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K3-spectrum

Context

Stable Homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 Φ X 2\Phi^2_{X} of K3 surfaces XX have height in {1,2,3,4,5,6,7,8,9,10,}\{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 Φ X 2\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 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

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex 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

A further discussion of K3-cohomology appears in Chapter 10 of

Lecture notes include

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

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;

The concepts of K3-spectrum as such as considered in

Revised on July 20, 2015 06:25:13 by Urs Schreiber (94.118.135.251)