The double-trace deformation effect in AdS/CFT

Question

Let me use this paper as the reference for this.

I want to understand better the argument at the bottom of page 6.

If the bulk $AdS$ metric is written as $\frac{1}{r^2}(dr^2 + A(r)ds_{boundary}(x)^2)$ then the free massive scalars fields on it are of the form, $\phi = r^{\Delta_{+}}(\alpha(x)) + r^{\Delta_{-}}(\beta(x))$,near $r=0$. (...and the numbers $\Delta_{\pm}$ depend on the dimensionality of the AdS and the mass of the field..)

• As per their notation their "regular" boundary condition in the bulk corresponds to setting $\beta(x)$ (the coefficient of $r^{\Delta_{-}}$) to zero on the boundary and "regular" boundary condition is setting $\alpha(x)$ (the coefficient of $r^{\Delta_{+}}$) to zero.

  How is this compatible with the fact that on using the regular boundary condition the dual CFT necessarily needs to have a term of the form $\int d^d x \beta(x)O(x)$ where $O$ has dimension $\Delta_{+}$ and $\alpha = \langle O \rangle$?

  I thought that they themselves said that in the regular scenario the $\beta$ is being set to $0$ at the boundary - then what is this $\beta(x)$ that is featuring in the boundary action?

• Below equation 3.19 they argue that the when the double-trace deformation of the boundary CFT is switched off it is seeing the "irregular" scenario and when it is turned to infinity it is seeing the regular scenario.

  But how does one argue that the regular scenario is the the IR fixed point and the irregular scenario is the UV fixed point for the boundary CFT? Where is that argument?

• Rethinking the section 4 of this paper in a different way - Suppose one wants to calculate the determinant of the bulk theory $det(-\nabla^2 + m^2)$ by taking product of eigenvalues. One wants to say calculate the determinant when the bulk has been quantized with the $\Delta_{+}$ boundary condition. (...one can presumably ask the same question with $\Delta_{-}$ as well..)

  If the bulk is $AdS_{d+1}$ then one can see that the small-r asymptotics ($r=0$ is the AdS boundary in the Poincare patch) of the harmonics are of the form, $# r^{a} + # r^{b}$ for some values $a$ and $b$ (which depend on the eigenvalue and $d$).

  Now knowing the above small-r asymptotics of the harmonics how does one pick out which of these will contribute to the above determinant with say the $\Delta_{+}$ boundary condition? Can someone schematically sketch how the calculation looks like?

Answer 1

A possible hint :

Following equations $2.5 \to 2.7$), we may define Kernels - Fourier transform of the $2$-point function - for the limiting cases ($f=0, f = +\infty$) :

$$G_{\pm}(k) \sim \int d^dk ~e^{ik.x} \frac{1}{x^{2 \Delta_\pm}}$$

We have then : $G_{\pm}(k) \sim ~ k^{\pm2\nu}$, where $\nu > 0$

We see, that in the UV, the kernel $G^+$ diverges, so it is not relevant in the UV, but converges in the IR. In the same manner, in the IR, the kernel $G^-$ diverges, so it is not relevant for the IR, but converges in the UV, so it would seem natural to associate the conformal dimension $\Delta^-$ ($f=0$), with the UV and the conformal dimension $\Delta^+$($f=+\infty$) with the IR.

We would have a RG flow which begins with $f=0$ in the UV, to finish at $f=+\infty$ in the IR

Finally, a list of the terms employed in the paper, which are not always clear:

$$\begin {matrix} UV & IR \\ f=0 & f=+\infty\\ "irregular" quantization & "regular" quantization\\ \Delta^- & \Delta^+\\ "irregular" boundary\, value & "regular" boundary \, value\\ \alpha = source & \beta = source\\ \beta = \langle O\rangle & \alpha = \langle O\rangle\\ \gamma = - \Delta^- &\gamma= + \infty\\ \end{matrix}$$