3D holography: from discretum to continuum

TL;DR

This paper investigates the one-loop partition function of 3D gravity on a solid torus using a discrete Regge calculus approach, revealing boundary degrees of freedom and dualities with a Liouville-type boundary field theory.

Contribution

It introduces a discrete Regge calculus method to analyze 3D gravity boundary effects, connecting boundary degrees of freedom with dual field theories at finite boundaries.

Findings

The one-loop correction matches continuum results for asymptotically flat boundaries.

Discrete approach identifies boundary degrees of freedom responsible for quantum corrections.

A dual boundary field theory with Liouville-type coupling is established.

Abstract

We study the one-loop partition function of 3D gravity without cosmological constant on the solid torus with arbitrary metric fluctuations on the boundary. To this end we employ the discrete approach of (quantum) Regge calculus. In contrast with similar calculations performed directly in the continuum, we work with a boundary at finite distance from the torus axis. We show that after taking the continuum limit on the boundary - but still keeping finite distance from the torus axis - the one-loop correction is the same as the one recently found in the continuum in Barnich et al. for an asymptotically flat boundary. The discrete approach taken here allows to identify the boundary degrees of freedom which are responsible for the non-trivial structure of the one-loop correction. We therefore calculate also the Hamilton-Jacobi function to quadratic order in the boundary fluctuations both in the discrete set-up and directly in the continuum theory. We identify a dual boundary field theory with a Liouville type coupling to the boundary metric. The discrete set-up allows again to identify the dual field with degrees of freedom associated to radial bulk edges attached to the boundary. Integrating out this dual field reproduces the (boundary diffeomorphism invariant part of the) quadratic order of the Hamilton-Jacobi functional. The considerations here show that bulk boundary dualities might also emerge at finite boundaries and moreover that discrete approaches are helpful in identifying such dualities.