  A relativistic action for scalar condensate-fermion mixture is considered
where both the scalar boson and the fermion fields are coupled to a $U(1)$
gauge field. The dynamics of the gauge field is governed by a linear
combination of the Maxwell term and the Lorentz invariant $\mathbf{E\cdot B}$
term with a constant coefficient $\theta$. We obtain an effective action
describing an emergent fermion-fermion interaction and fermion-vortex tube
interaction by using the particle-string duality, and find that the $\theta$
term can significantly affect the interaction of fermions and vortices. We also
perform a dimensional reduction to show a $\theta$ dependent flux attachment to
the itinerant fermions.
