't Hooft symbol explained

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.

Definition

η ^{a} _{\mu\nu} is the 't Hooft symbol:η ^{a} _{\mu\nu} = \begin{cases} \epsilon ^{a\mu\nu} & \mu,\nu =1 , 2 , 3 \ - \delta ^{a\nu} & \mu =4 \ \delta ^{a\mu} & \nu =4 \ 0 & \mu = \nu =4 \end{cases} Where \delta ^{a\nu} and \delta ^{a\mu} are instances of the Kronecker delta, and \epsilon ^{a\mu\nu} is the Levi-Civita symbol.

In other words, they are defined by

(

a =1 , 2 , 3 ;~ \mu,\nu =1 , 2 , 3 , 4 ;~ \epsilon _{1} =+1 )η _{a} = \epsilon _{a} + \delta _{a} \delta _{\nu} - \delta _{a} \delta _{\mu} \bar η _{a} = \epsilon _{a} - \delta _{a} \delta _{\nu} + \delta _{a} \delta _{\mu} where the latter are the anti-self-dual 't Hooft symbols.

Matrix Form

In matrix form, the 't Hooft symbols are

η _{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}, η _{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix}, η _{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix}, and their anti-self-duals are the following:\bar{ η } _{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}, \bar{ η } _{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}, \bar{ η } _{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.

Properties

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

η _{a\mu\nu} = (1)/(2) \epsilon _{\mu\nu\rho\sigma} η _{a\rho\sigma} , \bar η _{a\mu\nu} = - (1)/(2) \epsilon _{\mu\nu\rho\sigma} \bar η _{a\rho\sigma}

Some other properties are

\epsilon _{abc} η _{b\mu\nu} η _{c\rho\sigma} = \delta _{\mu\rho} η _{a\nu\sigma} + \delta _{\nu\sigma} η _{a\mu\rho} - \delta _{\mu\sigma} η _{a\nu\rho} - \delta _{\nu\rho} η _{a\mu\sigma} η _{a\mu\nu} η _{a\rho\sigma} = \delta _{\mu\rho} \delta _{\nu\sigma} - \delta _{\mu\sigma} \delta _{\nu\rho} + \epsilon _{\mu\nu\rho\sigma} , η _{a\mu\rho} η _{b\mu\sigma} = \delta _{ab} \delta _{\rho\sigma} + \epsilon _{abc} η _{c\rho\sigma} , \epsilon _{\mu\nu\rho\theta} η _{a\sigma\theta} = \delta _{\sigma\mu} η _{a\nu\rho} + \delta _{\sigma\rho} η _{a\mu\nu} - \delta _{\sigma\nu} η _{a\mu\rho} , η _{a\mu\nu} η _{a\mu\nu} = 12 , η _{a\mu\nu} η _{b\mu\nu} = 4 \delta _{ab} , η _{a\mu\rho} η _{a\mu\sigma} = 3 \delta _{\rho\sigma} .

The same holds for

\bar η except for\bar η _{a\mu\nu} \bar η _{a\rho\sigma} = \delta _{\mu\rho} \delta _{\nu\sigma} - \delta _{\mu\sigma} \delta _{\nu\rho} - \epsilon _{\mu\nu\rho\sigma} .

and

\epsilon _{\mu\nu\rho\theta} \bar η _{a\sigma\theta} = - \delta _{\sigma\mu} \bar η _{a\nu\rho} - \delta _{\sigma\rho} \bar η _{a\mu\nu} + \delta _{\sigma\nu} \bar η _{a\mu\rho} ,

Obviously

η _{a\mu\nu} \bar η _{b\mu\nu} = 0 due to differentduality 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

• Instanton
• 't Hooft anomaly
• 't Hooft–Polyakov monopole
• 't Hooft loop

Notes and References

1. '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 .
2. 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 .