On $AdS_2$ holography from redux, renormalization group flows and $c$-functions

TL;DR

This paper explores the holographic renormalization group flow in 2D dilaton-gravity-matter theories derived from extremal black branes, proposing a c-function that interpolates between UV and IR fixed points, with implications for $AdS_2$ holography.

Contribution

It introduces a holographic c-function in 2D dilaton-gravity-matter theories and discusses its properties, including a c-theorem and its relation to black brane entropy, extending the understanding of $AdS_2$ holography.

Findings

Proposed a c-function based on the 2D dilaton that satisfies a c-theorem.

Demonstrated the c-function reduces to black brane entropy at the IR fixed point.

Compared the c-function behavior in various explicit examples, including nonconformal branes.

Abstract

Extremal black branes upon compactification in the near horizon throat region are known to give rise to $AdS_2$ dilaton-gravity-matter theories. Away from the throat region, the background has nontrivial profile. We interpret this as holographic renormalization group flow in the 2-dim dilaton-gravity-matter theories arising from dimensional reduction of the higher dimensional theories here. The null energy conditions allow us to formulate a holographic c-function in terms of the 2-dim dilaton for which we argue a c-theorem subject to appropriate boundary conditions which amount to restrictions on the ultraviolet theories containing these extremal branes. At the infrared $AdS_2$ fixed point, the c-function becomes the extremal black brane entropy. We discuss the behaviour of this inherited c-function in various explicit examples, in particular compactified nonconformal branes, and compare it with other discussions of holographic c-functions. We also adapt the holographic renormalization group formulated in terms of radial Hamiltonian flow to 2-dim dilaton-gravity-scalar theories, which while not Wilsonian, gives qualitative insight into the flow equations and $\beta$-functions.