Small Black Holes and Near-Extremal CFTs

TL;DR

Pure theories of AdS$_3$ gravity are conjectured to be dual to CFTs with sparse light spectra, motivating Witten's extremal-CFT idea but requiring quantum corrections. The authors propose near-extremal CFTs that include small black holes with zero classical horizon area whose entropy is generated by loop effects, and they derive all-orders perturbative corrections to these entropies for non-supersymmetric, ${\rm N}=1$, and ${\rm N}=2$ theories, including charged variants. They test consistency through modular constraints, Rademacher-sum continuations, and positivity in Ramond sectors, and they connect bulk results to CFT bounds on threshold primaries, notably proving an exponential growth of threshold primaries in chiral CFTs with spin-1 currents. The work suggests a non-perturbative structure controlled by modularity that could delimit the space of holographic pure-gravity duals and guides future non-perturbative completions and explicit near-extremal CFT constructions.

Abstract

Pure theories of AdS$_3$ quantum gravity are conjectured to be dual to CFTs with sparse spectra of light primary operators. The sparsest possible spectrum consistent with modular invariance includes only black hole states above the vacuum. Witten conjectured the existence of a family of extremal CFTs, which realize this spectrum for all admissible values of the central charge. We consider the quantum corrections to the classical spectrum, and propose a specific modification of Witten's conjecture which takes into account the existence of "small" black hole states. These have zero classical horizon area, with a calculable entropy attributed solely to loop effects. Our conjecture passes various consistency checks, especially when generalized to include theories with supersymmetry. In theories with $\mathcal{N}=2$ supersymmetry, this "near-extremal CFT" proposal precisely evades the no-go results of Gaberdiel et al.

Figures (6)

• Figure 1: Blue dots: Ratio of the exact black hole entropy at $h=1$ to the prediction of Bekenstein-Hawking, computed up to $k=1000$. For the first few values, $k=1,2,3,4$, $N_{h=1}$ appears to be approaching the Bekenstein-Hawking prediction, but continuing to higher $k$ one sees that $N_{h=1}$ actually overshoots it. Black solid: The same ratio approximated with (\ref{['eq:OneLoop']}), with the sum cut off at $0$. This corresponds to taking only the perturbative part of the entropy at large $k$, and is an extremely good approximation. Red dashed: The asymptotic value of the ratio at large $k$, $\sqrt{\frac{25}{24}} \approx 1.02$; see (\ref{['eq:doitright']}).
• Figure 2: Black, Dashed Line: The leading order entropy due purely to vacuum gravitons. Blue, Solid Line: The entropy of black hole states including our shift. These meet at leading order in large $k$, $S_{\rm vac}(0)=S_{\rm BH}(0)=\pi\sqrt{2k/3}$. Red, Dot-dashed Line: The Cardy entropy, which vanishes at threshold.
• Figure 3: The four possible spin structures on the torus. "A" denotes that fermions have anti-periodic boundary conditions and "P" denotes periodic boundary conditions; the $x$-axis is the spatial direction. As noted, the first two spin structures correspond to the partition functions restricted to the Neveu-Schwarz or Ramond sectors, respectively. The last two spin structures contain an additional $(-1)^F$ inside the trace; for the Ramond sector, this produces the Witten index $\chi = {\rm Tr}_{{\cal H}_{\rm R}} \left((-1)^F q^{L_0}\right)$.
• Figure 4: $h=0$ states for an $\mathcal{N}=1$ near-extremal CFT. Red, solid: Equation (\ref{['eq:preciselog']}), which is the all-orders in large $k^*$ expression from the saddle point calculation for $S_{\rm BH}(0)$. Black dots: $\log{|\beta_{k^*}|}$. Note the strong agreement even for $k^* \sim 15$ - on this plot, the last few points at $k^* \sim 30$ appear to lie exactly on the asymptotic curve. To show the trend in the size of the difference more clearly, we plot the relative error in Figure \ref{['fig:n1']} at large $k^*$. Blue, dot-dashed: $\pi \sqrt{\frac{k^*}{2}}$, the leading perturbative piece (\ref{['eq:blah']}), is also shown for comparison.
• Figure 5: Here we plot the error $1-\frac{\log{|\beta_{k^*}|}}{S_{\rm BH}(0)}$, with $S_{\rm BH}(0)$ from (\ref{['eq:preciselog']}), as a function of $k^*$ for an $\mathcal{N}=1$ extremal CFT. The relative error decreases exponentially $\sim e^{-\frac{\pi}{2} \sqrt{\frac{k^*}{2}}}$, in accordance with the fact that all orders in the perturbative expansion are correctly reproduced by the gravity saddle point calculation.