'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.

η^a_μν is the 't Hooft symbol:

$\eta^a_{\mu\nu} = \begin{cases} \epsilon^{a\mu\nu} & \mu,\nu=1,2,3 \\ -\delta^{a\nu} & \mu=4 \\ \delta^{a\mu} & \nu=4 \\ 0 & \mu=\nu=4 \end{cases} .$

In other words they are defined by

($a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_{1 2 3 4}=+1$)

$\eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} + \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4}$

$\bar \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} + \delta_{a \nu} \delta_{\mu 4}$

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

$\epsilon_{abc} \eta_{b\mu\nu} \eta_{c\rho\sigma} = \delta_{\mu\rho} \eta_{a\nu\sigma} + \delta_{\nu\sigma} \eta_{a\mu\rho} - \delta_{\mu\sigma} \eta_{a\nu\rho} - \delta_{\nu\rho} \eta_{a\mu\sigma}$

$\eta_{a\mu\nu} \eta_{a\rho\sigma} = \delta_{\mu\rho} \delta_{\nu\sigma} - \delta_{\mu\sigma} \delta_{\nu\rho} + \epsilon_{\mu\nu\rho\sigma} \ ,$

$\eta_{a\mu\rho} \eta_{b\mu\sigma} = \delta_{ab} \delta_{\rho\sigma} + \epsilon_{abc} \eta_{c\rho\sigma} \ ,$

$\epsilon_{\mu\nu\rho\theta} \eta_{a\sigma\theta} = \delta_{\sigma\mu} \eta_{a\nu\rho} + \delta_{\sigma\rho} \eta_{a\mu\nu} - \delta_{\sigma\nu} \eta_{a\mu\rho} \ ,$

$\eta_{a\mu\nu} \eta_{a\mu\nu} = 12 \ ,\quad \eta_{a\mu\nu} \eta_{b\mu\nu} = 4 \delta_{ab} \ ,\quad \eta_{a\mu\rho} \eta_{a\mu\sigma} = 3 \delta_{\rho\sigma} \ .$

The same holds for $\bar\eta$ except for

$\bar\eta_{a\mu\nu} \bar\eta_{a\rho\sigma} = \delta_{\mu\rho} \delta_{\nu\sigma} - \delta_{\mu\sigma} \delta_{\nu\rho} - \epsilon_{\mu\nu\rho\sigma} \ .$

and

$\epsilon_{\mu\nu\rho\theta} \bar\eta_{a\sigma\theta} = -\delta_{\sigma\mu} \bar\eta_{a\nu\rho} - \delta_{\sigma\rho} \bar\eta_{a\mu\nu} + \delta_{\sigma\nu} \bar\eta_{a\mu\rho} \ ,$

Obviously $\eta_{a\mu\nu} \bar\eta_{b\mu\nu} = 0$ due to different duality properties.

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

See also

• Instanton