Self-dual ν=1 bosonic quantum Hall state in mixed-dimensional QED

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 $\ensuremath{\nu}=1$ quantum Hall state at the self-dual point. Using boson-fermion duality, we map the $\ensuremath{\nu}=1$ bosonic quantum Hall state to a Fermi sea of the dual fermion and observe that the exact relationships between transport coefficients at the bosonic self-dual point are reproduced by a simple random-phase approximation (RPA), 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 that the equivalence of the self-dual structure can be understood in terms of electromagnetic duality or modular invariance, and these features are not inherited by the nonrelativistic cousins of the present model.