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
η μ ν a {\displaystyle \eta _{\mu \nu }^{a}} is the 't Hooft symbol:
η μ ν a = { ϵ a μ ν μ , ν = 1 , 2 , 3 − δ a ν μ = 4 δ a μ ν = 4 0 μ = ν = 4 {\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 δ a ν {\displaystyle \delta ^{a\nu }} and δ a μ {\displaystyle \delta ^{a\mu }} are instances of the Kronecker delta, and ϵ a μ ν {\displaystyle \epsilon ^{a\mu \nu }} is the Levi–Civita symbol. In other words, the symbols are defined by ( a = 1 , 2 , 3 ; μ , ν = 1 , 2 , 3 , 4 ; ϵ 1234 = + 1 {\displaystyle a=1,2,3;~\mu ,\nu =1,2,3,4;~\epsilon _{1234}=+1} )
η a μ ν = ϵ a μ ν 4 + δ a μ δ ν 4 − δ a ν δ μ 4 η ¯ a μ ν = ϵ a μ ν 4 − δ a μ δ ν 4 + δ a ν δ μ 4 {\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
η 1 μ ν = [ 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ] , η 2 μ ν = [ 0 0 − 1 0 0 0 0 1 1 0 0 0 0 − 1 0 0 ] , η 3 μ ν = [ 0 1 0 0 − 1 0 0 0 0 0 0 1 0 0 − 1 0 ] , {\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:
η ¯ 1 μ ν = [ 0 0 0 − 1 0 0 1 0 0 − 1 0 0 1 0 0 0 ] , η ¯ 2 μ ν = [ 0 0 − 1 0 0 0 0 − 1 1 0 0 0 0 1 0 0 ] , η ¯ 3 μ ν = [ 0 1 0 0 − 1 0 0 0 0 0 0 − 1 0 0 1 0 ] . {\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:
η a μ ν = 1 2 ϵ μ ν ρ σ η a ρ σ , η ¯ a μ ν = − 1 2 ϵ μ ν ρ σ η ¯ a ρ σ {\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 η a μ ν = − η a ν μ , {\displaystyle \eta _{a\mu \nu }=-\eta _{a\nu \mu }\ ,}
ϵ a b c η b μ ν η c ρ σ = δ μ ρ η a ν σ + δ ν σ η a μ ρ − δ μ σ η a ν ρ − δ ν ρ η a μ σ {\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 }}
η a μ ν η a ρ σ = δ μ ρ δ ν σ − δ μ σ δ ν ρ + ϵ μ ν ρ σ , {\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 }\ ,}
η a μ ρ η b μ σ = δ a b δ ρ σ + ϵ a b c η c ρ σ , {\displaystyle \eta _{a\mu \rho }\eta _{b\mu \sigma }=\delta _{ab}\delta _{\rho \sigma }+\epsilon _{abc}\eta _{c\rho \sigma }\ ,}
ϵ μ ν ρ θ η a σ θ = δ σ μ η a ν ρ + δ σ ρ η a μ ν − δ σ ν η a μ ρ , {\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 }\ ,}
η a μ ν η a μ ν = 12 , η a μ ν η b μ ν = 4 δ a b , η a μ ρ η a μ σ = 3 δ ρ σ . {\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 η ¯ {\displaystyle {\bar {\eta }}} except for
η ¯ a μ ν η ¯ a ρ σ = δ μ ρ δ ν σ − δ μ σ δ ν ρ − ϵ μ ν ρ σ . {\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
ϵ μ ν ρ θ η ¯ a σ θ = − δ σ μ η ¯ a ν ρ − δ σ ρ η ¯ a μ ν + δ σ ν η ¯ a μ ρ . {\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 }\ .}
Furthermore, η a μ ν η ¯ b μ ν = 0 {\displaystyle \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
References