Bogomol'nyi–Prasad–Sommerfield bound

In a theory of non-abelian Yang–Mills–Higgs, the energy at a given time t is given by

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

where ${\displaystyle \pi }$ is the covariant derivative of the Higgs field 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, by virtue of the Yang–Mills Bianchi identity,

${\displaystyle {\begin{aligned}E&\geq \int d^{3}x\,\left[{\frac {1}{2}}\operatorname {Tr} \left[{\overrightarrow {D\varphi }}\cdot {\overrightarrow {D\varphi }}\right]+{\frac {1}{2g^{2}}}\operatorname {Tr} \left[{\vec {B}}\cdot {\vec {B}}\right]\right]\\&\geq \int d^{3}x\,\operatorname {Tr} \left[{\frac {1}{2}}\left({\overrightarrow {D\varphi }}\mp {\frac {1}{g}}{\vec {B}}\right)^{2}\pm {\frac {1}{g}}{\overrightarrow {D\varphi }}\cdot {\vec {B}}\right]\\&\geq \pm {\frac {1}{g}}\int d^{3}x\,\operatorname {Tr} \left[{\overrightarrow {D\varphi }}\cdot {\vec {B}}\right]\\&=\pm {\frac {1}{g}}\int _{S^{2}\ \mathrm {boundary} }\operatorname {Tr} \left[\varphi {\vec {B}}\cdot d{\vec {S}}\right].\end{aligned}}}$

Therefore,

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

Saturation of the inequality is obtained when the Bogomolny equations are satisfied.

${\displaystyle {\overrightarrow {D\varphi }}\mp {\frac {1}{g}}{\vec {B}}=0,}$

The other condition for saturation is that the Higgs mass and self-interaction are zero, which is the case in N=2 supersymmetric theories.

This quantity is the absolute value of the magnetic flux.

A slight generalization applying to dyons also exists. For that, the Higgs field needs to be a complex adjoint, not a real adjoint.

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.[3]

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