Exploring the twisted sector of $\mathbb{Z}_{L}$ orbifolds: Matching $\alpha'$-corrections to localisation

We consider type IIB string theory on $\mathrm{AdS}_5\times S^5/\mathbb{Z}_{L}$ orbifold spaces with generic $L$. Recent localisation results in the dual 4d $\mathcal{N}=2$ circular quiver gauge theories provide us with strong coupling expansions of certain correlators involving twisted half-BPS operators. To leading order, these results have been matched to an effective theory for massless twisted string states, which can be constructed by resolving the orbifold singularity and considering localised supergravity modes on the resolution cycles. Applying this reasoning to subleading order in strong coupling, we observe that for $L\neq 2,3,4,6$, a naive reduction of the 10d $(\alpha')^3$-correction does not result in the correct coefficients to match the localisation result. We explain this mismatch by the appearance of twisted sector resonances in string amplitudes involving external twisted sector states. We perform the low-energy expansion of a ``twisted'' Virasoro-Shapiro amplitude and recover the expected coefficients, suggesting that the orbifold resolution and the low-energy expansion can not be interchanged directly. Finally, we comment on the long-quiver limit, $L\to\infty$, in the context of the low-energy effective action.