# Does local physics depend on global topology?

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.

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You're thinking along the lines of - when you go to the IR limit, the modes can feel out the topology? I've no idea, but great question! – twistor59 Feb 20 at 17:52
@twistor59 Yeah that's along the lines of what I'm thinking of. – joshphysics Feb 20 at 17:56
Do you know about Kaluza-Klein modes (KK modes)? They are resonances that enter into the spectrum when you compactify your manifold (so for example $M_d \rightarrow M_{d-1} \times S^1$), and indeed to $(d-1)$-dimensional observers, these are just modes in the spectrum, so 'local physics' that can be probed with a sufficiently good accelerator. – Vibert Feb 20 at 21:49
@Vibert Yeah I have a bit of familiarity with KK modes. Thanks for the feedback; that seems reasonable. – joshphysics Feb 20 at 21:58