Genomic library

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. They are defined by

($a=1,2,3;\mu ,\nu =1,2,3,4;{ϵ}_{1234}=+1$)

${\eta }_{a\mu \nu }={ϵ}_{a\mu \nu 4}+{\delta }_{a\mu }{\delta }_{\nu 4}-{\delta }_{a\nu }{\delta }_{\mu 4}$

${\overline{\eta }}_{a\mu \nu }={ϵ}_{a\mu \nu 4}-{\delta }_{a\mu }{\delta }_{\nu 4}+{\delta }_{a\nu }{\delta }_{\mu 4}$

The (anti)self-duality properties are

${\eta }_{a\mu \nu }=\frac{1}{2}{ϵ}_{\mu \nu \rho \sigma }{\eta }_{a\rho \sigma },{\overline{\eta }}_{a\mu \nu }=-\frac{1}{2}{ϵ}_{\mu \nu \rho \sigma }{\overline{\eta }}_{a\rho \sigma }$

Some other properties are

${ϵ}_{abc}{\eta }_{b\mu \nu }{\eta }_{c\rho \sigma }={\delta }_{\mu \rho }{\eta }_{a\nu \sigma }+{\delta }_{\nu \sigma }{\eta }_{a\mu \rho }-{\delta }_{\mu \sigma }{\eta }_{a\nu \rho }-{\delta }_{\nu \rho }{\eta }_{a\mu \sigma }$

${\eta }_{a\mu \nu }{\eta }_{a\rho \sigma }={\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }+{ϵ}_{\mu \nu \rho \sigma },$

${\eta }_{a\mu \rho }{\eta }_{b\mu \sigma }={\delta }_{ab}{\delta }_{\rho \sigma }+{ϵ}_{abc}{\eta }_{c\rho \sigma },$

${ϵ}_{\mu \nu \rho \theta }{\eta }_{a\sigma \theta }={\delta }_{\sigma \mu }{\eta }_{a\nu \rho }+{\delta }_{\sigma \rho }{\eta }_{a\mu \nu }-{\delta }_{\sigma \nu }{\eta }_{a\mu \rho },$

${\eta }_{a\mu \nu }{\eta }_{a\mu \nu }=12,{\eta }_{a\mu \nu }{\eta }_{b\mu \nu }=4{\delta }_{ab},{\eta }_{a\mu \rho }{\eta }_{a\mu \sigma }=3{\delta }_{\rho \sigma }.$

The same holds for $\overline{\eta }$ except for

${\overline{\eta }}_{a\mu \nu }{\overline{\eta }}_{a\rho \sigma }={\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }-{ϵ}_{\mu \nu \rho \sigma }.$

Obviously ${\eta }_{a\mu \nu }{\overline{\eta }}_{b\mu \nu }=0$ due to different duality properties.

Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.

See also

• Instanton

References

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