Scheme dependence and instability of double-trace deformations for gauge fields in AdS$_5$

Abstract

In holography, double-trace deformations provide a general framework for deforming boundary field theories. In particular, they can be utilized to introduce dynamical gauge fields in the boundary theory through double-trace deformations of bulk gauge fields. In this work, we study this construction in the case where the bulk geometry is asymptotically AdS$_5$, and find that such a system involves tachyon and ghost modes. This instability originates from the logarithmic behavior of the gauge fields in the vicinity of the AdS boundary, which leads to a scheme-dependent ambiguity in the double-trace deformation. We investigate this instability by using both analytical and numerical methods in several holographic setups, including bottom-up models and the top-down D3-D7 construction.

Figures (7)

• Figure 1: Diagrammatic representation of Eq. (\ref{['eq:RPA']}). The cyan circle for $G^{mn}$ represents the holographic Green's function without dynamical gauge fields. The propagator for $V_{mn}$ is the free-photon propagator. Each vertex brings the coupling constant $\sqrt{\lambda}$.
• Figure 2: $1/\lambda$ as a function of $-i\omega$ with $u_{\rm ct} = 2^{-1/2}u_{0}$ in the SAdS$_{5}$ spacetime. The gray region indicates the first quadrant: $-i\omega\geq 0$ and $1/\lambda \geq 0$. The curve in this region implies the presence of the unstable modes for the corresponding choice of $\lambda$. Note that we set $k_{x} =0$.
• Figure 3: Locations of the quasinormal modes for $\lambda=2$ with $u_{\rm ct} = 2^{-1/2}u_{0}$. Note that $\omega=\omega_{\rm R} + i \omega_{\rm I}$.
• Figure 4: Real part of the AC conductivity with dynamical electromagnetic fields for $\mu/(\pi T) = 0, 1.0$ and $2.0$. We set $\lambda = 1/10$ and $u_{\rm ct}= 2^{-1/2} u_{0}$. Note that the plot is shown in the log-log scale.
• Figure 5: $1/\lambda$ as a function of $-i\omega$ with $u_{\rm ct}=2^{-1/2}u_{0}$ for $\mu/(\pi T) = 0, 1, 2$ in the RN-AdS$_{5}$ spacetime. Here, we set $k_{x}=0$.