028 arXiv:0903.0633 MR2011c:81201 doi