Contents

cohomology

# Contents

## Idea

Calabi-Yau cohomology is complex oriented generalized cohomology theory whose associated formal group is an Artin-Mazur formal group $\Phi^n_X$ of a Calabi-Yau variety $X$ of dimension $n$. (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 $n$-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

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.