nLab K3-spectrum

Idea

In higher dimensional analogy of how the formal Picard group of an elliptic curve gives the formal group of an elliptic spectrum representing an elliptic cohomology theory, so the formal Brauer group of a K3 surface gives the formal group of an complex oriented cohomology theory given by a spectrum hence called a K3-spectrum representing K3-cohomology (Szymik 10, section 4.2).

Properties

Existence

The formal Brauer groups $\Phi^2_{X}$ of K3 surfaces $X$ have height in $\{1,2,3,4,5,6,7,8,9,10,\infty\}$, and all values appear. (Artin 74, Artin-Mazur 77).

By the Landweber exact functor theorem there is a K3-spectrum associated with $\Phi^2_X$ if it is Landweber exact.

(Szymik 10, theorem 1) gives sufficient conditions for this to be the case and (Szymik 10, prop. 7, prop. 8) say that these condition are satisfied for enough K3 surfaces to realize all formal Brauer groups (…add details…).

Related concepts

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

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 spectrum
$n$	$n$th Morava K-theory	$K(n)$
	$n$th Morava E-theory	$E(n)$	BPR-theory
$n+1$	algebraic K-theory applied to chrom. level $n$	$K(E_n)$ (red-shift conjecture)
$\infty$	complex cobordism cohomology	MU	MR-theory

References

Discussion of the formal groups given as the formal Brauer groups of K3-surfaces originates in

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 numdam, MR56:15663

based on

• Michael Artin, Supersingular K3 Surfaces, Annal. Sc. d, l’Éc Norm. Sup. 4e séries, T. 7, fasc. 4, 1974, pp. 543-568

The general idea of Calabi-Yau cohomology apparently appears in

• Michael Hopkins, lectures at Midwest Topology Seminar, 1992

Textbook account:

• Charles Thomas, Chapter 10 of: Elliptic cohomology, Kluwer Academic, 2002 (doi:10.1007/b115001, pdf)

Lecture notes:

• Paul VanKoughnett, K3 cohomology theories, 2014 (pdf)

See also:

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

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)

A discussion of some kind of K3-cohomology in terms of differential geometry appears in

• Jorge Devoto, Quaternionic elliptic objects and K3-cohomology London Mathematical Society Lecture Note Series (No. 342) 05/2007 (doi:10.1017/CBO9780511721489.004)

The concepts of K3-spectrum as such as considered in

• Markus Szymik, K3 spectra, Bull. Lond. Math. Soc. 42 (2010) 137-148 (pdf, publisher)

• Markus Szymik, Crystals and derived local moduli for ordinary K3 surfaces, Adv. Math. 228 (2011) 1-21

See also:

• Oron Y. Propp, Constructing explicit K3-spectra (arXiv:1810.08953)