- Is there a reference which explains how the ⟨TT⟩ 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 CT as the coefficient of the leading singularity of the ⟨TT⟩ 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, gxy=gyx=ϵh(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 ∂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 (∂rh)(∂2rh) (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)=∑peipzHp(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, Hp(r)=cBesselK[d/2,(L2p)/(√f∞)]. And this c is determined by demanding that Hp(r→∞)=1 (where r=∞ is the boundary) [...here its not clear to me as to how will this f∞ be defined from just the GB Lagrangian - AFAIK this f∞ 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=∞ this boundary Lagrangian and then (for any of its Fourier mode k?) for a d-dimensional boundary if the d is even then this CT is given by the coefficient of the kdlogk terms and for d odd this is given by the kd 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