Bogomol'nyi-Prasad-Sommerfield bound

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The Bogomol'nyi-Prasad-Sommerfeld bound is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied (called "saturated"), one can come up with a simpler set of partial differential equations to solve. Solutions saturating the bound are called BPS states and play an important role in string theory.^[1]

Examples:

See instanton.
Incomplete: Yang-Mills-Higgs partial differential equations.

The energy at a given time t is given by

$E=\int d^3x\, \left[ \frac{1}{2}\overrightarrow{D\varphi}^T \cdot \overrightarrow{D\varphi} +\frac{1}{2}\pi^T \pi + V(\varphi) + \frac{1}{2g^2}\operatorname{Tr}\left[\vec{E}\cdot\vec{E}+\vec{B}\cdot\vec{B}\right]\right]$

where D is the covariant derivative and V is the potential. If we assume that V is nonnegative and is zero only for the Higgs vacuum and that the Higgs field is in the adjoint representation, then

$E \geq \int d^3x\, \left[ \frac{1}{2}\operatorname{Tr}\left[\overrightarrow{D\varphi} \cdot \overrightarrow{D\varphi}\right] + \frac{1}{2g^2}\operatorname{Tr}\left[\vec{B}\cdot\vec{B}\right] \right]$

$\geq \int d^3x\, \operatorname{Tr}\left[ \frac{1}{2}\left(\overrightarrow{D\varphi}\mp\frac{1}{g}\vec{B}\right)^2 \pm\frac{1}{g}\overrightarrow{D\varphi}\cdot \vec{B}\right]$

$\geq \pm \frac{1}{g}\int d^3x\, \operatorname{Tr}\left[\overrightarrow{D\varphi}\cdot \vec{B}\right]$

$= \pm\frac{1}{g}\int_{S^2\ \mathrm{boundary}} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right].$

Therefore,

$E\geq \left\|\int_{S^2} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right]\right \|.$

This quantity is the absolute value of the magnetic flux.

[edit] Supersymmetry

In supersymmetry, the BPS bound is saturated when half (or a quarter or an eighth) of the SUSY generators are unbroken. This happens when the mass is equal to the central extension, which is typically a topological charge.

In fact, most bosonic BPS bounds actually come from the bosonic sector of a supersymmetric theory and this explains their origin.