Asymptotic quasinormal frequencies of different spin fields in $d$-dimensional spherically-symmetric black holes

While Hod's conjecture is demonstrably restrictive, the link he observed between black hole (BH) area quantisation and the large overtone ($n$) limit of quasinormal frequencies (QNFs) motivated intense scrutiny of the regime, from which an improved understanding of asymptotic quasinormal frequencies (aQNFs) emerged. A further outcome was the development of the ``monodromy technique", which exploits an anti-Stokes line analysis to extract physical solutions from the complex plane. In this analysis of the large-$n$ limit, we apply the monodromy technique generalised by Nat{a}rio and Schiappa to higher-dimensional Schwarzschild, Reissner-Nordstr{o}m, and Schwarzschild (anti-)de Sitter BH spacetimes in order to demonstrate explicitly how the method is adjusted to accommodate BH charge and a non-zero $\Lambda$. We validate extant aQNF expressions for perturbations of integer spin, and provide new results for the aQNFs of half-integer spins within all BHs herewith explored. In doing so, we find that the monodromy technique produces reliable and generalisable results while avoiding the computational intricacies of other methods. Bar the Schwarzschild anti-de Sitter case, the spin-1/2 aQNFs are purely imaginary; the spin-3/2 aQNFs follow suit in Schwarzschild and Schwarzschild de Sitter BHs, but match the gravitational perturbations for most others. Particularly for Schwarzschild, extremal Reissner-Nordstr{o}m, and several Schwarzschild de Sitter cases, the application of $n \rightarrow \infty$ generally fixes $\mathbb{R}e \{ \omega \}$ and allows for the unbounded growth of $\mathbb{I}m \{ \omega \}$ in fixed quantities.