Monopole (mathematics)

In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.

Physical interpretation

Physically, such a monopole can be interpreted in a gauge theory coupled to a scalar field as a configuration of the scalar and gauge fields which satisfies the Bogomolny equations and has finite action. Due to the presence of a scalar field, this monopole is an example of an 't Hooft–Polyakov monopole^[1] and should not be confused with the singular monopole solutions to Maxwell's equations which are mathematically described by nontrivial principal bundles.

See also

• Nahm equations
• Instanton
• Magnetic monopole
• Yang–Mills theory

References

1. Katanaev (2023). "'t Hooft–Polyakov monopoles and a general spherically symmetric solution of the Bogomolny equations". Mod. Phys. Lett. A. 38. doi:10.1142/S0217732323500827.

• Hitchin, Nigel (1983). "On the construction of monopoles". Communications in Mathematical Physics. 89 (2): 145–190. Bibcode:1983CMaPh..89..145H. doi:10.1007/BF01211826. S2CID 120823242.
• Donaldson, Simon (1984). "Nahm's equations and the classification of monopoles". Communications in Mathematical Physics. 96 (3): 387–407. Bibcode:1984CMaPh..96..387D. doi:10.1007/BF01214583. S2CID 119959346.
• Atiyah, Michael; Hitchin, N. J. (1988). The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton, NJ: Princeton University Press. ISBN 0-691-08480-7.