Abstract
We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius l is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge c = 3l/2GN (GN is the 3D Newton constant) equals c = 1/2, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D85 (2012) 024032]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the c = 1/2 theory among all those with c < 1, extending the previous results of Castro et al. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”. But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge c < 1, the sum is finite and unique only when c = 1/2, corresponding to a dual Ising conformal field theory on the asymptotic boundary.
Highlights
Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”
One will need to both sum over all different geometries of the bulk 3-manifold X with the same equivalence class of conformal structures on the boundary torus, as well as integrate over all different boundary metrics connected by small diffeomorphisms, i.e., those isotopic to the identity
The key mathematical results that we prove in this paper and that underlie our physics conclusions on the quantum gravity partition function at c = 1/2 are: (1) The representation ρg, when viewed as a mapping from the mapping class group (MCG) Γg to a unitary group, has a finite image set for any genus g, and (2) the projective representation ρg of the MCG Γg is always irreducible for any genus g
Summary
The simplest Euclidean smooth 3-manifold X that contributes to the sum in (1.2) is that of thermal AdS3, topologically a solid torus. It is related to thermal AdS3 by exchanging the spatial and temporal cycles, consistent with the defining feature of a Euclidean BTZ black hole - the existence of a temporal cycle contractible in the bulk It was shown in [3] that the only smooth solutions to the equation of motion with torus boundary conditions are the ones above, but not all these solutions labeled by γ are inequivalent. Notice that all solid tori Mc,d share the same hyperbolic metric (2.1), because by a famous theorem of Sullivan [27,28,29], for a fixed conformal class of the asymptotic boundary, the bulk is a unique smooth and infinite-volume hyperbolic 3-manifold, with a rigid complete metric These saddle-point Euclidean spacetimes Mc,d can be obtained from the corresponding Lorentzian ones via analytical continuation, which amounts to taking the Schottky double. Γ∈Γ where Zvac is the partition function of the “vacuum seed”, by which we here mean here that of thermal AdS3 spacetime
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