Mixed 't Hooft Anomalies and the Witten Effect for AdS Black Holes

For a variety of BPS black holes in string theory, the supersymmetric index has provided a microscopic validation of the Bekenstein-Hawking formula. In the near-BPS limit, a gravitational path integral analysis previously revealed the semiclassical spectrum is modified, having a large extremal degeneracy (consistent with the index) and a mass gap up to a continuum of non-BPS black holes. Presently, we study examples in which these sharp features of the spectrum are altered due to the presence of anomalies in the form of $\vartheta$-angle terms in the action. These may appear generally, but we focus on near-BPS dyonic AdS$_4$ black holes in M-theory, dual to 3d $\mathcal{N}=2$ SCFTs of Class $R$ obtained by twisted compactification of $N$ wrapped M5 branes. Due to the Witten effect, the dyonic black holes receive quantum corrections to their charges, and when $\vartheta = \pi$ one may find a mixed `t Hooft anomaly between the $U(1)_R$ and $\mathbb{Z}_2$ time reversal symmetries. Using results from $\mathcal{N}=2$ JT supergravity, we find these effects result in a spectrum in which both the gap and index are reduced, and may even vanish. Surprisingly, for $\vartheta \rightarrow \pi$, neither the Bekenstein-Hawking formula nor the index correctly account for the extremal degeneracies.