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$\langle TT\rangle$ correlator of the boundary CFT from metric fluctuations in the bulk gravity

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  • Is there a reference which explains how the $\langle TT\rangle $ correlation of the boundary conformal field theory (CFT) can be holographically calculated from the bulk gravity? (..I am often getting referred to some paper by Skenderis et. al but I would like to see some more explicit explanation of the idea..)

In the following we defined $C_T$ as the coefficient of the leading singularity of the $\langle TT\rangle $ correlation.


Roughly that I understand the following possible algorithm works -

  • first expand the gravity Lagrangian about an AdS background making a fluctuation (say $h$ ) along two orthogonal spatial directions of the boundary. Like say add to the metric two off-diagonal components of the type, $g_{xy} = g_{yx} = \epsilon h (r,z)$ where $x,y,z$ are 3 randomly chosen orthogonal spatial directions on the boundary and $r$ is the bulk direction orthogonal to the boundary.

  • Now pick out all terms in the Lagrangian which are quadratic in $h$ and focus on all the terms which have a $\partial_r$ and rewrite those terms by pulling out a total derivative in $r$. (...I don't know how to make sense of this if one has terms of the kind $(\partial _r h)(\partial_r ^2 h)$ (and such terms do occur in examples I have tried)

  • Now in this "boundary Lagrangian" which is formed by the total derivatives in $r$ and quadratic in $h$ terms substitute a Fourier expansion for $h$ as $h(r,z) = \sum_p e^{ipz}H_p(r)$.

  • Now apparently there is (a universal? irrespective of the gravity Lagrangian?) value for this $H$ function which makes the fluctuation go on-shell and for Gauss-Bonnet gravity that apparently is, $H_p(r) = c BesselK[ d/2, (L^2p)/(\sqrt{f_\infty} )]$. And this $c$ is determined by demanding that $H_p (r \rightarrow \infty) = 1$ (where $r=\infty$ is the boundary) [...here its not clear to me as to how will this $f_\infty$ be defined from just the GB Lagrangian - AFAIK this $f_\infty$ is a parameter that is defined only when one is looking for asymptotically $AdS$ blackholes in GB gravity)

  • Now expand in a power series in $r$ about $r=\infty$ this boundary Lagrangian and then (for any of its Fourier mode $k$?) for a d-dimensional boundary if the $d$ is even then this $C_T$ is given by the coefficient of the $k^dlog k$ terms and for $d$ odd this is given by the $k^d$ term.


  • I would like to know if I understand the above algorithm right and if so then what is the reference for its derivation.
This post imported from StackExchange Physics at 2014-06-21 09:03 (UCT), posted by SE-user user6818
asked Jun 20, 2014 in Theoretical Physics by user6818 (955 points) [ no revision ]
Appendix B.5 in qpt.physics.harvard.edu/p234.pdf addresses this derivation. In principle what you are saying is OK, but it is the most non-invariant recipe I have ever seen. One takes the quadratic part of the action, evaluates it on the solutions to linearized equations (propagators). By definition the result is a total derivative, so it resides on the boundary. Careful regularization is needed since the boundary is not a true boundary, so one moves it to the finite distance $\epsilon$ and takes the limit at the end. arxiv.org/pdf/hep-th/0209067.pdf should be useful.

This post imported from StackExchange Physics at 2014-06-21 09:03 (UCT), posted by SE-user John
@John Thanks for the references! Can you explain what you mean by "evaluates it on the solutions to linearized equations" ? So one has truncated the Lagrangian to second order in fluctuations - now you want to find the equations of motion of the fluctuations for this truncated Lagrangian? (2) Also do you recognize this $H_p$ function that I have quoted? From where does this come? (3) Also how does one define the boundary term corresponding to these quadratic in fluctuation terms? I am not seeing an unambiguous way of doing it

This post imported from StackExchange Physics at 2014-06-21 09:03 (UCT), posted by SE-user user6818

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