't Hooft symbol

The 't Hooft symbol is a rank-3 tensor which mixes the properties of both the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft in 1976 in order to compute the properties of a dilute gas of instantons, specifically BPST instantons. It is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of the Lorentz group Lie algebra.

Definition

$\eta _{\mu \nu }^{a}$ is the 't Hooft symbol: $$\eta _{\mu \nu }^{a}={\begin{cases}\epsilon ^{a\mu \nu }&\mu ,\nu =1,2,3\\-\delta ^{a\nu }&\mu =4\\\delta ^{a\mu }&\nu =4\\0&\mu =\nu =4\end{cases}}$$ 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, the symbols are defined by

($a=1,2,3;~\mu ,\nu =1,2,3,4;~\epsilon _{1234}=+1$)

$${\begin{aligned}\eta _{a\mu \nu }&=\epsilon _{a\mu \nu 4}+\delta _{a\mu }\delta _{\nu 4}-\delta _{a\nu }\delta _{\mu 4}\\[1ex]{\bar {\eta }}_{a\mu \nu }&=\epsilon _{a\mu \nu 4}-\delta _{a\mu }\delta _{\nu 4}+\delta _{a\nu }\delta _{\mu 4}\end{aligned}}$$ 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 $$\epsilon _{\mu \nu \rho \theta }{\bar {\eta }}_{a\sigma \theta }=-\delta _{\sigma \mu }{\bar {\eta }}_{a\nu \rho }-\delta _{\sigma \rho }{\bar {\eta }}_{a\mu \nu }+\delta _{\sigma \nu }{\bar {\eta }}_{a\mu \rho }\ .$$

Furthermore, $\eta _{a\mu \nu }{\bar {\eta }}_{b\mu \nu }=0$ due to different duality properties.

See also

• Instanton
• 't Hooft anomaly
• 't Hooft–Polyakov monopole
• 't Hooft loop