= Spin connection of a conformally flat metric
{title2=$\omega_{a\mu\nu}=\eta_{a\mu}\partial_\nu\log\Omega-\eta_{a\nu}\partial_\mu\log\Omega$}
For $g_{ab}=\Omega^2\eta_{ab}$, use the <vierbein> $e^\mu{}_a=\Omega\delta^\mu{}_a$ and constant flat <gamma matrices>. The <tetrad postulate> and $\Gamma^c{}_{ab}=\delta^c_a\partial_b\log\Omega+\delta^c_b\partial_a\log\Omega-\eta_{ab}\eta^{cd}\partial_d\log\Omega$ give the displayed lowered <spin connection>. With $L=\widehat\gamma^c\partial_c\log\Omega$, its spinor action is $\nabla_a=\partial_a+[\widehat\gamma_a,L]/4$. The Lorentz index lowering matters in indefinite signature.
Back to article page
