Contents

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 vector bundle 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

The Möbius strip is the unique, up to isomorphism, non-trivial real line bundle over the circle.

###### 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.

Similarly:

###### Example

Every orientable complex manifold carries a complex line bundle of top-degree holomorphic differential forms. This is called its canonical line bundle.

###### Example

The line bundle on the 2-sphere whose first Chern class is a generator of $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$ is the pullback bundle of the universal complex line bundle (example ) along the map

$S^2 \longrightarrow B U(1) \simeq B^2 \mathbb{Z}$

which itself represents the generator in the second homotopy group $\pi_2(S^2) \simeq \mathbb{Z}$.

Beware that this is not the “canonical line bundle” from example , but “half” of it, its theta characteristic. See at geometric quantization of the 2-sphere for more on this.

###### Example

The classifying space/Eilenberg-MacLane space $B U(1)$$\simeq B^2 \mathbb{Z}$ carries the circle group-universal principal bundle. The corresponding associated bundle via the canonical action of $U(1)$ on $\mathbb{C}$ is the universal complex line bundle.

###### 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-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
$n = 0$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

Last revised on March 5, 2024 at 00:09:08. See the history of this page for a list of all contributions to it.