nLab
generalized Calabi-Yau manifold

Idea

A generalization of the notion of Calabi-Yau manifold in the context of generalized complex geometry.

Definition

In terms of $G$-structure

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 $TX\oplus T^{*}X$ along the inclusion

$$U(n,n)\hookrightarrow O(2n,2n)$$U(n,n) \hookrightarrow O(2n,2n)

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 $\mathrm{SU}(n)\hookrightarrow U(n)$ 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

$$\mathrm{SU}(n,n)\hookrightarrow U(n,n)\hookrightarrow O(2n,2n).$$SU(n,n) \hookrightarrow U(n,n) \hookrightarrow O(2n,2n) \,.

(Hitchin, section 4.5)

Related concepts

• Calabi-Yau object

  • Calabi-Yau algebra, Calabi-Yau manifold, generalized Calabi-Yau manifold

• supersymmetry and Calabi-Yau manifolds

• generalized G2-manifold

References

The notion was introduced in

• Nigel Hitchin, Generalised Calabi-Yau Manifolds, Quart. J. Math. Oxford Ser. 54:281-308, (2003) (math.DG/0209099)

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)