nLab Calabi-Yau cohomology

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:

• for $n = 0$: complex K-theory;

• for $n = 1$: elliptic cohomology;

• for $n = 2$: K3-cohomology.

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

• moduli space of Calabi-Yau spaces

References

The general idea of Calabi-Yau cohomology apparently appears in

• Michael Hopkins, lectures at Midwest Topology Seminar, 1992

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

• Hisham Sati, p. 18 of The Elliptic curves in gauge theory, string theory, and cohomology, JHEP 0603 (2006) 096 (arXiv:hep-th/0511087)

See also:

• Jumpei Nogami, On derived Calabi-Yau varieties, University of Illinois at Chicago 2010 (proquest)

For more see the references at K3-cohomology.