't Hooft symbol

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�;~ \epsilon_{1 2 3 4}=%2B1)

 \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} %2B \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4}
 \bar \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} %2B \delta_{a \nu} \delta_{\mu 4}

The (anti)self-duality properties are


\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


\epsilon_{abc} \eta_{b\mu\nu} \eta_{c\rho\sigma}
= \delta_{\mu\rho} \eta_{a\nu\sigma}
%2B \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}
%2B \epsilon_{\mu\nu\rho\sigma} \ ,

\eta_{a\mu\rho} \eta_{b\mu\sigma}
= \delta_{ab} \delta_{\rho\sigma} %2B \epsilon_{abc} \eta_{c\rho\sigma} \ ,

\epsilon_{\mu\nu\rho\theta} \eta_{a\sigma\theta}
= \delta_{\sigma\mu} \eta_{a\nu\rho}
%2B \delta_{\sigma\rho} \eta_{a\mu\nu}
- \delta_{\sigma\nu} \eta_{a\mu\rho} \ ,

\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 \bar\eta except for


\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} \ .

Obviously \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