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Deformations of the Almheiri-Polchinski model

Hideki Kyono, Suguru Okumura, Kentaroh Yoshida

Abstract

We study deformations of the Almheiri-Polchinski (AP) model by employing the Yang-Baxter deformation technique. The general deformed AdS$_2$ metric becomes a solution of a deformed AP model. In particular, the dilaton potential is deformed from a simple quadratic form to a hyperbolic function-type potential similarly to integrable deformations. A specific solution is a deformed black hole solution. Because the deformation makes the spacetime structure around the boundary change drastically and a new naked singularity appears, the holographic interpretation is far from trivial. The Hawking temperature is the same as the undeformed case but the Bekenstein-Hawking entropy is modified due to the deformation. This entropy can also be reproduced by evaluating the renormalized stress tensor with an appropriate counter term on the regularized screen close to the singularity.

Deformations of the Almheiri-Polchinski model

Abstract

We study deformations of the Almheiri-Polchinski (AP) model by employing the Yang-Baxter deformation technique. The general deformed AdS metric becomes a solution of a deformed AP model. In particular, the dilaton potential is deformed from a simple quadratic form to a hyperbolic function-type potential similarly to integrable deformations. A specific solution is a deformed black hole solution. Because the deformation makes the spacetime structure around the boundary change drastically and a new naked singularity appears, the holographic interpretation is far from trivial. The Hawking temperature is the same as the undeformed case but the Bekenstein-Hawking entropy is modified due to the deformation. This entropy can also be reproduced by evaluating the renormalized stress tensor with an appropriate counter term on the regularized screen close to the singularity.

Paper Structure

This paper contains 12 sections, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Yang-Baxter deformations of AdS$_2$
  3. Coset construction of AdS$_2$
  4. The general Yang-Baxter deformation
  5. A brief review of the AP model
  6. 1+1 D dilaton gravity system
  7. The AP model
  8. Deforming the AP model
  9. The deformed AP model
  10. A deformed black hole solution
  11. The boundary computation of entropy
  12. Conclusion and discussion

Figures (2)

  • Figure 1: Penrose diagram of the black hole solution. The global AdS$_2$ is parametrized by $\tau$ and $\theta$ . The largest red triangle describes the Poincaré patch of AdS$_2$ as usual. The coordinate system (\ref{['undeformed BH1']}) covers the inside of smaller triangle bounded by green lines. The right vertex corresponds to the black hole horizon which is specified as the point that $T$ is finite but $Z$ is infinity.
  • Figure 2: Penrose diagram of the deformed black hole. In this diagram, a curvature singularity is depicted in the global AdS$_2$ coordinates with $\alpha=1/2$, $\beta=0$ and $\gamma=\mu/2$ in (\ref{['def-para']}), where $\tau$ and $\theta$ are the same global coordinates as in the undeformed AdS$_2$ . The black curves represent the curvature singularities of the deformed spacetime. In the blue region, the scalar curvature is positive, while in the red and orange regions, it takes negative values. The black hole coordinates in (\ref{['d-BH']}) covers the interior bounded by the green lines. By employing a Schwarzschild-like coordinate system (\ref{['d-Sch']}) , we focus on the orange region in order to evaluate the black hole entropy.