  Whereas in the familiar Kondo effect the exchange interaction is dipolar, it
can also be multipolar, as has been realized in a recent experiment. Here we
study multipolar Kondo effect in a Fermi gas of cold $^{173}$Yb atoms. Making
use of different AC polarizability of the electronic ground state
Yb($^{1}$S$_{0}$) and the long-lived metastable state Yb$^{*}$($^{3}$P$_{2}$),
it is suggested that the latter atoms can be localized and serve as a dilute
concentration of magnetic impurities while the former ones remain itinerant.
The exchange mechanism between the itinerant Yb and the localized Yb$^{*}$
atoms is analyzed and shown to be antiferromagnetic. The quadruple and octuple
interactions act to enhance the Kondo temperature $T_K$ that is found to be
experimentally accessible. The bare exchange Hamiltonian needs to be decomposed
into dipole ($d$), quadruple ($q$) and octuple ($o$) interactions in order to
retain its form under renormalization group (RG) analysis, in which the
corresponding exchange constants ($\lambda_{\mathrm{d}}$,
$\lambda_{\mathrm{q}}$ and $\lambda_{\mathrm{o}}$) flow independently.
Numerical solution of the RG scaling equations reveals a few finite fixed
points, indicating an over-screening, which suggests a non-Fermi liquid phase.
The impurity contribution to the magnetic susceptibility is calculated in the
weak coupling regime (${T}\gg{T}_{K}$).
