nLab line bundle

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 $X$.

Related concepts

• complex vector space

  • complex line

• complex vector bundle

  • complex line bundle

• holomorphic line bundle, algebraic line bundle

• theta characteristic

• cubical structure on a line bundle

• fusion bundle

• bundle gerbe

• circle n-bundle with connection

• line 2-bundle

• line n-bundle

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

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian