't Hooft symbol

\eta^a_{\mu\nu} 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} Where \delta^{a\nu} and \delta^{a\mu} are instances of the Kronecker delta, and \epsilon^{a\mu\nu} is the Levi–Civita symbol. In other words, they are defined by ( a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_{1 2 3 4}=+1) \begin{align} \eta_{a \mu \nu} &= \epsilon_{a \mu \nu 4} + \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4} \\[1ex] \bar{\eta}_{a \mu \nu} &= \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} + \delta_{a \nu} \delta_{\mu 4} \end{align} where the latter are the anti-self-dual 't Hooft symbols. == Matrix form ==

In matrix form, the 't Hooft symbols are \eta_{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}, \quad \eta_{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix}, \quad \eta_{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix}, and their anti-self-duals are the following: \bar{\eta}_{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}, \quad \bar{\eta}_{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}, \quad \bar{\eta}_{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix}. == Properties ==

They satisfy the self-duality and the anti-self-duality properties: \eta_{a\mu\nu} = \tfrac{1}{2} \epsilon_{\mu\nu\rho\sigma} \eta_{a\rho\sigma} \ , \qquad \bar\eta_{a\mu\nu} = - \tfrac{1}{2} \epsilon_{\mu\nu\rho\sigma} \bar\eta_{a\rho\sigma} Some other properties are \eta_{a\mu\nu} = - \eta_{a\nu\mu} \ , \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==