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3d Calabi-Yau object

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)-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

Moduli spaces of line bundles

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate 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)

  • Claire Voisin (pdf)

  • 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.