Is it renormalization running? Or SU(N) Gauge transformation?
Or nothing, because AdS/CFT is only a correspondence up to gauge orbits of each side?
Or nothing, because a localized diffeomorphism in the bulk doesn't affect the AdS boundary where the correspondence lives?
This is the right answer. Bulk diffeomorphism invariance emerges along with the bulk space itself. From the conclusion of hep-th/0209104:
Another reference http://arxiv.org/abs/gr-qc/0602037
"In fact in most examples of duality there are gauge symmetries on both sides and these are unrelated to each other: the duality pertains only to the physical quantities."
Something else comes to mind. The dilaton inherent in string theory, and therefore AdS/CFT, breaks the equivalence principle. Does this mean diffeomorphism invariance is also broken? In other words, is it possible to have a diffeomorphism invariant theory which violates the equivalence principle?
I don't know what the situation is with the string dilaton, but in classical theories with a dynamic metric field, one can get equivalence principle violations by non-minimal coupling of matter and metric (eg. say coupling to higher derivatives). More precise definitions and extensive discussion here:
"As discussed in section 3, following Will’s book one can argue that the EEP can only be satisfied if there exists some metric and the matter fields are coupled to it not necessarily minimally but through a non-constant scalar..."
I will have to think about this for a while. There are a lot of prior issues I need to understand, for example the relationship between worldsheet diffeomorphism invariance and spacetime diffeomorphism invariance, and I have to find out which of the issues are just a problem for me and my incomplete understanding, and which are unsolved problems even for the experts! Meanwhile, there are no dilatons in these papers, but arXiV:0805.2203 and "staff.science.uva.nl/~jdeboer/publications/francqui.pdf"[/URL] look at some of the space-time diffeo issues.
As I understand:
The dilaton will mess up the equivalence principle only if it remains a massless scalar. But if it gains a vev and becomes massive, its long range effects cease to interfere with that of the gravitational field. There are ways in string theory to give such a vev (or equivalently, to have a PE term for the dilaton).
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