Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

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.

Partition functions of the tensionless string

“…For very small radii, the dual CFT can become a RCFT and the partition function obtained this way matches on the nose[25,26].…”

“…[77, page 46]. 26 Historically, it was often assumed that the putative dual CFT to pure 3d gravity is extremal and holomorphically factorized. In this holomorphic setting and for other holomorphic quantities like the elliptic genus, the sum over bulk geometries seems to work well [78][79][80][81].…”

Poincaré series, 3d gravity and averages of rational CFT

“…However, as seen in many of our tables, the identity seed often seems to return a positive linear combination of CFT partition functions, while other primary seeds often do not. 8 Related to this is the question of additional seeds that must be added to account for analogue wormholes in the multi-boundary case. We have found candidates that work correctly, which we think is remarkable, but we do not have a first-principles reason why the wormhole seeds should be what we propose.…”

Conical spaces, modular invariance and cp,1 holography

“…In this work we will instead consider general smooth solutions obeying the asymptotic conditions and therefore to include the other (r,r) sectors as well. 6 The energy of these solutions, computed from (2.19), is…”

“…The fact that the AdS group element behaves as g(φ + 2π) = −g was the reason for the Z2 quotient in our choice of the global gauge group SL(2, R)/Z2[16] 6. A different space of solutions which has been explored in the literature is to instead restrict to Chern-Simons gauge fields with hyperbolic holonomy.…”