Derivation of Eq. 7.12 in the review paper of Kraus

Question

I'm reading "Lectures on black holes and the $AdS_3/CFT_2$ 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 $AdS_{3}$ vacuum, one can evaluate the action at the $AdS_3$ vacuum by using the variation of the action with respect to the boundary metric $g^{(0)}$ and the gauge fields $A^{(0)}, \tilde A^{(0)}$

\begin{equation} \delta S= \int d^2x \sqrt{g}\,\left[ \frac12 T^{ij} \delta g_{ij}+\frac{i}{2\pi} J^{i} \delta A_{i} +\frac{i}{2\pi}\tilde J^{i} \delta \tilde A_{i} \right]~. \end{equation} where the superscript $(0)$ is omitted for brevity. Reexpressing this in complex coordinates of the boundary metric, we obtain:

\begin{equation} \delta S=4\pi i \left(T_{ww}\delta \tau+T_{\bar w\bar w}\delta \bar\tau +\frac{\tau_2}{\pi} J_{w}\delta A_{\bar w}+\frac{\tau_2}{\pi} \tilde J_{\bar w}\delta \tilde A_{w}\right)~. \end{equation}

One can integrate the above equation to get: \begin{eqnarray} \mathcal{S}(\tau) & = & - 2 \pi i \tau \big(L_{0} - \frac{c}{24} \big) + 2 \pi i \bar\tau \big(\tilde L_{0} - \frac{\tilde c}{24} \big) \cr && \; - \frac{i\pi}{2} k \big( \tau A_{w}^{2} + \bar \tau A_{\bar w}^{2} + 2 \bar \tau A_{w} A_{\bar w} \big) + \frac{i\pi}{2} \tilde k \big( \tau \tilde A_{w}^{2} + \bar\tau \tilde A_{\bar w}^{2} + 2 \tau \tilde A_{w} \tilde A_{\bar w} \big) \, . \end{eqnarray}

I would like to know the derivation of the last equation.

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Comments

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Answer 1

The answer is quite simple. You should use eqs.(6.8) in the paper. You put them into the last term of eq.(7.11) then, a straightforward integration, I mean something like $\delta\tau\rightarrow\tau$, $\delta A_{\bar w}\rightarrow A_{\bar w}$ and so on, should do the job.

So, let us consider (note that in your post there is a wrong sign)

$$ \delta S=(2\pi)^2 i \left(-T_{ww}\delta \tau+T_{\bar w\bar w}\delta \bar\tau +\frac{\tau_2}{\pi} J_{w}^I\delta A_{I\bar w}+\frac{\tau_2}{\pi} \tilde J_{\bar w}^I\delta \tilde A_{Iw}\right)_{constant}~. $$

(here "constant" means that only the zero mode is retained) and the corresponding eqs.(6.8) in Kraus' review

$$\eqalign{ T_{ww}&=-{k \over 8\pi} +{1 \over 8\pi} A_w^2+{1 \over 8\pi} {\tilde A}_w^2~, \cr T_{{\bar w}{\bar w}}&= -{{\tilde k} \over 8\pi} +{1 \over 8\pi}A_{{\bar w}}^2+{1 \over 8\pi}{\tilde A}_{{\bar w}}^2~, \cr J^I_w &= {i\over 2} k^{IJ} A_{Jw}~, \cr {\tilde J}^I_{{\bar w}}& = {i\over 2} {\tilde k}^{IJ} {\tilde A}_{J{\bar w}}~.}$$

By substitution one has

$$ \delta S = (2\pi)^2 i\left[-\left(-{k \over 8\pi} +{1 \over 8\pi} A_w^2+{1 \over 8\pi} {\tilde A}_w^2~\right)\delta\tau+\left(-{{\tilde k} \over 8\pi} +{1 \over 8\pi}A_{{\bar w}}^2+{1 \over 8\pi}{\tilde A}_{{\bar w}}^2~\right)\delta{\bar\tau}\right.$$ $$\left.+\frac{i\tau_2}{2\pi}k^{IJ} A_{Jw}\delta A_{I\bar w}+\frac{i\tau_2}{2\pi}{\tilde k}^{IJ}{\tilde A}_{J{\bar w}}\delta \tilde A_{Iw}\right]_{constant}.$$

From this you get immediately the result when you note that the variation with respect to the gauge field just cancels out the $\tau_1$ contribution, having ${\bar\tau}-\tau$ that comes from the squared terms, and is recovered upon integration.

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Comments

What would you substitute for the stress-energy tensors $T_{ww}, T_{\bar w \bar w}$ and the currents $J_w, \tilde J_{\bar w}$?

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Eqs.(6.8) in the Kraus' review.

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I thought plugging 6.8 into 7.11 would give us 7.12. But it does not. How would you explain the relative sign between $A^2_w$ and $\widetilde A^2_w$? Why there are no term like $\tau A_wA_{\bar w}$ even though you have $\tau_2$ in 7.11? –

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I have just expanded the computation in my previous answer.

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Thank you very much. But could you explain more elaborately how you obtain from the last equation you wrote to 7.12 in the paper? I have obtained the last equation you wrote before I post this question on the website. I would like to know the gap between this equation and 7.12. I am sorry about my stupid question, but I still don't understand it.

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The question is that eq.(7.11) would be right if instead of $\tau_2$ you will put $\tau$ and $\bar\tau$ respectively in the current terms. Besides, Kraus refers to a not existing sec. 4.4. I fear that this review should have some problems in the presentation.

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How would you explain the relative sign between $A_w^2$ and $\tilde A_w^2$ even though you substitute $\tau$ and $\bar\tau$ respectively as you mentioned. The section 4.4 should be the section 6.4 since this review paper largely depends on [the paper](http://arxiv.org/abs/hep-th/0607138) by Kraus and Larsen. So you cannot reproduce 7.12, can you?

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If you mean in the current terms, this sign difference should arise from $k$ and $\tilde k$ factors.

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Anyway, could you possibly write how you reproduce 7.12?

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If I look at the equation (4.14) in the paper by Kraus and Larsen my suspect that they did a mistake is enforced. At this point the best solution is to get in touch with them.

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OK. Thank you very much.

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