Deformed BTZ Radiance and Single Trace $T\bar{T}$ Holography

Abstract: We generalize the "holar wind" mechanism proposed in arXiv:2303.00234 to the case of rotating $λ$-deformed BTZ black holes. These backgrounds, which interpolate between $AdS_3$ in the infrared and a linear dilaton spacetime in the ultraviolet, are realized as an exact $\frac{SL(2,\mathbb{R})<em>k\times U(1)}{U(1)}$ gauged-WZW worldsheet theory. The long string sector of the theory provides a holographic dual to single-trace $T\overline{T}$ deformed symmetric product $\mathcal{M}<sup>p/S_p$ $CFT_2$ with a seed $\mathcal{M}$ carrying a central charge $c=6k$. By analyzing the geodesics and tunneling rates of probe particles and winding long strings, we show that the emission probabilities for both positive and negative deformation couplings are universally governed by the change in the Bekenstein-Hawking entropy, $ΔS</em>{BH}$. A central result of our analysis is that the consistency of long string emission within the grand canonical ensemble dictates a unique value for the background B-field at the origin. We demonstrate that this thermodynamically fixed value matches precisely the prediction required for the string excitation spectrum to agree with the $\mathbb{Z}_w$ twisted sector of a single-trace $T\overline{T}$ deformed symmetric product orbifold. Finally, we sketch the extension of these results to a broader class of black holes whose long string sector is dual to single-trace $T\overline{T} + J\overline{T} + T\overline{J}$ deformed theories.

• The paper shows that particle and winding-string emission from rotating λ-deformed BTZ black holes universally follows Γ ∼ e^ΔS_BH, using tunneling and WKB calculations for both deformation signs.
• Thermodynamic consistency in the grand canonical ensemble uniquely fixes the origin B-field integration constant, matching the value required by twisted-sector spectra in single-trace T̄T holography.
• For negative deformation, a UV wall limits particle escape while sufficiently wound strings can cross it; the proposed T̄T+J̄T+T̄J extension remains incomplete because thermodynamics cannot determine all B-field components.

Overview and context

This paper by Vainshtein extends the semiclassical analysis of black hole evaporation in single-trace $T\bar{T}$ holography to rotating backgrounds. The setting is string theory on $\lambda$-deformed BTZ black holes — geometries that interpolate between $AdS_3$ in the infrared and an asymptotically flat spacetime with a linear dilaton in the ultraviolet — realized as an exact $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$ gauged-WZW worldsheet theory. The long string sector of this background provides a holographic dual to a symmetric product $\mathcal{M}^p/S_p$ with seed central charge $c=6k$, deformed by an irrelevant $T\bar{T}$-type operator.

The paper generalizes the "holar wind" mechanism of Martinec (Martinec, 2023), in which winding long strings above an energy threshold escape to timelike infinity, providing a genuine decay channel (a black hole S-matrix) for BTZ black holes. The two main results are:

• The emission probability for both probe particles and winding long strings, for either sign of the deformation coupling $\lambda$, is universally governed by the change in Bekenstein-Hawking entropy, $\Gamma \sim e^{\Delta S_{BH}}$, computed via both tunneling (Parikh-Wilczek style) and WKB methods.
• Consistency of long string emission within the grand canonical ensemble fixes the integration constant $B_{\tau\theta}^{(0)}$ of the bulk $\lambda$0-field at the origin to a unique value, which coincides exactly with the value required for the excited long string spectrum to match the $\lambda$1 twisted sector of the single-trace $\lambda$2 deformed symmetric orbifold (Giveon et al., 2024).

The second point is the paper's most consequential claim: a purely thermodynamic bulk argument independently reproduces a value previously fixed only by boundary spectral data ("harmony with single-trace $\lambda$3 holography").

The positively deformed black string

For $\lambda$4, the background is a rotating black string with metric, $\lambda$5-field, and dilaton parametrized by inner and outer horizons $\lambda$6 and asymptotic radius $\lambda$7. The ADM mass, angular momentum, entropy, temperature, and angular velocity satisfy the first law $\lambda$8 in the microcanonical ensemble at fixed $\lambda$9. With entropy and angular momentum held fixed along the deformation line, the energy above extremality obeys

$AdS_3$0

which matches the maximally twisted $AdS_3$1 sector spectrum of the single-trace $AdS_3$2 deformed symmetric orbifold, with each seed carrying $AdS_3$3 and $AdS_3$4 of the total energy. The total entropy takes the Cardy form with the characteristic square-root combination $AdS_3$5.

