Английская Википедия:Bogomolny equations
Шаблон:Short description In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation
- <math>F_A = \star d_A \Phi,</math>
where <math>F_A</math> is the curvature of a connection <math>A</math> on a principal <math>G</math>-bundle over a 3-manifold <math>M</math>, <math>\Phi</math> is a section of the corresponding adjoint bundle, <math>d_A</math> is the exterior covariant derivative induced by <math>A</math> on the adjoint bundle, and <math>\star</math> is the Hodge star operator on <math>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>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
Шаблон:Mathapplied-stub
Шаблон:Differential-geometry-stub
