Circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle .
Oriented circle bundles are also known as principal U(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.
As 3-manifolds
Circle bundles over surfaces are an important example of 3-manifolds. A more general class of 3-manifolds is Seifert fiber spaces, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional orbifold.
Relationship to electrodynamics
The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with being cohomologous to zero. In particular, there always exists a 1-form A, the electromagnetic four-potential, (equivalently, the affine connection) such that
Given a circle bundle P over M and its projection
one has the homomorphism
where is the pullback. Each homomorphism corresponds to a Dirac monopole; the integer cohomology groups correspond to the quantization of the electric charge. The Bohm-Aharonov effect can be understood as the holonomy of the connection on the associated line bundle describing the electron wave-function. In essence, the Bohm-Aharonov effect is not a quantum-mechanical effect (contrary to popular belief), as no quantization is involved or required in the construction of the fiber bundles or connections.
Examples
- The Hopf fibration is an example of a non-trivial circle bundle.
- The unit normal bundle of a surface is another example of a circle bundle.
- The unit normal bundle of a non-orientable surface is a circle bundle that is not a principal bundle. Only orientable surfaces have principal unit tangent bundles.
- Another method for constructing circle bundles is using a complex line bundle and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from we have that it is a principal -bundle.[1] Moreover, the characteristic classes from Chern-Weil theory of the -bundle agree with the characteristic classes of .
- For example, consider the analytification a complex plane curve
Since and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf has Chern class .
Classification
The isomorphism classes of principal -bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps , where is called the classifying space for U(1). Note that is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space Such bundles are classified by an element of the second integral cohomology group of M, since
- .
This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because a circle is homotopically equivalent to , the complex plane with the origin removed; and so a complex line bundle with the zero section removed is homotopically equivalent to a circle bundle.)
A circle bundle is a principal bundle if and only if the associated map is null-homotopic, which is true if and only if the bundle is fibrewise orientable. Thus, for the more general case, where the circle bundle over M might not be orientable, the isomorphism classes are in one-to-one correspondence with the homotopy classes of maps . This follows from the extension of groups, , where .
Deligne complexes
The above classification only applies to circle bundles in general; the corresponding classification for smooth circle bundles, or, say, the circle bundles with an affine connection requires a more complex cohomology theory. Results include that the smooth circle bundles are classified by the second Deligne cohomology ; circle bundles with an affine connection are classified by while classifies line bundle gerbes.
See also
References
- ^ https://mathoverflow.net/questions/144092/is-every-orientable-circle-bundle-principal.
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- Chern, Shiing-shen (1977), "Circle bundles", Lecture Notes in Mathematics, vol. 597/1977, Springer Berlin/Heidelberg, pp. 114–131, ISBN 978-3-540-08345-0.