Oscillator Level for Black Holes and Black Rings

TL;DR

The paper addresses how to translate the CFT oscillator level $\hat{q}_0$ into a gravitational description for supersymmetric black holes with a dual 2D CFT. It introduces a horizon-adapted Komar integral $\mathcal{K}$ whose horizon value equals the oscillator level $\hat{q}_0$ and whose asymptotic value $\mathcal{K}_\infty$ equals the total CFT momentum, thereby providing a gravitational dual to Cardy’s entropy formula $S=2\pi\sqrt{\frac{c\,\hat{q}_0}{6}}$. By analyzing near-horizon geometries that factor into BTZ$_{ext}$ × $\mathcal{M}_{D-3}$ and using the relation $c=\frac{3l}{2G_3}$ with $G_3=G/V_{D-3}$, the authors demonstrate that $\hat{q}_0=\mathcal{K}_\mathrm{hor}$; at infinity, $\mathcal{K}_\infty$ yields the total momentum. The paper provides explicit checks for the BMPV black hole and the black ring, showing $\mathcal{K}_\mathrm{hor}=\hat{q}_0$ and $\mathcal{K}_\infty$ matching the CFT momentum, including local densities $\hat{q}_0(\psi)$ for inhomogeneous rings, and discusses broader implications for horizon topology and potential extensions to non-supersymmetric cases.

Abstract

Microscopic calculations of the Bekenstein-Hawking entropy of supersymmetric black holes in string theory are typically based on the application to a dual 2D CFT of Cardy's formula, S=2π\sqrt{c q /6}, where `c' is the central charge and `q' is the oscillator level. In the CFT, q is non-trivially related to the total momentum. We identify a Komar integral that equals q when evaluated at the horizon, and the total momentum when evaluated at asymptotic infinity, thus providing a gravitational dual of the CFT result.