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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 $ν=1$ quantum Hall state at the self-dual point. Using boson-fermion duality, we map the $ν=1$ 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.