Holography inside out: from 3D gravity to 2D statistical models

By Bianca Dittrich, Christophe Goeller, Etera R. Livine, and Aldo Riello

Despite many years of research, quantum gravity remains a challenge. One of the reasons is that the many tools developed for perturbative quantum field theory are, in general, not applicable to quantum gravity. On the other hand, non-perturbative approaches have a difficult time in finding and extracting computable observables. The foremost problem here is a lack of diffeomorphism-invariant observables.

The situation can be improved very much by considering space-time regions with boundaries. This is also physically motivated, since one would like to be able to describe the physics of a given bounded region in a quasi-local way, that is without requiring a detailed description of the rest of the space-time outside. The key point is that the boundary can be used as an anchor, allowing to define observables in relation to this boundary. Then we can consider different boundary conditions, which translates at the quantum level into a rich zoo of boundary wave-functions. These boundary states can correspond to semi-classical boundary geometries or superpositions of those. The states can also describe asymptotic flat boundaries, thus allowing us to compare with perturbative approaches. In this context, holography in quantum gravity aims to determine how much of the bulk geometry can be reconstructed from the data encoded in the boundary state.


The boundary wave function Ψ are described by  dual theories defined on the boundary of the solid torus. These 2D boundary theories, obtained by integrating over all the bulk degrees of freedom of the geometry, encode the full 3D quantum gravity partition function.

In our paper [1], we have applied this logic to 3D quantum gravity. Here it is possible to make many explicit calculations, allowing for a detailed comparison between the various approaches. The framework we have used is the Ponzano-Regge model [2] (see [3–5] for modern accounts), which, although relatively unknown, is arguably the first quantum gravity model. The crucial advantage of the Ponzano-Regge model is to propose an explicit local description of the quantum geometry. For instance, as opposed to e.g. Chern–Simons gravity, it allows for a straightforward implementation of boundary conditions which fix the intrinsic metric of the boundary at the quantum level. In our view, this is the most geometric choice possible. As shown in our work [1, 6, 8, 9] this allows us to derive holographic boundary field theories in a straightforward manner.

The Ponzano–Regge model proposes an intrinsically discrete description of the spacetime geometry and is formulated in terms of a discretization of both bulk and boundary. However it turns out that the corresponding partition function is independent from the choice of the discretization of the bulk, while it still depends on the boundary discretization. Our goal is to understand how this 3D quantum gravity partition function depends on the 2D boundary state, which includes a choice for the boundary discretization. This means integrating over all bulk degrees of freedom and computing the 3D partition function to express it as a 2D partition function on the boundary.

Exploring various classes of boundary states, representing different classes of boundary quantum geometries, leads to 2D discrete models, which we understand as holographic duals for 3D gravity [1, 6]. Interestingly some of these models are known in condensed matter. At criticality in the continuum limit, such 2D models are typically described by conformal field theories (CFT), so we interpret our framework as a quasi-local noncritical proposal for a bulk gravity – boundary CFT correspondence meant to lead back to the AdS/CFT correspondence in a suitable asymptotical regime.

We have focused on the quantum geometry of a twisted solid torus, whose boundary
is a twisted torus, see figure below. Considering boundary states on a lattice with the smallest length unit, we have identified the induced 2D theory as a version of the 6-vertex model. Another interesting class of boundary wave-functions are locally semi-classical states, which can be used to describe an asymptotically flat, semi-classical boundary. The corresponding partition function can be evaluated in a semi-classical (WKB) approximation and compared to perturbative (and continuum) results obtained by Barnich and collaborators [7]. Our detailed lengthy calculations are presented in [8, 9]. We find an interesting agreement, which also includes a relation of the partition function to the BMS3 character, which is consistent with the perspective of the AdS3/CFT2 correspondence. However, in our case, we also obtain additional contributions resulting from non-perturbative “quantum backgrounds” due to Planck-scale resonances in the bulk geometry.


The space time region we consider is a twisted solid torus. It can be obtained by taking a solid cylinder and by twisting one of the ends before gluing it to a solid torus.

This is to our knowledge the first time that a non-perturbative partition function for discrete quantum gravity has been evaluated for boundary states that allow for a continuum limit. The results are very encouraging: we can reproduce the results of continuum approaches, but also obtain fully non-perturbative effects. This shows that including boundaries can be a successful strategy, which can connect perturbative and non-perturbative approaches, and is worthwhile to be explored also for the four–dimensional theory.

Read the full Letter in CQG here.

[1] Bianca Dittrich, Christophe Goeller, Etera R. Livine and Aldo Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity, Class. Quantum Grav. 35 13LT01 (2018), [arXiv:1803.02759]
[2] Giorgio Ponzano and Tullio Regge, Semiclassical limit of Racah coefficients in Spectroscopic and group theoretical methods in physics (Bloch ed.), North-Holland, 1968
[3] Laurent Freidel and David Louapre, Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles, Class.Quant.Grav. 21 (2004) 5685-5726, [arXiv:hepth/0401076]
[4] Laurent Freidel and Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory, Class.Quant.Grav.23 (2006) 2021-2062
[5] JohnW. Barrett and Ileana Naish-Guzman, The Ponzano-Regge model, Class. Quantum Grav. 26(2009) 155014, [arXiv:0803.3319]
[6] Glenn Barnich, Hernan A. Gonzalez, Alexander Maloney, Blagoje Oblak, One-loop partition function of three-dimensional flat gravity, JHEP 04 (2015) 178, [arXiv:1502.06185]
[7] Bianca Dittrich, Christophe Goeller, Etera R. Livine and Aldo RIello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity I – Convergence of multiple approaches and examples of Ponzano-Regge statistical duals, arXiv:1710.04202
[8] Bianca Dittrich, Christophe Goeller, Etera R. Livine and Aldo RIello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS3 characters, arXiv:1710.04237
[9] Aldo Riello, Quantum edge modes in 3d gravity and 2+1d topological phases of matter,[arXiv:1802.02588]

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