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The 't Hooft η symbol is a symbol which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol . It was introduced by Gerard 't Hooft . It is used in the construction of the BPST instanton .
ηa μν is the 't Hooft symbol :
η
μ
ν
a
=
{
ϵ
a
μ
ν
μ
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1
,
2
,
3
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δ
a
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4
.
{\displaystyle \eta _{\mu \nu }^{a}={\begin{cases}\epsilon ^{a\mu \nu }&\mu ,\nu =1,2,3\\-\delta ^{a\nu }&\mu =4\\\delta ^{a\mu }&\nu =4\\0&\mu =\nu =4\end{cases}}.}
In other words they are defined by
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;
ϵ
1234
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1
{\displaystyle a=1,2,3;~\mu ,\nu =1,2,3,4;~\epsilon _{1234}=+1}
)
η
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{\displaystyle \eta _{a\mu \nu }=\epsilon _{a\mu \nu 4}+\delta _{a\mu }\delta _{\nu 4}-\delta _{a\nu }\delta _{\mu 4}}
η
¯
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4
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{\displaystyle {\bar {\eta }}_{a\mu \nu }=\epsilon _{a\mu \nu 4}-\delta _{a\mu }\delta _{\nu 4}+\delta _{a\nu }\delta _{\mu 4}}
The (anti)self-duality properties are
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{\displaystyle \eta _{a\mu \nu }={\frac {1}{2}}\epsilon _{\mu \nu \rho \sigma }\eta _{a\rho \sigma }\ ,\qquad {\bar {\eta }}_{a\mu \nu }=-{\frac {1}{2}}\epsilon _{\mu \nu \rho \sigma }{\bar {\eta }}_{a\rho \sigma }\ }
Some other properties are
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{\displaystyle \epsilon _{abc}\eta _{b\mu \nu }\eta _{c\rho \sigma }=\delta _{\mu \rho }\eta _{a\nu \sigma }+\delta _{\nu \sigma }\eta _{a\mu \rho }-\delta _{\mu \sigma }\eta _{a\nu \rho }-\delta _{\nu \rho }\eta _{a\mu \sigma }}
η
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,
{\displaystyle \eta _{a\mu \nu }\eta _{a\rho \sigma }=\delta _{\mu \rho }\delta _{\nu \sigma }-\delta _{\mu \sigma }\delta _{\nu \rho }+\epsilon _{\mu \nu \rho \sigma }\ ,}
η
a
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ρ
η
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c
η
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{\displaystyle \eta _{a\mu \rho }\eta _{b\mu \sigma }=\delta _{ab}\delta _{\rho \sigma }+\epsilon _{abc}\eta _{c\rho \sigma }\ ,}
ϵ
μ
ν
ρ
θ
η
a
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θ
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{\displaystyle \epsilon _{\mu \nu \rho \theta }\eta _{a\sigma \theta }=\delta _{\sigma \mu }\eta _{a\nu \rho }+\delta _{\sigma \rho }\eta _{a\mu \nu }-\delta _{\sigma \nu }\eta _{a\mu \rho }\ ,}
η
a
μ
ν
η
a
μ
ν
=
12
,
η
a
μ
ν
η
b
μ
ν
=
4
δ
a
b
,
η
a
μ
ρ
η
a
μ
σ
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3
δ
ρ
σ
.
{\displaystyle \eta _{a\mu \nu }\eta _{a\mu \nu }=12\ ,\quad \eta _{a\mu \nu }\eta _{b\mu \nu }=4\delta _{ab}\ ,\quad \eta _{a\mu \rho }\eta _{a\mu \sigma }=3\delta _{\rho \sigma }\ .}
The same holds for
η
¯
{\displaystyle {\bar {\eta }}}
except for
η
¯
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ν
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{\displaystyle {\bar {\eta }}_{a\mu \nu }{\bar {\eta }}_{a\rho \sigma }=\delta _{\mu \rho }\delta _{\nu \sigma }-\delta _{\mu \sigma }\delta _{\nu \rho }-\epsilon _{\mu \nu \rho \sigma }\ .}
and
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μ
ν
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θ
η
¯
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−
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{\displaystyle \epsilon _{\mu \nu \rho \theta }{\bar {\eta }}_{a\sigma \theta }=-\delta _{\sigma \mu }{\bar {\eta }}_{a\nu \rho }-\delta _{\sigma \rho }{\bar {\eta }}_{a\mu \nu }+\delta _{\sigma \nu }{\bar {\eta }}_{a\mu \rho }\ ,}
Obviously
η
a
μ
ν
η
¯
b
μ
ν
=
0
{\displaystyle \eta _{a\mu \nu }{\bar {\eta }}_{b\mu \nu }=0}
due to different
duality properties.
Many properties of these are tabulated in the appendix of 't Hooft's paper[ 1] and also in the article by Belitsky et al.[ 2]
See also
References