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, Z) images.

On the spectrum of pure higher spin gravity

“…Our aim here is to obtain explicit results for the W N case. Since we are ultimately interested in the density of states, we find it convenient to used the method of kernels discussed in [13]. For a given seed, the idea is to compute the contribution to the density of states coming from one of the PSL(2, Z) images of the seed, and then sum over all images.…”

“…where (w, γ) is a h, h -independent phase (see (A.6) of [13]). We will now consider a seed with arbitrary weights (h 0 ,h 0 ) and compute the resulting density for weights (h,h).…”

“…In the second part of the paper we interpret these states in terms of HS gravity on AdS 3 . For N = 2 it has been shown that these states have the interpretation of a conical defect JHEP12(2020)001 geometry, with deficit angle 2π 1 − 1 M , see [13], and it has been argued that they must be included in the Euclidean path integral of 3d gravity. In this paper we show that this interpretation extends to general W N , by studying the one-loop partition function of HS gravity on a quotient of thermal AdS 3 by Z M .…”

The first law of differential entropy and holographic complexity

“…A conical defect in AdS is dual to an excited state created, via the state/operator correspondence, by a heavy operator in the CFT [61][62][63][64]. There is substantial evidence that the CFT state dual to AdS 3 /Z N is excited by an operator which, at large c, has scaling dimension [65][66][67]…”

“…solutions to six-dimensional supergravity theories which reduce to the 3d conical defects upon compactification [62,91]. Moreover, the dual description of a conical defect spacetime in terms of a twist operator, discussed in [65], applies only for integer 1/γ. Therefore, in the context of AdS/CFT (and hence also in the present paper) the conical defect parameter 1/γ should most likely be taken to be integer.…”