'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

${\eta }_{\mu \nu }^a$ is the 't Hooft symbol:

${\eta }_{\mu \nu }^a=\{\begin{array}{ll}{ϵ}^{a\mu \nu }& \mu ,\nu =1,2,3\\ -{\delta }^{a\nu }& \mu =4\\ {\delta }^{a\mu }& \nu =4\\ 0& \mu =\nu =4\end{array}$

Where ${\delta }^{a\nu }$ and ${\delta }^{a\mu }$ are instances of the Kronecker delta, and ${ϵ}^{a\mu \nu }$ is the Levi-Civita symbol. In other words, they are defined by ($a=1,2,3;\mu ,\nu =1,2,3,4;{ϵ}_{1234}=+1$)

${\eta }_{a\mu \nu }={ϵ}_{a\mu \nu 4}+{\delta }_{a\mu }{\delta }_{\nu 4}-{\delta }_{a\nu }{\delta }_{\mu 4}$

${\overline{\eta }}_{a\mu \nu }={ϵ}_{a\mu \nu 4}-{\delta }_{a\mu }{\delta }_{\nu 4}+{\delta }_{a\nu }{\delta }_{\mu 4}$

where the latter are the anti-self-dual 't Hooft symbols.

Matrix Form

In matrix form, the 't Hooft symbols are

${\eta }_{1\mu \nu }=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ -1& 0& 0& 0\end{array}\right],{\eta }_{2\mu \nu }=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right],{\eta }_{3\mu \nu }=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right],$

and their anti-self-duals are the following:

${\overline{\eta }}_{1\mu \nu }=\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right],{\overline{\eta }}_{2\mu \nu }=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],{\overline{\eta }}_{3\mu \nu }=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right].$

Properties

They satisfy the self-duality and the anti-self-duality properties:

Failed to parse (syntax error): {\displaystyle \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

$$\begin{gathered} {\eta }_{a\mu \nu }=-{\eta }_{a\nu \mu } \\ , \end{gathered}$$

${ϵ}_{abc}{\eta }_{b\mu \nu }{\eta }_{c\rho \sigma }={\delta }_{\mu \rho }{\eta }_{a\nu \sigma }+{\delta }_{\nu \sigma }{\eta }_{a\mu \rho }-{\delta }_{\mu \sigma }{\eta }_{a\nu \rho }-{\delta }_{\nu \rho }{\eta }_{a\mu \sigma }$

$$\begin{gathered} {\eta }_{a\mu \nu }{\eta }_{a\rho \sigma }={\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }+{ϵ}_{\mu \nu \rho \sigma } \\ , \end{gathered}$$

$$\begin{gathered} {\eta }_{a\mu \rho }{\eta }_{b\mu \sigma }={\delta }_{ab}{\delta }_{\rho \sigma }+{ϵ}_{abc}{\eta }_{c\rho \sigma } \\ , \end{gathered}$$

$$\begin{gathered} {ϵ}_{\mu \nu \rho \theta }{\eta }_{a\sigma \theta }={\delta }_{\sigma \mu }{\eta }_{a\nu \rho }+{\delta }_{\sigma \rho }{\eta }_{a\mu \nu }-{\delta }_{\sigma \nu }{\eta }_{a\mu \rho } \\ , \end{gathered}$$

$$\begin{gathered} {\eta }_{a\mu \nu }{\eta }_{a\mu \nu }=12 \\ ,{\eta }_{a\mu \nu }{\eta }_{b\mu \nu }=4{\delta }_{ab} \\ ,{\eta }_{a\mu \rho }{\eta }_{a\mu \sigma }=3{\delta }_{\rho \sigma } \\ . \end{gathered}$$

The same holds for $\overline{\eta }$ except for

$$\begin{gathered} {\overline{\eta }}_{a\mu \nu }{\overline{\eta }}_{a\rho \sigma }={\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \sigma }{\delta }_{\nu \rho }-{ϵ}_{\mu \nu \rho \sigma } \\ . \end{gathered}$$

and

$$\begin{gathered} {ϵ}_{\mu \nu \rho \theta }{\overline{\eta }}_{a\sigma \theta }=-{\delta }_{\sigma \mu }{\overline{\eta }}_{a\nu \rho }-{\delta }_{\sigma \rho }{\overline{\eta }}_{a\mu \nu }+{\delta }_{\sigma \nu }{\overline{\eta }}_{a\mu \rho } \\ , \end{gathered}$$

Obviously ${\eta }_{a\mu \nu }{\overline{\eta }}_{b\mu \nu }=0$ 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
• 't Hooft anomaly
• 't Hooft–Polyakov monopole
• 't Hooft loop