Bogomolny equations

Template:Short description In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation

$F_A=\star d_A\mathrm{\Phi },$

where $F_A$ is the curvature of a connection $A$ on a principal $G$-bundle over a 3-manifold $M$, $\mathrm{\Phi }$ is a section of the corresponding adjoint bundle, $d_A$ is the exterior covariant derivative induced by $A$ on the adjoint bundle, and $\star$ is the Hodge star operator on $M$. These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.

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 $M$ 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

External links

• Bogomolny equation on nLab
• Template:Eom

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