# nLab K3-spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

cohomology

# Contents

## 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…).

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

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 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 cohomologyMUMR-theory

## References

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

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

Textbook account:

Lecture notes:

• 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

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