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generalized Calabi-Yau manifold

Contents

Context

Complex geometry

Manifolds and cobordisms

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)

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

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

Last revised on June 11, 2019 at 14:58:19. See the history of this page for a list of all contributions to it.