't Hooft symbol

Mathematical symbol used in algebras The t Hooft symbol is a rank-3 tensor which mixes the properties of both the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft in 1976 in order to compute the properties of a dilute gas of instantons, specifically BPST instantons. It is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of the Lorentz group Lie algebra. Definition is the 't Hooft symbol: Where and are instances of the Kronecker delta, and is the Levi–Civita symbol. In other words, the symbols are defined by () where the latter are the anti-self-dual 't Hooft symbols. Matrix form In matrix form, the 't Hooft symbols are and their anti-self-duals are the following: Properties They satisfy the self-duality and the anti-self-duality properties: Some other properties are 't Hooft G. Computation of the quantum effects due to a four-dimensional pseudoparticle Physical Review D 14 12 3432–3450 1976 10.1103/PhysRevD.14.3432 1976PhRvD..14.3432T Belitsky A. V. Vandoren S. Nieuwenhuizen P. V. 10.1088/0264-9381/17/17/305 Yang-Mills and D-instantons Classical and Quantum Gravity 17 17 3521–3570 2000 hep-th/0004186 2000CQGra..17.3521B 16107344 The same holds for except for and Furthermore, due to different duality properties. See also Instanton 't Hooft anomaly 't Hooft–Polyakov monopole 't Hooft loop References Category:Gauge theories Category:Mathematical symbols