On quantum corrections to semiclassical strings on $\mathrm{AdS}_5\times S^5/\mathbb{Z}_{L}$ orbifold backgrounds

We consider three families of semiclassical string solutions on AdS$_5\times S^5/\mathbb{Z}_L$ orbifold backgrounds which correspond to `twisted' single-trace operators in the SU$(2)$, SU$(3)$ and SL$(2)$ sectors, respectively, of the dual 4d $\mathcal{N}=2$ quiver gauge theory. The leading quantum correction to the classical energy of each solution is computed, and is matched, despite the lack of maximal supersymmetry, to the corresponding results from finite-size corrections to the twisted Bethe ansatz in the continuum $J\to\infty$ limit and the associated Landau-Lifshitz low-energy effective theory. The $\mathbb{Z}_L$ quotient allows the string to have fractional windings $\mu$ along orbifolded directions, resulting in novel stable subsectors of semiclassical states that are absent in the analogous semiclassical string solutions on AdS$_5\times S^5$. The energies of these stable states connect smoothly to those of states in relevant BPS sectors as $\mu\to0$. Motivated by recent localisation results on the dual gauge theory, we comment on the `long quiver' regime $L\gg\sqrt{\lambda}\gg 1$ from a semiclassical perspective, concluding that it reorganises the standard fixed-background semiclassical expansion.