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.
μ ν
a μ ν
μ = ν = 4
are instances of the Kronecker delta, and
a μ ν
μ , ν = 1 , 2 , 3 , 4 ;
a μ ν
a μ ν 4
a μ ν
a μ ν 4
where the latter are the anti-self-dual 't Hooft symbols.
In matrix form, the 't Hooft symbols are
1 μ ν
2 μ ν
3 μ ν
1 μ ν
2 μ ν
They satisfy the self-duality and the anti-self-duality properties:
μ ν ρ σ
μ ν ρ σ
μ ν ρ σ
μ ν ρ θ
μ ν ρ σ
μ ν ρ θ
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]