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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
In theoretical physics , massive representations of an extended supersymmetry algebra called BPS states have mass equal to the supersymmetry central charge Z . Quantum mechanically, if the supersymmetry remains unbroken, exact equality to the modulus of Z exists. Their importance arises as the supermultiplets shorten for generic massive representations, with stability and mass formula exact.
d = 4 N = 2[ edit ]
The generators for the odd part of the superalgebra have relations:[1]
{
Q
α
A
,
Q
¯
β
˙
B
}
=
2
σ
α
β
˙
m
P
m
δ
B
A
{
Q
α
A
,
Q
β
B
}
=
2
ϵ
α
β
ϵ
A
B
Z
¯
{
Q
¯
α
˙
A
,
Q
¯
β
˙
B
}
=
−
2
ϵ
α
˙
β
˙
ϵ
A
B
Z
{\displaystyle {\begin{aligned}\{Q_{\alpha }^{A},{\bar {Q}}_{{\dot {\beta }}B}\}&=2\sigma _{\alpha {\dot {\beta }}}^{m}P_{m}\delta _{B}^{A}\\\{Q_{\alpha }^{A},Q_{\beta }^{B}\}&=2\epsilon _{\alpha \beta }\epsilon ^{AB}{\bar {Z}}\\\{{\bar {Q}}_{{\dot {\alpha }}A},{\bar {Q}}_{{\dot {\beta }}B}\}&=-2\epsilon _{{\dot {\alpha }}{\dot {\beta }}}\epsilon _{AB}Z\\\end{aligned}}}
where:
α
β
˙
{\displaystyle \alpha {\dot {\beta }}}
are the Lorentz group indices, A and B are R-symmetry indices.
Take linear combinations of the above generators as follows:
R
α
A
=
ξ
−
1
Q
α
A
+
ξ
σ
α
β
˙
0
Q
¯
β
˙
B
T
α
A
=
ξ
−
1
Q
α
A
−
ξ
σ
α
β
˙
0
Q
¯
β
˙
B
{\displaystyle {\begin{aligned}R_{\alpha }^{A}&=\xi ^{-1}Q_{\alpha }^{A}+\xi \sigma _{\alpha {\dot {\beta }}}^{0}{\bar {Q}}^{{\dot {\beta }}B}\\T_{\alpha }^{A}&=\xi ^{-1}Q_{\alpha }^{A}-\xi \sigma _{\alpha {\dot {\beta }}}^{0}{\bar {Q}}^{{\dot {\beta }}B}\\\end{aligned}}}
Consider a state ψ which has 4 momentum
(
M
,
0
,
0
,
0
)
{\displaystyle (M,0,0,0)}
. Applying the following operator to this state gives:
(
R
1
1
+
(
R
1
1
)
†
)
2
ψ
=
4
(
M
+
R
e
(
Z
ξ
2
)
)
ψ
{\displaystyle {\begin{aligned}(R_{1}^{1}+(R_{1}^{1})^{\dagger })^{2}\psi &=4(M+Re(Z\xi ^{2}))\psi \\\end{aligned}}}
But because this is the square of a Hermitian operator, the right hand side coefficient must be positive for all
ξ
{\displaystyle \xi }
.
In particular the strongest result from this is
M
≥
|
Z
|
{\displaystyle {\begin{aligned}M\geq |Z|\\\end{aligned}}}
Example applications [ edit ]
Supersymmetric black hole entropies[2]
See also [ edit ]
References [ edit ]
^ Olsen, Kasper; Szabo, Richard (2000). "Constructing D-Branes from K-Theory" (PDF) . Advances in Theoretical and Mathematical Physics . 4 : 889–1025.
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Bogomol%27nyi–Prasad–Sommerfield_state&oldid=1191114790 "
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