Pure gravity and conical defects
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Springer
Received: June 16, 2020 Accepted: July 28, 2020 Published: September 4, 2020
Pure gravity and conical defects
a
Princeton Center for Theoretical Science, Princeton University, Washington Road, Princeton, NJ 08544, U.S.A. b Jefferson Physical Laboratory, Harvard University, Oxford Street, Cambridge, MA 02138, U.S.A. c Department of Physics, McGill University, Rue University, Montreal, QC H3A 2T8, Canada
E-mail: [email protected], [email protected], [email protected] Abstract: We revisit the spectrum of pure quantum gravity in AdS3 . 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. Keywords: AdS-CFT Correspondence, Conformal Field Theory, Models of Quantum Gravity ArXiv ePrint: 2004.14428
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)034
JHEP09(2020)034
Nathan Benjamin,a Scott Collierb and Alexander Maloneyc
Contents 1 Introduction
1
2 Non-unitarity of the pure gravity partition function 2.1 Review of the MWK spectrum 2.2 Minimalist spectrum
4 4 9 10
4 The one-loop determinant on orbifold backgrounds 4.1 Review of the gravitational one-loop determinant on thermal AdS3 4.2 The gravitational one-loop determinant on the conical defect backgrounds
14 15 17
5 Discussion
20
A MWK regularization
23
B A systematic analysis of the large-spin negativities
25
C Full orbifold sum
27
1
Introduction
One of the simplest possible theories of gravity is three dimensional general relativity with negative cosmological constant, governed by the Einstein Hilbert action Z 1 2 √ S= d3
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