# Why do we say that AdS/CFT is a background independent definition of string theory?

It is usually said that AdS/CFT is a background independent definition of string theory, how this concept emerge from the AdS/CFT correspondence?

We can define string theory on other manifolds beside those which are related to AdS/CFT paradigm.

Thanks

• Could you provide some references for this claim? I have heard it also and would like to read more about it. – pathintegral Mar 21 '19 at 19:14
• @pathintegral Polchinski and Horowitz have discussed about it in the book by Daniele Oriti "approachs to quantum gravity" part 2 chapter 10, but it is not too detailed. – Arian Mar 21 '19 at 21:43

In contrast to the ordinary definition of string theories by their perturbation series around a fixed background - which are hence not background-independent - a conformal field theory is well-defined non-perturbatively and need therefore not expand around any background. Hence taking the AdS/CFT correspondence as the definition of the string theory on the AdS side defines a background-independent string theory (but note that unless the AdS/CFT correspondence is rigorously established for the specific case in question, there is no guarantee that this "string theory" coincides with the usual peturbatively defined string theory).

No one is saying that the AdS/CFT correspondence is able to define string theories on non AdS spacetimes, unless you are referring to hopes that a generalization of the correspondence might also hold for string theories on other spacetimes.

• What do you mean by “background”? Is it a given topology of the string world sheet? – pathintegral Mar 21 '19 at 19:15
• @pathintegral The background is "flat target space". We really only can do the ordinary string path integral/sum over worldsheets when the target space (in the viewpoint of string theory being a $\sigma$-model) is flat. When the target space is curved with an aribtrary metric $G_{\mu\nu}$, the only known way to deal with it is to split it as $\eta_{\mu\nu} + \chi_{\mu\nu}$ and recognize the $+\chi_{\mu\nu}$ term inside the path integral as being the same as the insertion of a graviton vertex operator. The $\eta$ is the flat Minkowski metric and it is the "background" meant here. – ACuriousMind Mar 21 '19 at 19:28
• @ACuriousMind thanks for answer, background independence is limited to a class of backgrounds which are asymptotically AdS, right? From the derivation of correspondence, as I understand, it is a perfect AdS5×S5(for this specific case) that the string theory is defined on it and not on a asymptotically AdS5×S5. Does this mean that string theory depends on AdS background ? – Arian Mar 21 '19 at 22:03
• Adding to the above, there is also a bulk viewpoint of background independence: in AdS/CFT it is the boundary of the bulk that has AdS boundary conditions. In the bulk one imagines a path integral over all spacetime geometries (and other string fields) with those boundary conditions (and perturbations thereof) and it is only in a low energy semi-classical approximation that AdS makes any appearance at all in the bulk. So it is background independent also from a bulk viewpoint. – Wakabaloola Mar 22 '19 at 10:44
• @Arian You seem to be misunderstanding what "background" means. Yes, AdS/CFT only works for string theories whose target space is AdS. But that has nothing to do with a "background", it's just a subset of string theories. "background independence" w.r.t. to the metric means that we are able to describe the theory on a target space with metric $G_{\mu\nu}$ without having to split it into the Minkowski metric and a perturbation around it, see my first comment. – ACuriousMind Mar 22 '19 at 19:37