Pure Gravity and Conical Defects

Abstract

We revisit the spectrum of pure quantum gravity in AdS$_3$. The computation of the torus partition function will -- if computed using a gravitational path integral that includes only smooth saddle points -- lead to a density of states which is not physically sensible, as it has a negative degeneracy of states for some energies and spins. We consider a minimal cure for this non-unitarity of the pure gravity partition function, which involves the inclusion of additional states below the black hole threshold. We propose a geometric interpretation for these extra states: they are conical defects with deficit angle $2π(1-1/N)$, where $N$ is a positive integer. That only integer values of $N$ should be included can be seen from a modular bootstrap argument, and leads us to propose a modest extension of the set of saddle-point configurations that contribute to the gravitational path integral: one should sum over orbifolds in addition to smooth manifolds. These orbifold states are below the black hole threshold and are regarded as massive particles in AdS, but they are not perturbative states: they are too heavy to form multi-particle bound states. We compute the one-loop determinant for gravitons in these orbifold backgrounds, which confirms that the orbifold states are Virasoro primaries. We compute the gravitational partition function including the sum over these orbifolds and find a finite, modular invariant result; this finiteness involves a delicate cancellation between the infinite tower of orbifold states and an infinite number of instantons associated with $PSL(2,{\mathbb Z})$ images.

Figures (3)

• Figure 1: The spectrum of the MWK partition function. We start with the vacuum state, at $h=\bar{h} = 0$ and get a continuous spectra at all integer spins, with $h, \bar{h} \geq \frac{c-1}{24}$. The spectrum is not unitarity; the red regions represent the parts of the spectrum where the density of states is negative.
• Figure 2: Left: The Euclidean geometry is asymptotically $AdS_3$ but has an orbifold singularity in the interior (shown in green). Right: The orbifold singularity corresponds to a conical defect of $2\pi(1-1/N)$.
• Figure 3: Left: The spectrum of Virasoro primary operators of the pure gravity partition function endowed with the conical defect states. There are no discrete Regge trajectories with an asymptotic twist below the black hole threshold, and the spectrum of twists is continuous above the black hole threshold. Right: A rough sketch of the spectrum of Virasoro primary operators of a generic irrational CFT with a light scalar operator with weight $h_0$ below ${c-1\over 32}$. In this case there are a finite, discrete set of infinite towers of multi-twist operators with asymptotic twist below the black hole threshold. This sketch is meant to be schematic, and merely capture the existence of the discrete multi-twist Regge trajectories below the black hole threshold --- in a genuine CFT, the Regge trajectories are not exactly flat (i.e. the multi-twist operators have anomalous dimensions at finite spin) and the spectrum of twists above the black hole threshold is only continuous at large spin. Furthermore, the presence of the multi-twist Regge trajectories will serve to modify the black hole threshold in a spin-dependent way; see Maxfield:2019hdt for more details.