3D holography: from discretum to continuum

Abstract

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.

Holographic Formulation of 3D Metric Gravity with Finite Boundaries

“…These can be taken as boundary field variables, and one can thus easily integrate out all variables except these boundary field variables. [14] also computed the one-loop partition function for a finite boundary, which led to the same result as for asymptotically flat boundaries [18], see [19] for a corresponding result for asymptotic AdS boundaries.…”

“…Using metric boundary data [14] showed that a Liouville like dual field theory can also be identified more directly for finite boundaries. This work considered a specific background space time, so-called twisted thermal flat space [15], and employed (linearized) Regge calculus [16], a discretization of gravity, in which the variables are given by edge lengths in a piecewise flat geometry.…”

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

“…The BMS character is in fact well-defined on the upper complex plane, and can be identified as the Dedekind η-function up to a factor. This formula was also re-derived as an asymptotic limit of Regge calculus for discretized 3D gravity in [68]. It was further recovered as leading order of a WKB approximation of the Ponzano-Regge model with LS spin network boundary states in [3].…”

“…On the other hand, the less compact formula (95) gives the explicit expansion of the partition function in the momentum basis and shows the poles in γ. As we will discuss in the next section, this latter expression allows for a clearer continuum limit, where we will recover the BMS character formula for the 3D quantum gravity path integral, as derived in [22,23,68].…”

“…This Ponzano-Regge amplitude is given by the integral over the SU(2) holonomy along the non-contractible cycle of the evaluation of the boundary wave-function. We will show that it will be exactly computable and leads to a discretized, generalized and regularized version of the BMS character formula for the partition function of 3D gravity as derived in [22,23,68]. In particular, we will discuss the continuum and asymptotic limits, in which the usual BMS character is recovered.…”