[Paper Review] Comments on Single-Trace $T\bar T$ Holography

Soumangsu Chakraborty, Amit Giveon+1 more|arXiv (Cornell University)|Mar 22, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates the holographic duality between single-trace $T\bar{T}$ deformed CFTs and string theory in backgrounds interpolating between asymptotically linear dilaton spacetimes in the UV and $AdS_3$ in the IR. It shows that the $SL(2,\mathbb{R})$ invariant ground state energy becomes complex when the deformation coupling $\lambda$ exceeds $\lambda_c = 1/k$, signaling a maximal coupling beyond which the geometry becomes Euclidean, indicating a physical bound on the deformation strength.

ABSTRACT

We explore the holographic duality between string theory in backgrounds that interpolate between asymptotically linear dilaton spacetime in the UV and $AdS_3$ in the IR, and single-trace $T\bar T$ deformed CFT. In particular, we explain how the deformation of states in the boundary theory is reflected in the bulk geometry, and show that the coupling above which the deformed energy of the $SL(2,\mathbb{R})$ invariant ground state of the IR CFT becomes complex is a maximal coupling.

Motivation & Objective

• To understand how single-trace $T\bar{T}$ deformations in the boundary CFT are realized in the bulk geometry of string theory.
• To clarify the geometric and physical meaning of the maximal coupling $\lambda_c = 1/k$ in single-trace $T\bar{T}$ holography.
• To compare the behavior of the $SL(2,\mathbb{R})$ invariant vacuum in the deformed theory with that of higher-energy states and other deformation regimes.
• To establish a connection between the Hagedorn behavior of the boundary theory and the asymptotic linear dilaton geometry in the bulk.

Proposed method

• Analyzes the bulk geometry of string theory in backgrounds interpolating between linear dilaton spacetime (UV) and $AdS_3$ (IR), derived from single-trace $T\bar{T}$ deformed CFTs.
• Uses the boundary CFT relation $E(\lambda) = \frac{1}{\lambda R}\left[\sqrt{1 + 2\lambda R E(0) + (\lambda R P)^2} - 1\right]$ to study energy deformation and identify the critical energy scale $E_c \sim 1/\lambda R$.
• Examines the behavior of the $SL(2,\mathbb{R})$ invariant vacuum state in the deformed theory and computes its energy as a function of $\lambda$.
• Analyzes the spacetime metric, dilaton, and $B$-field in the bulk, showing that as $\lambda \to \lambda_c = 1/k$, the $B$-field and $e^{\Phi}$ vanish and the signature flips from Lorentzian to Euclidean.
• Compares the single-trace $T\bar{T}$ deformed background with the double-trace case and with the $\lambda < 0$ regime, highlighting differences in singularities and complex energy formation.
• Uses modular invariance and Euclidean continuation to interpret the maximal coupling as a bound on temperature, consistent with Hagedorn growth in the boundary theory.

Experimental results

Research questions

• RQ1 What is the geometric realization of the single-trace $T\bar{T}$ deformation in the bulk, and how does it differ from the double-trace case?
• RQ2 At what coupling does the $SL(2,\mathbb{R})$ invariant vacuum state in the deformed CFT develop complex energy, and what does this imply for the bulk geometry?
• RQ3 How does the Hagedorn behavior of the boundary CFT relate to the asymptotic linear dilaton geometry in the bulk?
• RQ4 Why does the bulk geometry become Euclidean when $\lambda > \lambda_c = 1/k$, and what does this imply for the physical validity of the theory?
• RQ5 How does the maximal coupling $\lambda_c = 1/k$ compare to the Hagedorn temperature $T_H = 1/(2\pi r_5)$ in the boundary theory?

Key findings

• The $SL(2,\mathbb{R})$ invariant ground state energy in the single-trace $T\bar{T}$ deformed CFT becomes complex when the coupling $\lambda$ exceeds $\lambda_c = 1/k$, indicating a physical upper bound on the deformation.
• At $\lambda = \lambda_c = 1/k$, the $B$-field and $e^{\Phi}$ in the bulk geometry vanish uniformly, and the spacetime signature flips from Lorentzian to Euclidean, signaling a breakdown of the Lorentzian description.
• The maximal coupling $\lambda_c = 1/k$ corresponds to the Hagedorn temperature $T_H = 1/(2\pi r_5)$, and the condition $\lambda < 1/k$ is equivalent to $R > r_5$, ensuring the temperature remains below the Hagedorn temperature.
• For $\lambda < \lambda_c$, the bulk geometry interpolates smoothly between linear dilaton in the UV and $AdS_3$ in the IR, with the $AdS_3$ region shrinking as $\lambda$ increases.
• The critical behavior at $\lambda_c$ is distinct from the $\lambda < 0$ case, where complex energies arise for high-energy states but the $AdS_3$ region remains intact; here, the pathology affects the entire spacetime.
• The result is consistent with modular invariance and the ability to Wick-rotate spatial and Euclidean time, supporting the interpretation that $\lambda_c$ is a fundamental upper bound on the coupling in this class of $T\bar{T}$ deformed CFTs.