't Hooft symbol

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.

ηaμν is the 't Hooft symbol:

In other words, they are defined by


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

More explicitly, these symbols are



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

Some other properties are

The same holds for except for


Obviously 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]

See also[]


  1. ^ 't Hooft, G. (1976). "Computation of the quantum effects due to a four-dimensional pseudoparticle". Physical Review D. 14 (12): 3432–3450. Bibcode:1976PhRvD..14.3432T. doi:10.1103/PhysRevD.14.3432.
  2. ^ Belitsky, A. V.; Vandoren, S.; Nieuwenhuizen, P. V. (2000). "Yang-Mills and D-instantons". Classical and Quantum Gravity. 17 (17): 3521–3570. arXiv:hep-th/0004186. Bibcode:2000CQGra..17.3521B. doi:10.1088/0264-9381/17/17/305. S2CID 16107344.