Girish Mahajan (Editor)

'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.

References

't Hooft symbol Wikipedia


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