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Finite Cut-Off Holography and the DBI Counter-Term

Dileep P. Jatkar, Upamanyu Moitra

Abstract

We demonstrate some very special features of the Dirac-Born-Infeld--like (DBI) gravitational counter-term in AdS$_4$ spacetime, in the context of holography with a sharp radial cut-off. We show that the three-sphere partition function is not only independent of a constant radial cut-off, but also remains unchanged under deformations of the cut-off surface. We also consider the renormalized holographic entanglement entropy for an equatorial Ryu-Takayanagi surface with a cut-off with an arbitrary shape and show that it can also be independent of the cut-off under a special condition. We also numerically study the behavior of the renormalized entropy with different counter-terms and relate the results to monotonicity properties under holographic renormalization group flow. The DBI counter-term is always seen to be associated with integrating out fewer degrees of freedom compared to other counter-terms.

Finite Cut-Off Holography and the DBI Counter-Term

Abstract

We demonstrate some very special features of the Dirac-Born-Infeld--like (DBI) gravitational counter-term in AdS spacetime, in the context of holography with a sharp radial cut-off. We show that the three-sphere partition function is not only independent of a constant radial cut-off, but also remains unchanged under deformations of the cut-off surface. We also consider the renormalized holographic entanglement entropy for an equatorial Ryu-Takayanagi surface with a cut-off with an arbitrary shape and show that it can also be independent of the cut-off under a special condition. We also numerically study the behavior of the renormalized entropy with different counter-terms and relate the results to monotonicity properties under holographic renormalization group flow. The DBI counter-term is always seen to be associated with integrating out fewer degrees of freedom compared to other counter-terms.
Paper Structure (11 sections, 72 equations, 2 figures)

This paper contains 11 sections, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The DBI Counter-Term and Topological Partition Function
  3. The 3-Sphere Partition Function at a Finite Radial Cut-Off
  4. Is $F_{\text{DBI}}$ a Topological Invariant?
  5. Ryu-Takayanagi Surfaces and the DBI Counter-Term
  6. A Flat Boundary
  7. Renormalized Entropy in Global AdS Spacetime
  8. Arbitrary Cut-Out Equatorial Disk
  9. Numerical Results for the Polar Caps
  10. Discussion
  11. Components of the Ricci Tensor for the Equatorial Cut-Out

Figures (2)

  • Figure 1: An illustration of the set-up under consideration in §\ref{['subsec-rtcut']}. The sphere represents the asymptotic infinity. The surface in blue is a non-spherical surface at finite cut-off, which intersects the equatorial Ryu-Takayanagi surface along the red curve.
  • Figure 2: Variation of the renormalized entanglement entropy with the radial FG cut-off $\epsilon_{\mathrm{FG}}/L$ for a fixed angle $\theta_0 = \pi/4$ for the two different counter-terms.