On String theory on deformed BTZ and $T\bar{T} + J\bar{T} + T\bar{J}$

1. Problem Statement

The paper addresses the challenge of understanding the holographic dual of string theory on deformed BTZ black hole backgrounds within the framework of single-trace $T\bar{T} + J\bar{T} + \bar{T}J$ holography.

• Context: Standard $AdS_3/CFT_2$ correspondence relates string theory on $AdS_3$ to a 2D CFT. Deformations of the worldsheet sigma model (specifically marginal deformations of the $SL(2, \mathbb{R})$ WZW model) lead to irrelevant deformations in the boundary theory, known as $T\bar{T}$, $J\bar{T}$, and $\bar{T}J$ deformations.
• The Gap: While the spectrum of long strings in massless BTZ (global $AdS_3$) and undeformed massive BTZ is well understood, the behavior of the spectrum in massive BTZ black holes subjected to the full family of $T\bar{T} + J\bar{T} + \bar{T}J$ deformations was not fully established.
• Specific Question: What is the required asymptotic value of the $B$-field in a deformed massive BTZ background such that the excitation energy of a long string (plus its contribution to the black hole energy) evolves exactly according to the single-trace $T\bar{T} + J\bar{T} + \bar{T}J$ deformed CFT formulas?

2. Methodology

The authors employ perturbative string theory in a flat spacetime limit with a linear dilaton to derive the spectrum, rather than solving the full non-linear worldsheet CFT.

• Background Setup: They consider superstring theory on a background formed by $k$ NS5-branes and $p$ fundamental strings (F1) wrapping a circle ($S^1_x$). The geometry is a deformed BTZ black hole times a circle ($S^1_y$) and an internal space ($T^4$ or $S^3$).
• Deformation Parameters: The geometry is deformed by parameters $(\lambda, \epsilon_\pm)$, corresponding to the $T\bar{T}$ and $J\bar{T}/\bar{T}J$ couplings respectively.
• Asymptotic Analysis:
  • The authors analyze the asymptotic behavior of the metric, $B$-field, and dilaton as the radial coordinate $\phi \to \infty$ (the linear dilaton regime).
  • They utilize vertex operators in the $(-1, -1)$ picture to describe long string excitations.
  • They impose the on-shell condition for the string states, which relates the scaling dimensions of the operators to the energy and momenta.
• Key Technique: Instead of solving the full deformed geometry, they use the asymptotic data (constant metric, $B$-field, and dilaton slope) to calculate the energy spectrum. This relies on the fact that long strings are delta-normalizable and probe the entire asymptotic region.
• Derivation Strategy:
  1. Derive the mass-shell condition for a probe long string in the asymptotic flat space with a linear dilaton.
  2. Formulate a quadratic equation for the total energy $E_{total}$.
  3. Impose the constraint that this energy must match the evolution predicted by single-trace $T\bar{T} + J\bar{T} + \bar{T}J$ holography (specifically, the $T\bar{T}$ flow equation for the seed CFT and its symmetric product structure).
  4. Solve for the necessary asymptotic $B$-field components ($B_{tx}, B_{ty}, B_{xy}$) that satisfy this condition.

3. Key Contributions and Results

A. The $B$-Field Condition

The primary result is the identification of a specific asymptotic value for the $B$-field component $B_{tx}$ required for consistency.

• For the deformed massive BTZ, the asymptotic $B$-field must approach:
  $$B_{tx} \to 1 + \frac{1}{\Psi r_x^2} \left( -1 + \sqrt{1 + \frac{2\Psi r_5 M_{BTZ}}{p}} \right)$$
  where $\Psi = \lambda - (\epsilon_- + \epsilon_+)^2$, $r_x$ is a deformation-dependent radius, $r_5 = \sqrt{\alpha' k}$, and $M_{BTZ}$ is the BTZ mass.
• The other components $B_{ty}$ and $B_{xy}$ remain fixed to their standard deformed values.

B. Energy Spectrum and Black Hole Contribution

The authors define the total energy $E_{total}$ of a system consisting of a probe long string with winding $w$ and the background black hole:
$$E_{total} \equiv E + w \frac{E_{BH}}{p}$$
where:

• $E$ is the excitation energy of the string.
• $E_{BH}$ is the energy of the black hole above extremality.
• $p$ is the number of F1 strings forming the background.

The Result: With the specific $B$-field derived above, $E_{total}$ evolves exactly as predicted by single-trace $T\bar{T} + J\bar{T} + \bar{T}J$ holography:

1. $w=1$ Sector: The energy evolves according to the $T\bar{T} + J\bar{T} + \bar{T}J$ flow equation for a CFT with central charge $c=6k$.
2. $w>1$ Sector: The energy evolves as states in the $\mathbb{Z}_w$ twisted sector of a $p$-fold symmetric product of the seed CFT ($M^p/S_p$).

C. Black Hole Thermodynamics

The paper confirms that the deformed black hole entropy and energy are consistent with the holographic picture:

• Entropy: The entropy $S_{BH}$ remains invariant under the deformation and matches the entropy of the dual CFT states.
• Energy: The black hole energy $E_{BH}$ is related to the BTZ mass $M_{BTZ}$ via a specific deformation-dependent factor, ensuring the ground state energy vanishes in the extremal limit ($M_{BTZ}=0$).

D. Constraints on Negative Deformations

The authors analyze the case where the deformation parameters lead to $\Psi < 0$ (negatively deformed theories).

• They find that both the black hole energy and the total energy of long strings have a maximal bound:
  $$E_{max}^{BH} = -\frac{\sqrt{\Psi} R_x}{p}, \quad E_{max}^{total} = -w \frac{\sqrt{\Psi} R_x}{p}$$
• This suggests a physical limit where long strings cannot be excited beyond a certain energy fraction of the black hole in these specific deformation regimes.

4. Significance

1. Validation of Single-Trace Holography: The paper provides a concrete derivation confirming that the single-trace $T\bar{T} + J\bar{T} + \bar{T}J$ holography conjecture holds for massive black hole backgrounds, not just massless (global $AdS_3$) cases.
2. Resolution of the $B$-Field Ambiguity: In string theory, adding a constant to the $B$-field is a gauge redundancy that usually doesn't affect local physics. However, this work demonstrates that in the context of holographic deformations, the asymptotic value of the $B$-field is physically significant and must be tuned precisely to match the boundary CFT deformation.
3. Universality of the Symmetric Product: It reinforces the picture that the spectrum of long strings in string theory on $AdS_3$ (and its deformations) is universally described by a symmetric product orbifold, where the seed CFT is the specific deformed theory.
4. Technical Framework: The methodology of using asymptotic flat-space data (metric, $B$-field, dilaton) to derive the spectrum of deformed black holes offers a powerful tool for future investigations into other holographic deformations and non-AdS geometries.

In summary, the paper bridges the gap between the microscopic string theory description of deformed black holes and the macroscopic effective field theory description of $T\bar{T}$-deformed CFTs, proving their consistency through the precise tuning of the asymptotic $B$-field.