nLab
generalized Calabi-Yau manifold

Contents

Idea

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

Definition

In terms of GG-structure

For XX a 2n2n-dimensional smooth manifold, a generalized complex structure on XX is a reduction of the structure group of the generalized tangent bundle TXT *XT X \oplus T^* X along the inclusion

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

into the Narain group.

Recall that for XX an ordinary compact complex manifold of real dimension 2n2n, a Calabi-Yau manifold structure on XX is a reduction of the structure group along the inclusion SU(n)U(n)SU(n) \hookrightarrow U(n) of the special unitary group into the unitary group.

A generalized Calabi-Yau structure on a generalized complex manifold XX is a further reduction of the structure group along

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

(Hitchin, section 4.5)

References

The notion was introduced in

The role of generalized CY-manifolds as (factors of) target spaces in string theory is discussed for instance in

Revised on December 14, 2012 07:37:41 by Urs Schreiber (71.195.68.239)