  We present first-principles many-body perturbation theory calculations of the
quasiparticle electronic structure and of the optical response of HfO$_2$
polymorphs. We use the $GW$ approximation including core electrons by the
projector augmented wave (PAW) method and performing a quasiparticle
self-consistency also on wavefunctions (QS$GW$). In addition, we solve the
Bethe-Salpeter equation on top of $GW$ to calculate optical properties
including excitonic effects. For monoclinic HfO$_2$ we find a fundamental band
gap of $E_g = 6.33$ eV (with the direct band gap at $E_g^d = 6.41$ eV), and an
exciton binding energy of 0.57 eV, which situates the optical gap at $E^o_g =
5.85$ eV. The latter is in the range of spectroscopic ellipsometry (SE)
experimental estimates (5.5-6 eV), whereas our electronic band gap is well
beyond experimental photoemission (PE) estimates ($< 6$ eV) and previous $GW$
works. Our calculated density of states and optical absorption spectra compare
well to raw PE and SE spectra. This suggests that our predictions of both
optical and electronic gaps are close to, or at least lower bounds of, the real
values.