Geodesics and particle emission

Solving the worldline Hamiltonian constraint yields closed-form radial geodesics organized by an effective mass squared $AdS_3$6, with solutions given by trigonometric or hyperbolic functions depending on the sign of $AdS_3$7 and $AdS_3$8. The qualitative picture differs sharply from pure BTZ: timelike particles escape to timelike infinity provided $AdS_3$9 (requiring $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$0); heavier particles are bounded, reaching a finite maximum radius $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$1 before falling back. This escape channel exists precisely because the AdS throat opens into an asymptotically flat region.

For emission, the author works in Gullstrand–Painlevé coordinates, which regularize the horizon. The near-horizon radial momentum behaves as $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$2, independent of the particle mass — reflecting that horizon crossing is locally null. Treating the black hole as a microcanonical system whose geometry adjusts during emission, the imaginary part of the Hamiltonian action reduces, via the Sokhotski–Plemelj theorem and the first law, to

$\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$3

so that $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$4. A fixed-background WKB computation and an explicit linear-order expansion of the Cardy-form entropy difference give identical results. Physically, low-energy quanta are emitted far more readily than high-energy ones, since large emissions substantially reduce the remnant entropy.

Long strings and the thermodynamic fixation of the B-field

Long string trajectories are generated from particle geodesics by spectral flow, $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$5, $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$6. The author argues this operation remains valid across the full interpolating geometry because it holds at both asymptotic extremes — as a global $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$7 automorphism of the $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$8 WZW model in the IR, and as the trivial winding symmetry of the free compact boson in the UV. This argument is qualitative rather than derived from the exact coset worldsheet theory, a point the paper acknowledges indirectly by noting supporting evidence from TsT analyses.

Emission of a $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}$9-winding string fluctuates the fundamental string charge $\mathcal{M}^p/S_p$0, so the appropriate ensemble is grand canonical:

$\mathcal{M}^p/S_p$1

with chemical potential $\mathcal{M}^p/S_p$2, which depends directly on the undetermined constant $\mathcal{M}^p/S_p$3. Repeating the tunneling calculation with the string Hamiltonian constraint (with radial $\mathcal{M}^p/S_p$4-components gauged away and dilaton effects suppressed by Weyl gauge choice) again yields $\mathcal{M}^p/S_p$5.

The decisive step comes from expanding the exact Cardy-form entropy difference after emission to linear order in $\mathcal{M}^p/S_p$6, $\mathcal{M}^p/S_p$7, and $\mathcal{M}^p/S_p$8, then matching against the first law. This matching demands a specific chemical potential,

$\mathcal{M}^p/S_p$9

and hence a unique value of the B-field at the origin,

$c=6k$0

This coincides precisely with the value fixed in (Giveon et al., 2024) by requiring the $c=6k$1-excited long string energies to reproduce the $c=6k$2 twisted sector spectrum. Notably, the value vanishes in both the non-rotating limit ($c=6k$3) and the undeformed limit ($c=6k$4), consistent with earlier claims; it is nonzero only for simultaneously deformed and rotating backgrounds. The nonzero value implies a localized source at the origin producing a singular three-form flux $c=6k$5, which does not contradict the sourceless nature of global $c=6k$6, where the value vanishes.

Negative deformation

For $c=6k$7 (obtained formally by $c=6k$8, $c=6k$9), the structure changes qualitatively. The spectrum acquires a maximal energy $T\bar{T}$0 and maximal entropy beyond which energies become complex; the black hole has maximal size $T\bar{T}$1, and the IR and UV regimes are separated by a naked singularity at $T\bar{T}$2 acting as a UV cutoff, beyond which the signatures of $T\bar{T}$3 and $T\bar{T}$4 flip and closed timelike curves appear.

Particle geodesics map directly onto the positive-case solutions under parameter replacements, but the critical mass condition inverts: timelike particles escape when $T\bar{T}$5. Crucially, although the radial solution is analytically smooth through $T\bar{T}$6, the coordinate velocities of $T\bar{T}$7 and $T\bar{T}$8 freeze there, so particles emitted from the horizon never reach true infinity — unlike the positive case. For maximal black holes ($T\bar{T}$9, $\lambda$0), the radial momentum vanishes and the semiclassical tunneling approach breaks down; the temperature diverges there.

