geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
manifolds and cobordisms
cobordism theory, Introduction
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
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
The notion was introduced in
Relation to (non-integrable) G-structure for $G =$ SU(n) (see also at MSU):
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)
Last revised on December 18, 2020 at 12:01:08. See the history of this page for a list of all contributions to it.