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formal Brauer group

Contents

Contents

Idea

A formal group-analog of the Brauer group. The special case of the concept of Artin-Mazur formal group for n=2n = 2. The n=1n = 1-analog is the formal Picard group.

Properties

Relation to K3K3-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.

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

References

The original account of the construction of formal Picard groups is

Modern reviews include

  • Markus Szymik, section 3 of K3 spectra (pdf)

  • Christian Liedtke, around p. 40 of Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem (arXiv.1403.2538)

Last revised on May 21, 2014 at 21:23:58. See the history of this page for a list of all contributions to it.