In
mathematics, a monopole is a
connection over a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
''G'' with a section of the associated
adjoint bundle In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles ha ...
.
Physical interpretation
Physically, the section can be interpreted as a ''
Higgs field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Standa ...
'', where the connection and Higgs field should satisfy the
Bogomolny equations
In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation
:F_A = \star d_A \Phi,
where F_A is the curvature of a connection A on a principal G-bundle over a 3-manifold M, \Phi is a section of the ...
and be of
finite action
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
.
See also
*
Nahm equations In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the ''Nahm transform'' – an alternative to Ward's twistor construction of monopoles. Th ...
*
Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
*
Magnetic monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
*
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
References
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Differential geometry
Mathematical physics
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