Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

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

+ 5 like - 0 dislike
777 views
  • 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 (960 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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...