  We consider a (2+1) dimensional Wilson-Fisher boson coupled to a (3+1)
dimensional U(1) gauge field. This theory possesses a strong-weak duality in
terms of the coupling constant e, and is self-dual at a particular value of
$e$. We derive exact relations between transport coefficients for a $\nu=1$
quantum Hall state at the self-dual point. Using boson-fermion duality, we map
the $\nu=1$ bosonic quantum Hall state to a Fermi sea of the dual fermion, and
observed that the exact relationships between transport coefficients at the
bosonic self-dual point are reproduced by a simple random phase approximation,
coupled with a Drude formula, in the fermionic theory. We explain this success
of the RPA by pointing out a cancellation of a parity-breaking term in the
fermion theory which occurs only at the self-dual point, resulting in the
fermion self-dual theory explored previously. In addition, we argue n that the
equivalence of self-dual structure can be understood in terms of
electromagnetic duality or modular invariance, and these features are not
inherited by the non-relativistic cousins of the present model.
