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Suppose I have some Lagrangian of some higher derivative gravity coupled to a may be matter fields. Now I want to fluctuate it to quadratic order about an AdS background and calculate the 1-loop partition function.

Can someone point to a reference where such a thing might be done? This sounds like something standard which someone would have done...

I can't see how one can even write down the fluctuated Lagrangian to quadratic order...And even if I wrote that down, how does one impose that the background is AdS? Then the issues of gauge fixing and ghosts are only added complication..


I can imagine calculating the first order variation in the Christoffel symbols and Riemann and Ricci tensors but that goes only so far.

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    $\begingroup$ What is the problem to take Lagrangian and expand it to any order? as an example of how people expand you may have a look at arxiv.org/abs/hep-th/9901121 or other classical papers like Freedman, D'Hocker,... There are not so many results about one-loop diagrams with legs in AdS. Fast googling gives arxiv.org/abs/hep-th/0506185 and arxiv.org/abs/1007.2653 $\endgroup$ – John Jun 6 '14 at 20:13
  • $\begingroup$ @John Thanks! Let me see the references. What I am confused about is how is the gauge fixed and ghosts introduced for doing this path-integral over metric fluctuations...also has anyone calculated 4-point stress-tensor correlations from the gravity side? $\endgroup$ – user6818 Jun 7 '14 at 5:27
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    $\begingroup$ Before Faddeev and Popov discovered their ghosts people had already computed a lot in various gauges. It seems that AdS/CFT computations are still difficult enough for people, so that they do want to be so general as to introduce ghosts. At least you cannot see ghosts in any of the classical papers. The best you can find is four-point of scalars from exchange of something in the bulk arxiv.org/pdf/1404.5625.pdf and refs therein $\endgroup$ – John Jun 7 '14 at 6:58
  • $\begingroup$ @John Are you saying that there is a way to do gravity path integrals without worrying about gauge fixing ghosts? I have added a more specific question here - which you might want to help with - physics.stackexchange.com/questions/118424/… $\endgroup$ – user6818 Jun 8 '14 at 20:01

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