bundles

cohomology

# Contents

## Definition

A line bundle is a vector bundle of rank (or dimension) $1$, i.e. a vector bundle whose typical fiber is a $1$-dimensional vector space (a line).

For complex vector bundles, complex line bundles are canonically associated bundles of circle group-principal bundles.

## Properties

The class of line bundles has a nicer behaviour (in some ways) than the class of vector bundles in general. In particular, the dual? of a line bundle $L$ is a weak inverse of $L$ under the tensor product of line bundles. Thus the isomorphism classes of line bundles form a group.

## Examples

###### Example

Over any manifold there is canonically the density line bundle which is the associated bundle to the principal bundle underlying the tangent bundle by the determinant homomorphism.

###### Example

The product of any space $X$ with the moduli stack $Pic_X$ of line bundles over it (its Picard stack) carries a tautological line bundle. This is called the Poincaré line bundle of $X$.

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$Calabi-Cau 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

Revised on May 31, 2014 06:00:38 by Urs Schreiber (88.128.80.68)