Abstract
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 BMS3 character. We furthermore show how certain choices of boundary states lead to known statistical models as dual field theories – but with a twist.
Highlights
This is the first of a series of papers exploring the holographic nature of 3d quantum gravity
1 First defined in 1968 [sic], the latter is an instantiation of by the first-order (BF) topological quantum field theory, and it has been rigorously related to other approaches to 3d quantum gravity, notably to the combinatorial quantization of Chern–Simons theory and to Loop Quantum Gravity
Our focus is on the properties of 3d quantum gravity as described by the Ponzano–Regge model [3], and on its relation to other approaches
Summary
This is the first of a series of papers exploring the holographic nature of 3d quantum gravity. Being formulated in terms of a local state-sum, the Ponzano–Regge model allows one to compute the amplitude of quantum gravitational processes within finite, i.e. quasi-local, regions. This is to be contrasted to the AdS/CFT framework, which intrinsically refers to the asymptotic boundary of AdS. It implies, again differently from the AdS/CFT philosophy, that each amplitude is associated to one given spacetime topology, just as in Chern–Simons theory.
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