The 't Hooft η symbol is a symbol 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.
ηaμν 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 . In other words, they are defined by
( a = 1 , 2 , 3 ; μ , ν = 1 , 2 , 3 , 4 ; ϵ 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 The (anti)self-duality properties are
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 Some other properties are
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 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 , η a μ ρ η b μ σ = δ a b δ ρ σ + ϵ a b c η c ρ σ , 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 , 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 . The same holds for η ¯ 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 . 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 , Obviously η a μ ν η ¯ b μ ν = 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.
