Editing Magnetic monopole (section)

=== Mathematical formulation ===
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In mathematics, a (classical) gauge field is defined as a [[connection form|connection]] over a [[principal bundle|principal G-bundle]] over spacetime. {{math|G}} is the gauge group, and it acts on each fiber of the bundle separately.

A ''connection'' on a {{math|G}}-bundle tells you how to glue fibers together at nearby points of {{math|M}}. It starts with a continuous symmetry group {{math|G}} that acts on the fiber {{math|F}}, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by having the {{math|G}} element associated to a path act on the fiber {{math|F}}.

In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of [[characteristic class]]es in [[algebraic topology]] is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over ''any'' connection over it. Note that a connection over a trivial bundle can never give us a nontrivial principal bundle.

If spacetime is <math>\mathbb{R}^4</math> the space of all possible connections of the {{math|G}}-bundle is [[connected space|connected]]. But consider what happens when we remove a [[timelike]] [[worldline]] from spacetime. The resulting spacetime is [[homotopy|homotopically equivalent]] to the [[topological sphere]] {{math|''S''<sup>2</sup>}}.

A principal {{math|G}}-bundle over {{math|''S''<sup>2</sup>}} is defined by covering {{math|''S''<sup>2</sup>}} by two [[chart (topology)|charts]], each [[homeomorphic]] to the open 2-ball such that their intersection is homeomorphic to the strip {{math|''S''<sup>1</sup>×''I''}}. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle {{math|''S''<sup>1</sup>}}.  So a topological classification of the possible connections is reduced to classifying the transition functions. The transition function maps the strip to {{math|G}}, and the different ways of mapping a strip into {{math|G}} are given by the first [[homotopy group]] of {{math|G}}.

So in the {{math|G}}-bundle formulation, a gauge theory admits Dirac monopoles provided {{math|G}} is not [[simply connected]], whenever there are paths that go around the group that cannot be deformed to a constant path (a path whose image consists of a single point). U(1), which has quantized charges, is not simply connected and can have Dirac monopoles while <math>\mathbb{R}</math>, its [[universal covering group]], ''is'' simply connected, doesn't have quantized charges and does not admit Dirac monopoles. The mathematical definition is equivalent to the physics definition provided that—following Dirac—gauge fields are allowed that are defined only patch-wise, and the gauge field on different patches are glued after a gauge transformation.

The total magnetic flux is none other than the first [[Chern number]] of the principal bundle, and depends only upon the choice of the principal bundle, and not the specific connection over it. In other words, it is a topological invariant.

This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to {{math|''d'' + 1}} dimensions with {{math|''d'' ≥ 2}} in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension {{math|''d'' − 3}}. Another way is to examine the type of topological singularity at a point with the homotopy group {{math|{{pi}}<sub>''d''−2</sub>(G)}}.