Abstract
We have argued previously, based on the analysis of two-dimensional stringy black holes, that information in stringy versions of four-dimensional Schwarzschild black holes (whose singular regions are represented by appropriate Wess-Zumino-Witten models) is retained by quantum $W$-symmetries when the horizon area is not preserved due to Hawking radiation. It is key that the exactly-marginal conformal world-sheet operator representing a massless stringy particle interacting with the black hole requires a contribution from $W_\infty$ generators in its vertex function. The latter correspond to delocalised, non-propagating, string excitations that guarantee the transfer of information between the string black hole and external particles. When infalling matter crosses the horizon, these topological states are excited via a process: (Stringy black hole) + infalling matter $\rightarrow $ (Stringy black hole)$^\star$, where the black hole is viewed as a stringy state with a specific configuration of $W_\infty$ charges that are conserved. Hawking radiation is then the reverse process, with conservation of the $W_\infty$ charges retaining information. The Hawking radiation spectrum near the horizon of a Schwarzschild or Kerr black hole is specified by matrix elements of higher-order currents that form a phase-space $W_{1+\infty}$ algebra. We show that an appropriate gauging of this algebra preserves the horizon two-dimensional area classically, as expected because the latter is a conserved Noether charge.
Highlights
AND SUMMARYThe black-hole information problem was posed by the discoveries by Bekenstein [1] and Hawking [2] that fourdimensional black holes have thermodynamical properties such as temperature and nonzero entropy and so must be described by mixed quantum-mechanical states
We have argued previously, based on the analysis of two-dimensional stringy black holes, that information in stringy versions of four-dimensional Schwarzschild black holes is retained by quantum W symmetries when the horizon area is not preserved due to Hawking radiation
That two-dimensional black holes carry an infinite set of quantum numbers associated with a W∞ symmetry and that these W charges preserve the lost information in principle, though this information could not in practice be extracted
Summary
The black-hole information problem was posed by the discoveries by Bekenstein [1] and Hawking [2] that fourdimensional black holes have thermodynamical properties such as temperature and nonzero entropy and so must be described by mixed quantum-mechanical states. [22,23] where the fluctuation of the black-hole horizon induced by infalling matter is argued to play an important role in retaining information In this connection, we recall that in string theory the interaction of a massless particle with a black hole is represented by a conformal operator on the world sheet of the string, which is exactly marginal if and only if contributions from W∞ generators are included in its vertex function [6,24]. [25], we consider W1þ∞ symmetry [26,27] to be essential for “balancing the blackhole information books.” This symmetry is manifest in the effective two-target-space dimensional string theories that describe the excitations in the near-horizon geometry of a spherically symmetric stringy black hole. V where, in view of the role of W symmetries in preserving information during matter infall or Hawking radiation, the evaporation of the stringy black hole is viewed as successive two-body decays
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