On external backgrounds and linear potential in three dimensions

For a three-dimensional theory with a coupling $\phi \epsilon ^{\mu \nu \lambda} v_\mu F_{\nu \lambda}$,where$v_\mu$ is an external constant background, we compute the interaction potential within the structure of the gauge-invariant but path-dependent variables formalism. While in the case of a purely timelike vector the static potential remains Coulombic, in the case of a purely spacelike vector the potential energy is the sum of a Bessel and a linear potentials, leading to the confinement of static charges. This result may be considered as another realization of the known Polyakov’s result.