I'm reading "Lectures on black holes and the AdS3/CFT2 correspondence" by Kraus.
http://arxiv.org/abs/hep-th/0609074
I don't know how one can obtain Eq.7.12. My stupid question is how to obtain this equation. After this equation, it is stated that "one has to take care to consider only variations consistent with the equations of motion and the assumed boundary conditions". What are the variations consistent with the equations of motion and the assumed boundary conditions? What Krasu says is as follows.
To compute the bulk functional integral, we need to evaluate the bulk action for the solutions
which contribute, including boundary counterterms if necessary. For an on-shell solution around the AdS3 vacuum, one can evaluate
the action at the AdS3 vacuum by using
the variation of the action with respect to the boundary metric g(0) and the
gauge fields A(0),˜A(0)
δS=∫d2x√g[12Tijδgij+i2πJiδAi+i2π˜Jiδ˜Ai] .
where the superscript
(0) is omitted for brevity.
Reexpressing this in complex coordinates of the boundary metric, we obtain:
δS=4πi(Twwδτ+Tˉwˉwδˉτ+τ2πJwδAˉw+τ2π˜Jˉwδ˜Aw) .
One can integrate the above equation to get:
S(τ)=−2πiτ(L0−c24)+2πiˉτ(˜L0−˜c24)−iπ2k(τA2w+ˉτA2ˉw+2ˉτAwAˉw)+iπ2˜k(τ˜A2w+ˉτ˜A2ˉw+2τ˜Aw˜Aˉw).
I would like to know the derivation of the last equation.
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