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

TL;DR

The paper shows that pure AdS$_3$ gravity at strong coupling ($c=\tfrac{1}{2}$) is dual to the 2D Ising CFT for spacetimes with boundary genus $g$ arbitrarily large, by constructing the gravity partition function as a finite modular sum over the mapping class group acting on a genus-$g$ vacuum seed identified with the Ising vacuum block. It provides a complete all-genus argument: the vacuum seed undergoes a finite, irreducible projective representation of the MCG, yielding a finite, unique sum that reproduces the Ising CFT partition function on the boundary; this establishes the duality and connects the bulk theory to 3D Ising TQFT data. The work also clarifies why extending to the Tricritical Ising case ($c=\tfrac{7}{10}$) fails at higher genus due to infinite image representations (via Fib and related MTCs). Overall, the results tie strong-coupling 3D gravity to a topological and conformal-field-theory framework, with implications for black-hole interpretations and entanglement-related corrections in holography.

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 $c=3l/2G_N$ ($G_N$ 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. D {\bf 85}, 024032 (2012)]. 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 {\it only} when $c=1/2$, corresponding to a dual Ising conformal field theory on the asymptotic boundary.

Figures (12)

• Figure 1: From left to right: (a) Geometry for Euclidean thermal AdS$_3$, with time going along the longitudinal direction; (b) The constant time slice of (a) is a disk; (c) Geometry for Euclidean BTZ black hole, with the event horizon being the dashed line in the core of the solid torus; (d) Constant time slice of (c) is an annulus,where the inner boundary is the event horizon.
• Figure 2: Different Euclidean saddles with $\mathbb{Z}_2$ time-reflection symmetry. They analytically continue to three copies of thermal AdS$_3\,\,$ (left), three-sided wormhole (middle), and one thermal AdS$_3\,\,$ plus a BTZ black hole (right). The areas encircled by the green lines are the sets of fixed points of the action of the $\mathbb{Z}_2$ symmetry.
• Figure 3: The canonical homology basis for $\Sigma_g$.
• Figure 4: The low-temperature or long-cylinder limit of the genus-two geometry.
• Figure 5: Generators of the Torelli group $\mathcal{I}_g$. Left: Dehn twist along a separating curve. Right: the bounding pair map.