nLab
line n-bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For 𝔾 m\mathbb{G}_m the multiplicative group, a B n1𝔾 m\mathbf{B}^{n-1}\mathbb{G}_m principal infinity-bundle, modulated by a cocycle XB n𝔾 mX\to \mathbf{B}^n \mathbb{G}_m, classified by a cohomology class in in H n(X,𝔾 m)H^n(X,\mathbb{G}_m).

Examples

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

Revised on September 4, 2014 06:59:17 by Urs Schreiber (141.0.8.142)