nLab
Calabi-Yau cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Calabi-Yau cohomology is complex oriented generalized cohomology theory whose associated formal group is an Artin-Mazur formal group Φ X n\Phi^n_X of a Calabi-Yau variety XX of dimension nn. (Which means e.g. complex dimension if working over the complex numbers).

As special cases this includes:

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

The general idea of Calabi-Yau cohomology apparently appears in

The suggestion that from the point of view of string theory/F-theory K3-cohomology and further Calabi-Yau cohomology this is the required generalization of elliptic cohomology appears in

Last revised on July 19, 2015 at 18:19:27. See the history of this page for a list of all contributions to it.