group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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).
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 $X$
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 |
Discussion of the formal groups given as the formal Brauer groups of K3-surfaces originates in
based on
The general idea of Calabi-Yau cohomology apparently appears in
Textbook account:
Lecture notes:
See also:
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
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:
Last revised on January 20, 2021 at 13:22:38. See the history of this page for a list of all contributions to it.