[Paper Review] Classification of black holes in three dimensional spacetime by the $W_{1+\infty}$ symmetry

Jingbo Wang|arXiv (Cornell University)|Apr 25, 2018

Black Holes and Theoretical Physics3 citations

TL;DR

This paper demonstrates that the near-horizon symmetry algebra of three-dimensional black holes is a subalgebra of the $W_{1+∞}$ algebra, a quantum symmetry of the quantum Hall effect. By classifying black holes via $W_{1+∞}$ quantum numbers, it shows that black hole radii are quantized and provides new evidence that black holes can be viewed as topological insulators.

ABSTRACT

The BMS symmetry and related near horizon symmetry play important roles in holography in asymptotically flat spacetimes. They may also be crucial for solving the information paradox. But we still don't fully understand those infinite-dimensional symmetries. On the other hand, $W_{1+\infty}$ symmetry is the quantum version of area-preserving diffeomorphism of a plane. It is a dynamical symmetry of quantum Hall liquid and can be used to classify the quantum Hall universality classes. In this paper, we will show that the near horizon symmetry can be obtained from the $W_{1+\infty}$ symmetry. Based on this result, the black holes in three dimensional spacetime can be classified just as in the quantum Hall liquid. It also gives the result that the radius of black hole are quantized. This gives another evidence that our early claim "black hole can be considered as a kind of topological insulator" is correct.

Motivation & Objective

• To establish a connection between the near-horizon symmetry of 3D black holes and the $W_{1+\infty}$ algebra of quantum Hall systems.
• To investigate whether the infinite-dimensional symmetry of black holes can be understood as arising from area-preserving diffeomorphisms, as in quantum Hall liquids.
• To classify black holes using the quantum numbers of the $W_{1+\infty}$ algebra, analogous to classifying quantum Hall universality classes.
• To provide further evidence for the claim that black holes are topological insulators by linking their symmetry structure to that of topological quantum matter.
• To explore the implications of $W_{1+\infty}$ symmetry for black hole entropy, quantized radii, and the information paradox via 'W-hair'.

Proposed method

• Identify the near-horizon symmetry algebra of 3D black holes as a semidirect sum of the Witt algebra ($Y_n$) and an abelian current ($T_n$), with commutation relations $[Y_m, Y_n] = (m-n)Y_{m+n}$, $[Y_m, T_n] = -nT_{m+n}$, $[T_m, T_n] = 0$.
• Show that this algebra is isomorphic to a subalgebra of the $W_{1+\infty}$ algebra by identifying $T_n = L^{(0)}_n$ and $Y_n = L^{(1)}_n$ in the $W_{1+\infty}$ generator relations.
• Use the $W_{1+\infty}$ algebra's structure to define quantum numbers $Q$ and $J$ as eigenvalues of $L^{(0)}_0$ and $L^{(1)}_0$, corresponding to charge and spin of quasi-particles in the quantum Hall analogy.
• Relate the $W_{1+\infty}$ generators to physical black hole parameters by re-scaling $L^{(0)}_0$ to match entropy, using $K = \text{diag}(2k, -2k)$ and $k = l/(4G)$.
• Derive the black hole radius quantization condition by solving $n_1 = \frac{1}{4G}(r_+ + r_-)$, $n_2 = \frac{1}{4G}(r_+ - r_-)$, leading to $r_+ = 2(n_1 + n_2)L_{PL}$, $r_- = 2(n_1 - n_2)L_{PL}$.
• Interpret the resulting integer quantum numbers $(n_1, n_2)$ as 'W-hairs' and relate them to generalized soft hair, suggesting a mechanism for information storage in black holes.

Experimental results

Research questions

• RQ1 Can the near-horizon symmetry algebra of 3D black holes be embedded as a subalgebra within the $W_{1+\infty}$ algebra?
• RQ2 How does the $W_{1+\infty}$ symmetry of quantum Hall systems relate to the dynamics of black hole horizons?
• RQ3 What is the physical interpretation of the quantum numbers $n_1$ and $n_2$ derived from the $W_{1+\infty}$ algebra in the context of black hole classification?
• RQ4 Does the emergence of $W_{1+\infty}$ symmetry from the near-horizon algebra imply that black hole radii are quantized?
• RQ5 Can the $W_{1+\infty}$-generated quantum numbers serve as a mechanism for resolving the black hole information paradox?

Key findings

• The near-horizon symmetry algebra of 3D black holes is a subalgebra of the $W_{1+\infty}$ algebra, with $T_n = L^{(0)}_n$ and $Y_n = L^{(1)}_n$.
• Black holes in three-dimensional spacetime can be classified using the $W_{1+\infty}$ algebra, with classification labels given by integer quantum numbers $n_1$ and $n_2$.
• The black hole radius is quantized, with $r_+ = 2(n_1 + n_2)L_{PL}$ and $r_- = 2(n_1 - n_2)L_{PL}$, where $L_{PL} = G$ is the Planck length.
• The quantum numbers $n_1$ and $n_2$ correspond to the eigenvalues of $L^{(0)}_0$ and $L^{(1)}_0$, interpreted as charge and spin of quasi-particles in the quantum Hall analogy.
• The $W_{1+\infty}$ symmetry provides a framework for 'W-hair'—generalized soft hair—offering a potential resolution to the black hole information paradox.
• The result supports the claim that black holes are topological insulators, as the symmetry and classification structure mirror those of quantum Hall systems.