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. 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. The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with being cohomologous to zero, i.e. exact. 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 Aharonov–Bohm effect can be understood as the holonomy of the connection on the associated line bundle describing the electron wave-function. In essence, the Aharonov–Bohm 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. The Hopf fibration is an example of a non-trivial circle bundle. The unit tangent bundle of a surface is another example of a circle bundle. The unit tangent 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. Moreover, the characteristic classes from Chern-Weil theory of the -bundle agree with the characteristic classes of .