Holographic $a$-functions and Boomerang RG Flows

TL;DR

This paper constructs a monotonic holographic $a$-function for non-relativistic RG flows using the null energy condition, demonstrating its behavior in geometries with Boomerang RG flows and linking it to fixed point properties.

Contribution

It introduces a new $a$-function for non-relativistic holographic RG flows and tests its monotonicity in geometries with Boomerang RG flows, connecting it to fixed point characteristics.

Findings

The $a$-function decreases monotonically along the flow.

It approaches a constant at asymptotic regimes.

The $a$-function encodes fixed point central charge and speed of light.

Abstract

We use the radial null energy condition to construct a monotonic $a$-function for a certain type of non-relativistic holographic RG flows. We test our $a$-function in three different geometries that feature a Boomerang RG flow, characterized by a domain wall between two AdS spaces with the same AdS radius, but with different (and sometimes directions dependent) speeds of light. We find that the $a$-function monotonically decreases and goes to a constant in the asymptotic regimes of the geometry. Using the holographic dictionary in this asymptotic AdS spaces, we find that the $a$-function not only reads the fixed point central charge but also the speed of light, suggesting what the correct RG charge might be for non-relativistic RG flows.