This is the first of a series of papers dedicated to the study of the partition function of three-dimensional quantum gravity on the twisted solid torus with the aim to deepen our understanding of holographic dualities from a non-perturbative quantum gravity perspective. Our aim is to compare the Ponzano-Regge model for non-perturbative three-dimensional quantum gravity with the previous perturbative calculations of this partition function. We begin by reviewing the results obtained in the past ten years via a wealth of different approaches, and then introduce the Ponzano-Regge model in a self-contained way. Thanks to the topological nature of three-dimensional quantum gravity we can solve exactly for the bulk degrees of freedom and identify dual boundary theories which depend on the choice of boundary states, that can also describe finite, non-asymptotic boundaries. This series of papers aims precisely at the investigation of the role played by the different quantum boundary conditions leading to different boundary theories. Here, we will describe the spin network boundary states for the Ponzano-Regge model on the twisted torus and derive the general expression for the corresponding partition functions. We identify a class of boundary states describing a tessellation with maximally fuzzy squares for which the partition function can be explicitly evaluated. In the limit case of a large, but finely discretized, boundary we find a dependence on the Dehn twist angle characteristic for the BMS 3 character. We furthermore show how certain choices of boundary states lead to known statistical models as dual field theories -but with a twist. * Electronic address: bdittrich@perimeterinstitute.ca † Electronic address: christophe.goeller@ens-lyon.fr ‡ Electronic address: etera.livine@ens-lyon.fr § Electronic address: ariello@perimeterinstitute.ca arXiv:1710.04202v1 [hep-th]
We analyze the partition function of three-dimensional quantum gravity on the twisted solid tours and the ensuing dual field theory. The setting is that of a non-perturbative model of three dimensional quantum gravity-the Ponzano-Regge model, that we briefly review in a self-contained manner-which can be used to compute quasi-local amplitudes for its boundary states. In this second paper of the series, we choose a particular class of boundary spin-network states which impose Gibbons-Hawking-York boundary conditions to the partition function. The peculiarity of these states is to encode a two-dimensional quantum geometry peaked around a classical quadrangulation of the finite toroidal boundary. Thanks to the topological properties of three-dimensional gravity, the theory easily projects onto the boundary while crucially still keeping track of the topological properties of the bulk. This produces, at the non-perturbative level, a specific non-linear sigma-model on the boundary, akin to a Wess-Zumino-Novikov-Witten model, whose classical equations of motion can be used to reconstruct different bulk geometries: the expected classical one is accompanied by other "quantum" solutions. The classical regime of the sigma-model becomes reliable in the limit of large boundary spins, which coincides with the semiclassical limit of the boundary geometry. In a 1-loop approximation around the solutions to the classical equations of motion, we recover (with corrections due to the non-classical bulk geometries) results obtained in the past via perturbative quantum General Relativity and through the study of characters of the BMS 3 group. The exposition is meant to be completely 11 This pair of variables is the one identified by Kapovich and Millson [77]. 12 Indeed, in the 3-dimensional context in which they were originally developed, such intertwiners are interpreted as quantum tetrahedra in the four-valent case m = 4, and more generally as quantum polyhedra for higher valencies [80][81][82][83]. See also [78,84] for a discussion of polyhedra in homogeneously curved space. In the context of canonical 3d gravity, the interpretation of SU(2) intertwiners as polygons was never truly developed beyond the interpretation of 3-valent intertwiners as quantum triangles. 13 If non-closing configuration are excluded, their contribution is suppressed in the large-spin limit [32], which admits in turn a semiclassical interpretation in terms of Regge geometries.14 Computing the expectation values of the su(2) generators on the state |j, j , we get J = (0, 0, j). In turn, we can compute the variance J 2 = j(j + 1), which is simply given by the su(2) Casimir. Thus the state |j, j corresponds to a semi-classical vector of length j in the z-direction, peaked on (0, 0, j) with spread 1 J J 2 − J 2 ∼ 1 √ j . The corresponding polar angle θ can be estimated to be θ ≈ arccos j √ j(j+1) ≈ 1 √ j → 0. 15 Indeed, the S2 in which n lives is better understood as SU(2)/U(1). This corresponds precisely to the Hopf fibration S2 ∼ = S3/S1.
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