't Hooft symbol

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)

\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.

References

References

1. (1976). "Computation of the quantum effects due to a four-dimensional pseudoparticle". Physical Review D.
2. (2000). "Yang-Mills and D-instantons". Classical and Quantum Gravity.