Quasi-local holographic dualities in non-perturbative 3D quantum gravity

Abstract

We present a line of research aimed at investigating holographic dualities in the context of three dimensional quantum gravity within finite bounded regions. The bulk quantum geometrodynamics is provided by the Ponzano-Regge state-sum model, which defines 3d quantum gravity as a discrete topological quantum field theory (TQFT). This formulation provides an explicit and detailed definition of the quantum boundary states, which allows a rich correspondence between quantum boundary conditions and boundary theories, thereby leading to holographic dualities between 3d quantum gravity and 2d statistical models as used in condensed matter. After presenting the general framework, we focus on the concrete example of the coherent twisted torus boundary, which allows for a direct comparison with other approaches to 3d/2d holography at asymptotic infinity. We conclude with the most interesting questions to pursue in this framework. I. OVERVIEW

Non-perturbative 3D quantum gravity: quantum boundary states and exact partition function

“…Here we push further the investigations of holographic dualities in non-perturbative 3d quantum gravity as initiated in [1][2][3]. Gravity in three space-time dimensions is indeed a topological field theory [4,5], for which gauge-invariant bulk observables should be entirely determined by the space-time topology and appropriate boundary conditions.…”

“…In the present work, we aimed at pursuing the exploration of holographic dualities in non-perturbative 3D quantum gravity initiated in [1][2][3]17]. Using the Ponzano-Regge model as the definition of a discrete quantum path integral for 3D Euclidean gravity (for vanishing cosmological constant) as a topological quantum field theory (TQFT), we focus on the case of 3d manifolds with boundaries in order to explore the correspondence between bulk geometry and boundary observables.…”

“…Then invariant tensors intertwining between representations are associated to each face and finally an amplitude constructed from the representations and intertwiners is associated to every 3D cell and ultimately to the whole 3D cellular complex. 1 Instead of integrating over the frame field e, one could instead integrate over the connection ω. Solving for the connection in terms of e gives the Levi-Civita connection.…”

“…We would like to consider the more general case of a manifold with boundaries, in order to study the boundary theories induced by the Ponzano-Regge model and possible holographic dualities between bulk and boundary theories along the lines sketched in [1,17]. Let us consider a 3D cellular complex ∆ with a 2D boundary ∂∆.…”

“…Following this line of research developed in [1][2][3]17], we propose here to use a class of coherent boundary states, defined as quantum superpositions of spins-thus lengths-admitting a critical regime with a manifest scale invariance in the semi-classical limit. Introduced in [18][19][20][21], we show that these coherent spin network wave-packets allow for an exact evaluation of the Ponzano-Regge amplitudes with quantum boundary.…”