nLab
line bundle

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Cohomology

cohomology

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Special notions

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Contents

Definition

A line bundle is a vector bundle of rank (or dimension) 11, i.e. a vector bundle whose typical fiber is a 11-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 LL is a weak inverse of LL 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,)H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z} is the pullback bundle of the universal complex line bundle (example 5) along the map

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

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

Beware that this is not the “canonical line bundle” from example 3, 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 BU(1)B 2B 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)U(1) on \mathbb{C} is the universal complex line bundle.

Example

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

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-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=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 May 29, 2017 15:22:01 by Urs Schreiber (178.6.236.87)