 # 't Hooft symbol

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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.

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