Long strings behave differently from particles: spectral flow allows them to cross the UV wall in finite time, provided their winding rate exceeds a threshold $\lambda$1. Only a small fraction of emitted strings satisfy this threshold and reach the asymptotic boundary. The tunneling and WKB computations again yield $\lambda$2, and the first-law matching in the grand canonical ensemble fixes

$\lambda$3

again agreeing with the value demanded by negative single-trace $\lambda$4 twisted-sector spectral matching. The sign flip relative to the positive case follows from the analytic continuation.

Extension to $\lambda$5 deformations

The final section sketches the generalization to $\lambda$6-deformed BTZ$\lambda$7 backgrounds, dual to single-trace $\lambda$8 deformed symmetric products. The long string spectrum takes the quadratic form $\lambda$9 with coefficients involving $\Gamma \sim e^{\Delta S_{BH}}$0; real energies require $\Gamma \sim e^{\Delta S_{BH}}$1, while $\Gamma \sim e^{\Delta S_{BH}}$2 produces a maximal energy, extending the negative-$\Gamma \sim e^{\Delta S_{BH}}$3 behavior. The regime $\Gamma \sim e^{\Delta S_{BH}}$4 gives a complex asymptotic radius and requires careful treatment analogous to negative strings.

Using a Cardy-like entropy with a correction functional $\Gamma \sim e^{\Delta S_{BH}}$5 incorporating the charges $\Gamma \sim e^{\Delta S_{BH}}$6, the WKB expansion determines the intensive quantities $\Gamma \sim e^{\Delta S_{BH}}$7, $\Gamma \sim e^{\Delta S_{BH}}$8, and the chemical potentials $\Gamma \sim e^{\Delta S_{BH}}$9 in terms of extensive variables. However, a structural limitation emerges: with three independent B-field components at the origin ($B_{\tau\theta}^{(0)}$0), the thermodynamic constraints provide one fewer equation than needed, so the B-field cannot be uniquely fixed by thermodynamics alone in this richer class — in contrast to the clean uniqueness result for the $B_{\tau\theta}^{(0)}$1-deformed BTZ. Moreover, no explicit supergravity solution with the required parameters is constructed here, and the mapping between these conventions and the tri-TsT solutions of (Apolo et al., 2021) is left open. The universal emission law is assumed rather than proven for these backgrounds, since the spectral-flow derivation has not been carried out.

Limitations and open questions

Several caveats bear directly on the results. First, the validity of spectral flow for long strings on the full deformed background is argued by symmetry at the IR and UV limits plus unbroken target-space isometries, not derived from the exact gauged-WZW worldsheet theory; Appendix B does show, however, that the coset construction naturally generates the same preferred $B_{\tau\theta}^{(0)}$2 assuming a vanishing B-field at the origin of the underlying BTZ solution. Second, the claim of a unique thermodynamically favored $B_{\tau\theta}^{(0)}$3 stands in tension with recent work (Apolo, 26 Aug 2025) arguing, from partition function matching, that deformed BTZ backgrounds are compatible with single-trace $B_{\tau\theta}^{(0)}$4 holography under an assumption of vanishing chemical potential. The paper explicitly flags this contradiction and leaves its resolution open, noting also that whether worldsheet consistency restricts the choice of $B_{\tau\theta}^{(0)}$5 remains unexamined. Third, the information problem persists: semiclassically, emitted quanta are entangled only with their behind-the-horizon partners, and while the boundary theory evolves unitarily, the bulk mechanism for information recovery is not identified. Finally, the greybody factors governing how the near-horizon emission spectrum appears to a distant observer — and specifically how the B-field modifies long-string versus point-particle scattering amplitudes — are not computed.

Conclusion

This work establishes that Hawking-type radiation from rotating $B_{\tau\theta}^{(0)}$6-deformed BTZ black holes, whether by particles in the microcanonical ensemble or by winding long strings in the grand canonical ensemble, is universally controlled by the change in Bekenstein-Hawking entropy, for either sign of the deformation. Its principal contribution is the demonstration that black hole thermodynamics alone — through the chemical potential conjugate to winding charge — selects a unique, nontrivial value of the B-field at the origin, in exact agreement with the value independently required by matching the long string spectrum to the twisted sectors of a single-trace $B_{\tau\theta}^{(0)}$7 deformed symmetric orbifold. The extension to $B_{\tau\theta}^{(0)}$8 deformed backgrounds is sketched but incomplete, with the missing thermodynamic constraint on the third B-field component and the unresolved tension with partition-function-based analyses constituting the clearest open questions.