Contents

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 comple 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 May 29, 2017 at 15:22:01. See the history of this page for a list of all contributions to it.