Bogomolny equations
In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation
- [math]\displaystyle{ F_A = \star d_A \Phi, }[/math]
where [math]\displaystyle{ F_A }[/math] is the curvature of a connection [math]\displaystyle{ A }[/math] on a principal [math]\displaystyle{ G }[/math]-bundle over a 3-manifold [math]\displaystyle{ M }[/math], [math]\displaystyle{ \Phi }[/math] is a section of the corresponding adjoint bundle, [math]\displaystyle{ d_A }[/math] is the exterior covariant derivative induced by [math]\displaystyle{ A }[/math] on the adjoint bundle, and [math]\displaystyle{ \star }[/math] is the Hodge star operator on [math]\displaystyle{ M }[/math]. These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.[1][2]
The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If [math]\displaystyle{ M }[/math] is closed, there are only trivial (i.e. flat) solutions.
See also
- Monopole moduli space
- Ginzburg–Landau theory
- Seiberg–Witten theory
- Bogomol'nyi–Prasad–Sommerfield bound
References
- ↑ Atiyah, Michael; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, ISBN 978-0-691-08480-0
- ↑ Hitchin, N. J. (1982), "Monopoles and geodesics", Communications in Mathematical Physics 83 (4): 579–602, doi:10.1007/bf01208717, ISSN 0010-3616, Bibcode: 1982CMaPh..83..579H, https://projecteuclid.org/download/pdf_1/euclid.cmp/1103920970
- Hazewinkel, Michiel, ed. (2001), "Magnetic_monopole", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
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