Editing Magnetic monopole (section)

== Topological interpretation ==

=== Dirac string ===
{{Main|Dirac string}}
A [[gauge theory]] like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.

In electrodynamics, the group is [[U(1)]], unit complex numbers under multiplication. For infinitesimal paths, the group element is {{math|1 + ''iA''<sub>''μ''</sub>''dx''<sup>''μ''</sup>}} which implies that for finite paths parametrized by {{math|''s''}}, the group element is:
{{block indent|1=<math>\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} \, ds \right) = \exp \left( ie\int A\cdot dx \right) . </math>}}

The map from paths to group elements is called the [[Wilson loop]] or the [[holonomy]], and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

{{block indent|1=<math>e \oint_{\partial D} A\cdot dx  = e \int_D (\nabla \times A) \, dS = e \int_D B \, dS.</math>}}

So that the phase a charged particle gets when going in a loop is the [[magnetic flux]] through the loop. When a small [[solenoid]] has a magnetic flux, there are [[Aharonov–Bohm effect|interference fringes]] for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

But if all particle charges are integer multiples of {{math|''e''}}, solenoids with a flux of {{math|2{{pi}}/''e''}} have no interference fringes, because the phase factor for any charged particle is {{math|1=exp(2{{pi}}''i'') = 1}}. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of {{math|2{{pi}}/''e''}}, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.

Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.

=== Grand unified theories ===
{{main|'t Hooft–Polyakov monopole}}
In a U(1) gauge group with quantized charge, the group is a circle of radius {{math|2{{pi}}/''e''}}. Such a U(1) gauge group is called [[compact space|compact]]. Any U(1) that comes from a [[grand unified theory]] (GUT) is compact – because only compact higher gauge groups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of a large-volume gauge group, the interaction of any fixed representation goes to zero.

The case of the U(1) gauge group is a special case because all its [[irreducible representations]] are of the same size – the charge is bigger by an integer amount, but the field is still just a complex number – so that in U(1) gauge field theory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small, but each charged particle has a huge number of charge quanta so its charge stays finite. In a non-compact U(1) gauge group theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantization is an experimental certainty, it is clear that the U(1) gauge group of electromagnetism is compact.

GUTs lead to compact U(1) gauge groups, so they explain charge quantization in a way that seems logically independent from magnetic monopoles. However, the explanation is essentially the same, because in any GUT that breaks down into a U(1) gauge group at long distances, there are magnetic monopoles.

The argument is topological:
# The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to group elements infinitesimally close to the identity.
# If you imagine a big sphere in space, you can deform an infinitesimal loop that starts and ends at the north pole as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts and ends at the north pole) then let it shrink back to a little loop while going over the eastern hemisphere. This is called [[Poincaré conjecture|''lassoing the sphere'']].
# Lassoing is a sequence of loops, so the holonomy maps it to a sequence of group elements, a continuous path in the gauge group. Since the loop at the beginning of the lassoing is the same as the loop at the end, the path in the group is closed.
# If the group path associated to the lassoing procedure winds around the U(1), the sphere contains magnetic charge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere.
# Since the holonomy at the beginning and at the end is the identity, the total magnetic flux is quantized. The magnetic charge is proportional to the number of windings {{math|''N''}}, the magnetic flux through the sphere is equal to {{math|2{{pi}}''N''/''e''}}. This is the Dirac quantization condition, and it is a topological condition that demands that the long distance U(1) gauge field configurations be consistent.
# When the U(1) gauge group comes from breaking a [[compact Lie group]], the path that winds around the U(1) group enough times is topologically trivial in the big group. In a non-U(1) compact Lie group, the [[covering space]] is a [[Lie group]] with the same [[Lie algebra]], but where all closed loops are [[contractible]]. Lie groups are homogeneous, so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the covering group ends at {{math|''P''}}, which is a lift of the identity. Going around the loop twice gets you to {{math|''P''<sup>2</sup>}}, three times to {{math|''P''<sup>3</sup>}}, all lifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. This number of times one has to traverse the loop to make it contractible is small, for example if the GUT group is SO(3), the covering group is SU(2), and going around any loop twice is enough.
# This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopole configuration to unwind itself at short distances, at the cost of not staying in the U(1). To do this with as little energy as possible, you should leave only the U(1) gauge group in the neighborhood of one point, which is called the '''core''' of the monopole. Outside the core, the monopole has only magnetic field energy.

Hence, the Dirac monopole is a [[topological defect]] in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity – the core shrinks to a point. But when there is some sort of short-distance regulator on spacetime, the monopoles have a finite mass. Monopoles occur in [[lattice gauge theory|lattice U(1)]], and there the core size is the lattice size. In general, they  are expected to occur whenever there is a short-distance regulator.

=== String theory ===
In the universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by [[Hawking radiation]], the lightest charged particles cannot be too heavy.<ref>{{Cite journal |arxiv = hep-th/0601001|doi = 10.1088/1126-6708/2007/06/060|bibcode = 2007JHEP...06..060A|title = The string landscape, black holes and gravity as the weakest force|year = 2007|last1 = Arkani-Hamed|first1 = Nima|last2 = Motl|first2 = Luboš|last3 = Nicolis|first3 = Alberto|last4 = Vafa|first4 = Cumrun|journal = Journal of High Energy Physics|volume = 2007|issue = 6|pages = 060|s2cid = 16415027}}</ref> The lightest monopole should have a mass less than or comparable to its charge in [[natural units]].

So in a consistent holographic theory, of which [[string theory]] is the only known example, there are always finite-mass monopoles. For ordinary electromagnetism, the upper mass bound is not very useful because it is about same size as the [[Planck mass]].

=== Mathematical formulation ===
{{unreferenced section|date=January 2021}}
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)}}.