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Fix the dual geometries of $T\bar{T}$ deformed CFT$_2$ and highly excited states of CFT$_2$

Peng Wang, Houwen Wu, Haitang Yang

TL;DR

The paper develops a holographic, kinematics-driven method to fix bulk dual geometries from entanglement entropies, without relying on bulk dynamics. Applied to $T\bar{T}$-deformed CFT$_2$, it yields a deformed BTZ metric with a finite radial cutoff tied to the deformation parameter and reproduces the theory’s energy spectrum and propagation speeds. The authors extend the framework to highly excited states, obtaining dual geometries that are conical defects or BTZ black holes depending on the heavy operator dimension. This approach provides a universal way to infer bulk spacetime structure from boundary entanglement data while clarifying the role of a finite holographic region in the AdS$_3$/CFT$_2$ correspondence.

Abstract

In previous works, we have developed an approach to fix the leading behaviors of the pure AdS$_3$ and BTZ black hole from the entanglement entropies of the free CFT$_2$ and finite temperature CFT$_2$, respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the $T\bar{T}$ deformed CFT$_2$, which breaks the conformal symmetry. In terms of the physical arguments of the $T\bar{T}$ deformed CFT$_2$, the derived metric is a deformed BTZ black hole. The requirement that the CFT$_2$ lives on a conformally flat boundary leads to $r_{c}^{2}=\ 6R_{AdS}^{4}/(πcμ)$ naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from $T\bar{T}$ deformed CFT$_2$. We furthermore fix the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.

Fix the dual geometries of $T\bar{T}$ deformed CFT$_2$ and highly excited states of CFT$_2$

TL;DR

The paper develops a holographic, kinematics-driven method to fix bulk dual geometries from entanglement entropies, without relying on bulk dynamics. Applied to -deformed CFT, it yields a deformed BTZ metric with a finite radial cutoff tied to the deformation parameter and reproduces the theory’s energy spectrum and propagation speeds. The authors extend the framework to highly excited states, obtaining dual geometries that are conical defects or BTZ black holes depending on the heavy operator dimension. This approach provides a universal way to infer bulk spacetime structure from boundary entanglement data while clarifying the role of a finite holographic region in the AdS/CFT correspondence.

Abstract

In previous works, we have developed an approach to fix the leading behaviors of the pure AdS and BTZ black hole from the entanglement entropies of the free CFT and finite temperature CFT, respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the deformed CFT, which breaks the conformal symmetry. In terms of the physical arguments of the deformed CFT, the derived metric is a deformed BTZ black hole. The requirement that the CFT lives on a conformally flat boundary leads to naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from deformed CFT. We furthermore fix the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.

Paper Structure

This paper contains 5 sections, 59 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. The dual geometry of the $T\bar{T}$ deformed CFT$_2$
  3. The dual geometry of the $T\bar{T}$ deformed CFT$_2$
  4. The dual geometry of the highly excited states in CFT$_{2}$
  5. Conclusion

Figures (2)

  • Figure 1: The left picture is given by BCFT with $\Delta x\to\infty$. The middle picture is obtained from $L_{\mathrm{bulk}}$ by setting $y=a$, $y^{\prime}=\beta_{H}$ and $\triangle x/2\rightarrow\infty$. The right picture is also given by $L_{\mathrm{bulk}}$ from a different point of view, by setting $y=y^{\prime}=a$ and $\Delta x\to\infty$. The solid lines in all three pictures describe the same object.
  • Figure 2: The geodesic between $y=a$ and $y^{\prime}=\beta_{H}$ at a finite temperature system is mapped to a finite size system, corresponding to the radius of a circle from $y=a$ to $y=\frac{L_S}{2\pi}$.