Contents

# Contents

## 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

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)-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)$

### Moduli spaces of line bundles

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

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

## References

### Intermediate Jacobian

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

• 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)

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

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

### Hall algebra

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

Last revised on March 30, 2019 at 10:00:17. See the history of this page for a list of all contributions to it.