nLab 3d Calabi-Yau object

Idea

A Calabi-Yau object (Calabi-Yau manifold, Calabi-Yau category) of (complex) dimension 3.

Properties

As exceptional geometry

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

Moduli spaces of line bundles

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

Related concepts

• Calabi-Yau cohomology

References

Algebraic topology

Discussion of the algebraic topology (e.g. the topological K-theory) of CY3s:

• Charles F. Doran, John W. Morgan: Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties, Geom. Topol. 11 (2007) 597-642 [arXiv:math/0605074, arXiv:10.2140/gt.2007.11.597]

In this context, generally on stable splitting of closed 6-manifolds:

• Ruizhi Huang: Suspension homotopy of 6-manifolds, Algebr. Geom. Topol. 23 (2023) 439-460 [arXiv:2104.04994, doi:10.2140/agt.2023.23.439]

Intermediate Jacobian

Discussion of intermediate Jacobians of Calabi-Yau 3-folds:

• C. Herbert Clemens, Phillip Griffith, The intermediate Jacobian of the cubic threefold, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 (JSTOR)

• Claire Voisin (pdf)

• Andreas Höring, Minimal classes on the intermediate Jacobian of a generic cubic threefold, 2008 (pdf)

Discussion of the Artin-Mazur formal groups of CY3s in positive characteristic:

• Gerard van der Geer, T. Katsura, On the height of Calabi-Yau varieties in positive characteristic (arXiv:math/0302023)

Hall algebra

Discussion of motivic Hall algebras of CY 3-folds is in

• Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (arXiv:0811.2435)