't Hooft symbol

Short description: Matrices concerning the SU(2) Lie algebra

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

[math]\displaystyle{ \eta^a_{\mu\nu} }[/math] is the 't Hooft symbol:

[math]\displaystyle{ \eta^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} }[/math]

Where [math]\displaystyle{ \delta^{a\nu} }[/math] and [math]\displaystyle{ \delta^{a\mu} }[/math] are instances of the Kronecker delta, and [math]\displaystyle{ \epsilon^{a\mu\nu} }[/math] is the Levi-Civita symbol.

In other words, they are defined by

([math]\displaystyle{ a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_{1 2 3 4}=+1 }[/math])

[math]\displaystyle{ \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} + \delta_{a \mu} \delta_{\nu 4} - \delta_{a \nu} \delta_{\mu 4} }[/math]

[math]\displaystyle{ \bar \eta_{a \mu \nu} = \epsilon_{a \mu \nu 4} - \delta_{a \mu} \delta_{\nu 4} + \delta_{a \nu} \delta_{\mu 4} }[/math]

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

Matrix Form

In matrix form, the 't Hooft symbols are

[math]\displaystyle{ \eta_{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}, \quad \eta_{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix}, \quad \eta_{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix}, }[/math]

and their anti-self-duals are the following:

[math]\displaystyle{ \bar{\eta}_{1\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}, \quad \bar{\eta}_{2\mu\nu} = \begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}, \quad \bar{\eta}_{3\mu\nu} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix}. }[/math]

Properties

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

[math]\displaystyle{ \eta_{a\mu\nu} = \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} \eta_{a\rho\sigma} \ , \qquad \bar\eta_{a\mu\nu} = - \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} \bar\eta_{a\rho\sigma} \ }[/math]

Some other properties are

[math]\displaystyle{ \epsilon_{abc} \eta_{b\mu\nu} \eta_{c\rho\sigma} = \delta_{\mu\rho} \eta_{a\nu\sigma} + \delta_{\nu\sigma} \eta_{a\mu\rho} - \delta_{\mu\sigma} \eta_{a\nu\rho} - \delta_{\nu\rho} \eta_{a\mu\sigma} }[/math]

[math]\displaystyle{ \eta_{a\mu\nu} \eta_{a\rho\sigma} = \delta_{\mu\rho} \delta_{\nu\sigma} - \delta_{\mu\sigma} \delta_{\nu\rho} + \epsilon_{\mu\nu\rho\sigma} \ , }[/math]

[math]\displaystyle{ \eta_{a\mu\rho} \eta_{b\mu\sigma} = \delta_{ab} \delta_{\rho\sigma} + \epsilon_{abc} \eta_{c\rho\sigma} \ , }[/math]

[math]\displaystyle{ \epsilon_{\mu\nu\rho\theta} \eta_{a\sigma\theta} = \delta_{\sigma\mu} \eta_{a\nu\rho} + \delta_{\sigma\rho} \eta_{a\mu\nu} - \delta_{\sigma\nu} \eta_{a\mu\rho} \ , }[/math]

[math]\displaystyle{ \eta_{a\mu\nu} \eta_{a\mu\nu} = 12 \ ,\quad \eta_{a\mu\nu} \eta_{b\mu\nu} = 4 \delta_{ab} \ ,\quad \eta_{a\mu\rho} \eta_{a\mu\sigma} = 3 \delta_{\rho\sigma} \ . }[/math]

The same holds for [math]\displaystyle{ \bar\eta }[/math] except for

[math]\displaystyle{ \bar\eta_{a\mu\nu} \bar\eta_{a\rho\sigma} = \delta_{\mu\rho} \delta_{\nu\sigma} - \delta_{\mu\sigma} \delta_{\nu\rho} - \epsilon_{\mu\nu\rho\sigma} \ . }[/math]

and

[math]\displaystyle{ \epsilon_{\mu\nu\rho\theta} \bar\eta_{a\sigma\theta} = -\delta_{\sigma\mu} \bar\eta_{a\nu\rho} - \delta_{\sigma\rho} \bar\eta_{a\mu\nu} + \delta_{\sigma\nu} \bar\eta_{a\mu\rho} \ , }[/math]

Obviously [math]\displaystyle{ \eta_{a\mu\nu} \bar\eta_{b\mu\nu} = 0 }[/math] due to different duality properties.

Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.

See also

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