Motivating Example
In standard treatments of AdS/CFT (MAGOO for example), one defines $\mathrm{AdS}_{p+2}$ as a particular embedded submanifold of $\mathbb R^{2,p+1}$ which gives it topology $S^1\times \mathbb R^{p+1}$ where the $S^1$ factor is "timelike." This leads to the property that there are closed timelike curves, so to obtain a causal spacetime, one "decompactifies" the timelike direction by unwrapping the $S^1$ factor to $\mathbb R^1$ while retaining the same expression for the metric (in other words one considers the universal cover of $\mathrm{AdS}_{p+2}$
It seems to me that since the metric has not changed, but the topology has, that at least at the classical level, one cannot locally distinguish between $\mathrm{AdS}_{p+2}$ and its universal cover.
The Question
Can one locally distinguish between the two at the quantum level?
Namely, if we were working with fully quantum AdS/CFT, would quantum effects due to global topology become apparent in local physics? I would think that the answer is yes in some sense based on intuition from a free particle moving in 1D on a circle, for example, versus on the entire real line.
Apologies for the vagueness of parts of the question.
