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

Kedar S. Kolekar, K. Narayan

TL;DR

This work develops a 2D dilaton-gravity framework, descended from dimensional reduction of extremal branes with AdS$_2$ near-horizon regions, to formulate holographic RG flows and a dilatonic c-function ${\\cal C}(u)=\\frac{\\Phi(u)^2}{4G_2}$. Using null energy conditions, the authors argue that ${\\cal C}(u)$ decreases along the flow and approaches the extremal black brane entropy $S_{BH}=\\frac{\\Phi_h^2}{4G_2}$ in the IR AdS$_2$ limit, with explicit checks in nonconformal hvLif backgrounds such as D2–M2 and D4–M5 systems. They compare this dilatonic c-function to entropic c-functions and to holographic c-functions in other settings, highlighting its extensivity and dependence on transverse volumes. A 2D radial Hamiltonian (deBoer–Verlinde–Verlinde) analysis is adapted to derive RG flow equations and beta-functions for 2D couplings, revealing that placing the AdS$_2$ throat inside nonconformal phases generally yields nonvanishing beta-functions at the IR fixed point, while conformal reductions consistently accommodate AdS$_2$ as a fixed point. Overall, the paper provides a coherent 2D holographic RG picture for AdS$_2$ holography, clarifies when a c-theorem can hold, and outlines directions toward a Wilsonian 2D holographic RG and potential 2D de Sitter extensions.

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 $β$-functions.

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

TL;DR

This work develops a 2D dilaton-gravity framework, descended from dimensional reduction of extremal branes with AdS near-horizon regions, to formulate holographic RG flows and a dilatonic c-function . Using null energy conditions, the authors argue that decreases along the flow and approaches the extremal black brane entropy in the IR AdS limit, with explicit checks in nonconformal hvLif backgrounds such as D2–M2 and D4–M5 systems. They compare this dilatonic c-function to entropic c-functions and to holographic c-functions in other settings, highlighting its extensivity and dependence on transverse volumes. A 2D radial Hamiltonian (deBoer–Verlinde–Verlinde) analysis is adapted to derive RG flow equations and beta-functions for 2D couplings, revealing that placing the AdS throat inside nonconformal phases generally yields nonvanishing beta-functions at the IR fixed point, while conformal reductions consistently accommodate AdS as a fixed point. Overall, the paper provides a coherent 2D holographic RG picture for AdS holography, clarifies when a c-theorem can hold, and outlines directions toward a Wilsonian 2D holographic RG and potential 2D de Sitter extensions.

Abstract

Extremal black branes upon compactification in the near horizon throat region are known to give rise to 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 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 -functions.

Paper Structure

This paper contains 16 sections, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reviewing $AdS_2$ dilaton gravity from higher dimensional redux
  3. The 2-dim theory and the attractor conditions
  4. Null energy conditions and a c-function
  5. A holographic c-function
  6. Null energy conditions from the $2$-dim perspective
  7. The c-function in the $M2$-$D2$ system
  8. c-function in the $M5$-$D4$ system
  9. On dilatonic and entropic c-functions
  10. 2-dim radial Hamiltonian formalism and $\beta$-functions
  11. $\beta$ functions for conformal/non-conformal theories
  12. Examples
  13. Discussion
  14. Effective potential in $D$-dim gravity-scalar theory
  15. Radial Lagrangian \ref{['2dradialL']} from dimensional reduction
  16. ...and 1 more sections

Figures (1)

  • Figure 1: A cartoon of the bulk spacetime with the holographic RG flow in the radial direction to the infrared $AdS_2$ throat region from the far (UV) region through possible intermediate regions (and associated RG transition scales).