geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A generalization of the notion of Calabi-Yau manifold in the context of generalized complex geometry.
For $X$ a $2n$-dimensional smooth manifold, a generalized complex structure on $X$ is a reduction of the structure group of the generalized tangent bundle $T X \oplus T^* X$ along the inclusion
into the Narain group.
Recall that for $X$ an ordinary compact complex manifold of real dimension $2n$, a Calabi-Yau manifold structure on $X$ is a reduction of the structure group along the inclusion $SU(n) \hookrightarrow U(n)$ of the special unitary group into the unitary group.
A generalized Calabi-Yau structure on a generalized complex manifold $X$ is a further reduction of the structure group along
The notion was introduced in
The role of generalized CY-manifolds as (factors of) target spaces in string theory is discussed for instance in
Mariana Graña, Ruben Minasian, Michela Petrini, Alessandro Tomasiello, Type II Strings and Generalized Calabi-Yau Manifolds (arXiv:hep-th/0409176)
Jan Louis, Generalized Calabi-Yau compactifications with D-branes and fluxes, Forthschr. Phys. 53, no 7-8 (2005) (pdf)