The black hole S-matrix in gauge/gravity duality

Abstract: A variety of examples of gauge/gravity duality have a Coulomb branch for the gauge theory dynamics. We exploit this feature to construct an S-matrix, in which the initial state is a shell of branes converging on the origin of the Coulomb branch. In the gravitational dual, the shell of branes sources a geometry with a capped throat; the cap descends from the asymptotic region to larger and larger redshift. A trapped surface forms when the redshift of the cap reaches the point where the excitation of strings stretching between the branes is unsuppressed, and the dual gauge theory deconfines. The intermediate state is a long-lived black hole, which then decays via the slow emission of branes back onto the Coulomb branch. We compare and contrast the descriptions of this process in the bulk effective field theory and the dual gauge theory; and discuss the consequences of this construction for the black hole information paradox.

• The paper constructs an explicit black hole S-matrix by modeling collapse as a shell of separated D-branes that forms a horizon and later emits its constituent branes through a unitary gauge theory.
• It finds a universal tunneling relation, Im S = −½ΔS_BH, giving an emission rate Γ ∼ e^(S_f−S_i), while graybody thresholds imply each brane carries roughly an O(1/N) fraction of the black hole mass.
• The analysis argues that semiclassical Hawking radiation misses O(N²) non-abelian degrees of freedom near the horizon, requiring a non-vacuum interior or phase-boundary description while leaving its direct bulk formulation unresolved.

Overview

This paper constructs an explicit black hole S-matrix within gauge/gravity duality by exploiting the Coulomb branch of the dual gauge theory. The central idea is to assemble a black hole from a collapsing shell of D-branes that are initially well-separated on the Coulomb branch, let it evolve as a long-lived intermediate state, and then follow its decay through the slow emission of constituent branes back onto the Coulomb branch. Because the entire process is described by a unitary gauge theory whose holographic dictionary is under control in the relevant regimes, the construction provides a setting in which the black hole information paradox can be analyzed without invoking auxiliary systems or non-local manipulations of boundary operators (2607.18393).

The work builds on earlier matrix model analyses (BFSS and its generalizations), where Hawking radiation was identified with the leakage of D0-branes onto the Coulomb branch, but circumvents their principal limitation: in the strongly coupled regime, brane wavefunctions spread over scales parametrically larger than the objects of interest. By working instead with toroidally or hyperbolically wrapped D3-branes, where the geometric description is valid to arbitrarily large radius, the radiated branes become cleanly separated from the remnant black hole.

Capped throats from separated brane sources

The technical foundation is an effective action formalism coupling $N$ individual DBI brane actions to bulk supergravity, valid when the branes are sufficiently separated that stretched strings (the non-abelian W-strings) are heavy and integrable out. The paper first reviews this framework in the BPS F1–NS5 system, where slightly separated fivebranes source capped $AdS_2$ throats whose depth tracks the transverse separation of the sources. Compressing the source distribution deepens the throat; when it reaches the "non-abelian scale," little string dynamics deconfines and a horizon forms, with throat depth approximately $R_{AdS_2}\log S_{BH}$ — matching the scale where quantum fluctuations of near-extremal $AdS_2$ horizons become large.

The same mechanism operates for D3-branes: Gaussian spheres inside the shell enclose less five-form flux, so the transverse $S^5$ shrinks smoothly and the geometry caps off at the shell's inner edge. The cap's radial position is dual, in the gauge theory, to the absence of excitations below the scale of the scalar vevs.

Collapse dynamics and horizon formation

Treating the shell as smeared over $S^5$ into a thin domain wall in $AdS_5$, the Israel junction conditions determine the shell trajectory self-consistently. For spatial curvature $k=0$ (torus), the effective potential is flat and collapse always forms an apparent horizon; for $k=-1$ (compact hyperbolic $\Sigma_3$), the conformal coupling destabilizes the Coulomb branch, and sub-threshold shells ($AdS_2$0) bounce back to the asymptotic region without forming a black hole. Above threshold, a trapped surface forms precisely when the redshift makes stretched-string excitations unsuppressed — a dynamical realization of the deconfinement/Hawking–Page correspondence. At horizon formation, the shell's proper velocity satisfies $AdS_2$1, i.e., of order the temperature scale.

