't Hooft symbol

The 't Hooft symbol is a collection of numbers 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.

Definition
$$\eta^a_{\mu\nu}$$ 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

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


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


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

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} = \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.