General properties of pole-skipping points

Abstract

Pole-skipping is a phenomenon in holography theory, making the corresponding retarded Green's function undetermined. This phenomenon is emphasized through the idea to link quantum chaos and gravity theory, especially using the energy-density Green's function. The pole-skipping points can be also defined for other Green's functions, and the general properties of pole-skipping points are studied by dealing each cases. It is found that EOM(Equation Of Motion)s of each field perturbations have general forms for bosonic and fermionic cases. Considering the EOMs come from multiple equations of various field components, having a general equation form deserves attention. The EOM structures lead the general properties of pole-skipping points such as Matsubara frequency. The frequency of the leading pole-skipping points depends on the type of perturbation field and spin number. In near horizon analysis, even though the most of pole-skipping points are obtained by Frobenius method, the leading pole-skipping points can not be found by the same method. These exceptional pole-skipping points are divided into two cases(special case I and II). This is a result of attempts to make a standard way to get pole-skipping points with the exceptions. In AdS_{d+2}- RN background with axion, the complex pole-skipping points are also obtained. Considering the Green's functions G live in 4D Fourier space, the complex results seem like natural. However, it is not trivial story to get a pole-skipping point(0D) by overlap of 2D zero and pole surfaces of G. The interpretations of pole-skipping point with 1D zero and pole lines are considered carefully.