Brane Hawking radiation

The emission rate is computed via the tunneling formalism in Painlevé–Gullstrand coordinates. The imaginary part of the reduced worldline action yields a strikingly universal result:

$AdS_2$2

The paper argues this follows independently from Fermi's Golden Rule combined with the Eigenstate Thermalization Hypothesis, with the absorptive nature of black holes fixing the choice of $AdS_2$3 scaling. Since each emitted D3-brane changes the entropy by $AdS_2$4 against a total entropy of $AdS_2$5, emission is exponentially suppressed in $AdS_2$6, and quanta are produced essentially at threshold, in their ground state.

Graybody analysis sharpens the picture: for both $AdS_2$7 and $AdS_2$8, escape requires energy above a threshold of order $AdS_2$9. Consequently each emitted brane carries away an $R_{AdS_2}\log S_{BH}$0 fraction of the black hole mass, so an order-one fraction of the initial mass is radiated before end-stage decay — avoiding the pathological scenario in which the black hole loses entropy but not mass. In the hyperbolic case, once past the graybody barrier, the unstable potential sweeps the brane rapidly away from the thermal atmosphere, ensuring clean separation between radiation and remnant.

Consequences for unitarity

The construction exposes a sharp conflict between the bulk EFT description and the exact gauge theory. In classical supergravity, the emitted brane arises from vacuum brane–antibrane pair creation near a smooth horizon, uncorrelated with the original infalling branes sequestered deep inside; Mathur's small corrections theorem then forces a monotonically rising Page curve. The Page time here occurs when $R_{AdS_2}\log S_{BH}$1, i.e., when roughly $R_{AdS_2}\log S_{BH}$2 branes remain — a modified profile relative to standard evaporation because the only open channel emits $R_{AdS_2}\log S_{BH}$3-entropy chunks, at most $R_{AdS_2}\log S_{BH}$4 times.

In the gauge theory, by contrast, the emitted brane is one of the original constituents, highly scrambled; emission reduces the rank $R_{AdS_2}\log S_{BH}$5 rather than creating new charge. The paper argues that locality plus unitarity of charged Hawking radiation requires the emerging brane to be on-shell at the horizon, with the interior not in the vacuum state — an inference drawn from Gauss law, the small corrections theorem, and unitarity rather than a direct calculation. The proposed minimal modification of bulk gravity is to retain the $R_{AdS_2}\log S_{BH}$6 light non-abelian strings at the temperature scale and below as explicit degrees of freedom; these are active only near the horizon and cannot mediate non-local communication with distant radiated branes, since W-bosons connecting separated clusters sit in their ground state. The paper also argues against hidden strong-coupling non-locality via wormholes, using the throat-splitting geometry of two well-separated brane clusters as evidence.

Limitations and open questions

Several caveats are stated plainly. The thin-shell, angularly-smeared approximation introduces numerical ambiguities; notably, the turning point for near-threshold hyperbolic shells does not exactly match the extremal horizon radius $R_{AdS_2}\log S_{BH}$7, which the author attributes to the approximations. Graybody coefficients are estimated only parametrically, so the precise fraction of mass radiated before end stage remains undetermined. The description of the black hole interior lies outside the scope of the exterior effective action: the claim that the interior is non-vacuum is inferential, not computational. The appendix examines the proposal that the Milne-space gauge theory describes the interior smoothly, concluding that no bulk EFT of the interior exists in that frame — all $R_{AdS_2}\log S_{BH}$8 degrees of freedom participate, with no approximation recovering smooth vacuum geometry. Whether a controlled bulk description of the interior compatible with the on-shell-at-the-horizon picture exists is left open, as is the extension to hyperbolic counterparts of non-conformal $R_{AdS_2}\log S_{BH}$9-brane theories, whose gravity solutions are unknown.

Conclusion

The paper demonstrates that gauge/gravity dualities with a Coulomb branch support a genuine black hole S-matrix: collapse of a brane shell forms a capped throat that deepens until deconfinement and trapped-surface formation, after which the black hole decays exponentially slowly by emitting its constituent branes at rate $AdS_2$0. The comparison of the two descriptions indicates that the failure of the semiclassical Hawking calculation lies in its coarse-graining over the non-abelian entropic degrees of freedom, and that consistency requires the horizon to be a phase boundary behind which the geometry is non-vacuum — while leaving the direct bulk characterization of that interior phase unresolved.