'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

${\displaystyle \eta _{\mu \nu }^{a}}$ is the 't Hooft symbol: $${\displaystyle \eta _{\mu \nu }^{a}={\begin{cases}\epsilon ^{a\mu \nu }&\mu ,\nu =1,2,3\\-\delta ^{a\nu }&\mu =4\\\delta ^{a\mu }&\nu =4\\0&\mu =\nu =4\end{cases}}}$$ Where ${\displaystyle \delta ^{a\nu }}$ and ${\displaystyle \delta ^{a\mu }}$ are instances of the Kronecker delta, and ${\displaystyle \epsilon ^{a\mu \nu }}$ is the Levi–Civita symbol.

In other words, they are defined by

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

$${\displaystyle {\begin{aligned}\eta _{a\mu \nu }&=\epsilon _{a\mu \nu 4}+\delta _{a\mu }\delta _{\nu 4}-\delta _{a\nu }\delta _{\mu 4}\\[1ex]{\bar {\eta }}_{a\mu \nu }&=\epsilon _{a\mu \nu 4}-\delta _{a\mu }\delta _{\nu 4}+\delta _{a\nu }\delta _{\mu 4}\end{aligned}}}$$ where the latter are the anti-self-dual 't Hooft symbols.

Matrix Form

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

Properties

They satisfy the self-duality and the anti-self-duality properties: $${\displaystyle \eta _{a\mu \nu }={\tfrac {1}{2}}\epsilon _{\mu \nu \rho \sigma }\eta _{a\rho \sigma }\ ,\qquad {\bar {\eta }}_{a\mu \nu }=-{\tfrac {1}{2}}\epsilon _{\mu \nu \rho \sigma }{\bar {\eta }}_{a\rho \sigma }}$$

Some other properties are

$${\displaystyle \eta _{a\mu \nu }=-\eta _{a\nu \mu }\ ,}$$$${\displaystyle \epsilon _{abc}\eta _{b\mu \nu }\eta _{c\rho \sigma }=\delta _{\mu \rho }\eta _{a\nu \sigma }+\delta _{\nu \sigma }\eta _{a\mu \rho }-\delta _{\mu \sigma }\eta _{a\nu \rho }-\delta _{\nu \rho }\eta _{a\mu \sigma }}$$$${\displaystyle \eta _{a\mu \nu }\eta _{a\rho \sigma }=\delta _{\mu \rho }\delta _{\nu \sigma }-\delta _{\mu \sigma }\delta _{\nu \rho }+\epsilon _{\mu \nu \rho \sigma }\ ,}$$$${\displaystyle \eta _{a\mu \rho }\eta _{b\mu \sigma }=\delta _{ab}\delta _{\rho \sigma }+\epsilon _{abc}\eta _{c\rho \sigma }\ ,}$$$${\displaystyle \epsilon _{\mu \nu \rho \theta }\eta _{a\sigma \theta }=\delta _{\sigma \mu }\eta _{a\nu \rho }+\delta _{\sigma \rho }\eta _{a\mu \nu }-\delta _{\sigma \nu }\eta _{a\mu \rho }\ ,}$$$${\displaystyle \eta _{a\mu \nu }\eta _{a\mu \nu }=12\ ,\quad \eta _{a\mu \nu }\eta _{b\mu \nu }=4\delta _{ab}\ ,\quad \eta _{a\mu \rho }\eta _{a\mu \sigma }=3\delta _{\rho \sigma }\ .}$$

The same holds for ${\displaystyle {\bar {\eta }}}$ except for

$${\displaystyle {\bar {\eta }}_{a\mu \nu }{\bar {\eta }}_{a\rho \sigma }=\delta _{\mu \rho }\delta _{\nu \sigma }-\delta _{\mu \sigma }\delta _{\nu \rho }-\epsilon _{\mu \nu \rho \sigma }\ .}$$

and $${\displaystyle \epsilon _{\mu \nu \rho \theta }{\bar {\eta }}_{a\sigma \theta }=-\delta _{\sigma \mu }{\bar {\eta }}_{a\nu \rho }-\delta _{\sigma \rho }{\bar {\eta }}_{a\mu \nu }+\delta _{\sigma \nu }{\bar {\eta }}_{a\mu \rho }\ ,}$$

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

Many properties of these are tabulated in the appendix of 't Hooft's paper ^[1] and also in the article by Belitsky et al. ^[2]

See also

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

References

1. 't Hooft, G. (1976). "Computation of the quantum effects due to a four-dimensional pseudoparticle". Physical Review D. 14 (12): 3432–3450. Bibcode:1976PhRvD..14.3432T. doi:10.1103/PhysRevD.14.3432.
2. Belitsky, A. V.; Vandoren, S.; Nieuwenhuizen, P. V. (2000). "Yang-Mills and D-instantons". Classical and Quantum Gravity. 17 (17): 3521–3570. arXiv: hep-th/0004186 . Bibcode:2000CQGra..17.3521B. doi:10.1088/0264-9381/17/17/305. S2CID   16107344.