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 $T X \oplus T^* X$ along the inclusion

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

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 $SU(n) \hookrightarrow U(n)$ 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

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

(Hitchin, section 4.5)

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

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)

Relation to (non-integrable) G-structure for $G =$ SU(n) (see also at MSU):

• Daniël Prins, On flux vacua, $SU(n)$-structures and generalised complex geometry, Université Claude Bernard – Lyon I, 2015. (arXiv:1602.05415, tel:[phone])

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)

• Mariana Grana, Ruben Minasian, Michela Petrini, Alessandro Tomasiello, Generalized structures of N=1 vacua, hep-th/0505212

• Jan Louis, Generalized Calabi-Yau compactifications with D-branes and fluxes, Forthschr. Phys. 53, no 7-8 (2005) (pdf)