't Hooft symbol

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.

η^a_μν is the 't Hooft symbol:

In other words, they are defined by

(${\displaystyle a=1,2,3;~\mu ,\nu =1,2,3,4;~\epsilon _{1234}=+1}$)

where the latter are the anti-self-dual 't Hooft symbols.

More explicitly, these symbols are

and

They satisfy the self-duality and the anti-self-duality properties:

Some other properties are

The same holds for ${\displaystyle {\bar {\eta }}}$ except for

and

Obviously ${\displaystyle \eta _{a\mu \nu }{\bar {\eta }}_{b\mu \nu }=0}$ due to different duality properties.

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