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This is the first of a series of papers dedicated to the study of thepartition function of three-dimensional quantum gravity on the twisted solidtorus with the aim to deepen our understanding of holographic dualities from anon-perturbative quantum gravity perspective. Our aim is to compare thePonzano-Regge model for non-perturbative three-dimensional quantum gravity withthe previous perturbative calculations of this partition function. We begin byreviewing the results obtained in the past ten years via a wealth of differentapproaches, and then introduce the Ponzano--Regge model in a self-containedway. Thanks to the topological nature of three-dimensional quantum gravity wecan solve exactly for the bulk degrees of freedom and identify dual boundarytheories which depend on the choice of boundary states, that can also describefinite, non-asymptotic boundaries. This series of papers aims precisely at theinvestigation of the role played by the different quantum boundary conditionsleading to different boundary theories. Here, we will describe the spin networkboundary states for the Ponzano-Regge model on the twisted torus and derive thegeneral expression for the corresponding partition functions. We identify aclass of boundary states describing a tessellation with maximally fuzzy squaresfor which the partition function can be explicitly evaluated. In the limit caseof a large, but finely discretized, boundary we find a dependence on the Dehntwist angle characteristic for the BMS3 character. We furthermore show howcertain choices of boundary states lead to known statistical models as dualfield theories-but with a twist.

nuclear physics b

Quasi-local holographic dualities in non-perturbative 3d quantum gravity  I - Convergence of multiple approaches and examples of Ponzano-Regge  statistical duals