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| {{technical|date=June 2012}}
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| {{Refimprove|date=November 2013}}
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| }}
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| The '''Bogomol'nyi–Prasad–Sommerfield bound''' (named after [[Eugène Bogomolny]],<ref>E. B. Bogomolny, Sov.J.Nucl.Phys. 24 (1976) 449; Yad.Fiz. 24 (1976) 861</ref> [[Manoj Prasad]], and [[Charles Sommerfield]]<ref>M.K. Prasad & C. M. Sommerfield, Phys.Rev.Lett. 35 (1975) 760.</ref>) is a series of [[inequality (mathematics)|inequalities]] for solutions of [[partial differential equation]]s depending on the [[homotopy class]] of the solution at infinity. This set of inequalities is very useful for solving [[soliton (topological)|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, the Bogomol'nyi equations. Solutions saturating the bound are called '''BPS states''' and play an important role in field theory and [[string theory]].
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| Examples:
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| *[[Instanton]].
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| *''Incomplete:'' [[Yang-Mills-Higgs equations|Yang-Mills-Higgs partial differential equations]].
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| The energy at a given time ''t'' is given by
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| :<math>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]</math>
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| 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
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| :<math> | |
| \begin{align}
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| 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] \\
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| & \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] \\
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| & \geq \pm \frac{1}{g}\int d^3x\, \operatorname{Tr}\left[\overrightarrow{D\varphi}\cdot \vec{B}\right] \\
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| & = \pm\frac{1}{g}\int_{S^2\ \mathrm{boundary}} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right].
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| \end{align}
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| </math>
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| Therefore,
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| :<math>E\geq \left\|\int_{S^2} \operatorname{Tr}\left[\varphi \vec{B}\cdot d\vec{S}\right]\right \|.</math> | |
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| Saturation happens when <math>\pi = 0</math> and
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| :<math>\overrightarrow{D\varphi}\mp\frac{1}{g}\vec{B} = 0</math>
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| The Bogomol'nyi equation. The other condition for saturation is the Higgs mass and self-interaction are zero, which is the case in N=2 supersymmetric theories.
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| This quantity is the absolute value of the [[magnetic flux]].
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| A slight generalization applying to dyons also exists. For that, the Higgs field needs to be a complex adjoint, not a real adjoint.
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| ==Supersymmetry==
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| 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 [[Group extension%23Central extension|central extension]], which is typically a [[topological charge]].<ref>Weinberg, Steven (2000). ''The Quantum Theory of Fields: Volume 3,'' p 53. Cambridge University Press, Cambridge. ISBN 0521660009.</ref>
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| In fact, most bosonic BPS bounds actually come from the bosonic sector of a supersymmetric theory and this explains their origin.
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Bogomol'nyi-Prasad-Sommerfield bound}}
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| [[Category:Partial differential equations]]
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| [[Category:Quantum field theory]]
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| [[Category:Solitons]]
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