Ponzano Regge Research Articles

Self-duality of the $6j$-symbol and Fisher zeros for the tetrahedron

The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner’s 6j -symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j -symbol and Fisher zeros. Using the Ponzano–Regge formula for the asymptotics for the 6j -symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well known that the 2d intrinsic geometry can be used to parametrize the critical point of the Ising model, e.g., on isoradial graphs, it is the first time to our knowledge that the extrinsic geometry is found to also be relevant. This outlines a method towards a more general geometric parametrization of the Fisher zeros for the 2d Ising model on arbitrary graphs.

The Ponzano–Regge cylinder and propagator for 3d quantum gravity

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano–Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes subspaces with different total recoupled spin. This may select the vanishing overall spin sector at late time depending on the chosen cylinder boundary data, leading to an emergent symmetry scenario in the continuum limit. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.

The quantum gravity disk: Discrete current algebra

We study the quantization of the corner symmetry algebra of 3D gravity, that is, the algebra of observables associated with 1D spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double DSU(2). Those discrete currents depend on an integer N, a discreteness parameter, understood as the number of quanta of geometry on the 1D boundary: low N is the deep quantum regime, while large N should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano–Regge state-sum model, which provides discrete path integral amplitudes for 3D quantum gravity. The integer N then counts the flux lines attached to the boundary. Second, we analyze the refinement, coarse-graining, and fusion processes as N changes, and we show that the N → ∞ limit is a classical limit where we recover the Poincaré current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step toward understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity.

Searching for classical geometries in spin foam amplitudes: a numerical method

We develop a numerical method to investigate the semiclassical limit of spin foam amplitudes with many vertices. We test it using the Ponzano–Regge model, a spin foam model for three-dimensional euclidean gravity, and a transition amplitude with three vertices. We study the summation over bulk spins, and we identify the stationary phase points that dominate it and that correspond to classical geometries. We complement with the numerical analysis of a four vertex transition amplitude and with a modification of the model that includes local curvature. We discuss the generalization of our results to the four-dimensional EPRL spin foam model, and we provide suggestions for new computations.

Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS3 characters

We analyze the partition function of three-dimensional quantum gravity on the twisted solid torus and the ensuing dual field theory. The setting is that of a non-perturbative model of three-dimensional quantum gravity, the Ponzano–Regge model, which is here reviewed in a self-contained manner and then 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 semi-classical 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 BMS3 group. The exposition is meant to be completely self-contained.

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

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.

Quasi-local holographic dualities in non-perturbative 3D quantum gravity

We present a line of research aimed at investigating holographic dualities in the context of three dimensional quantum gravity within finite bounded regions. The bulk quantum geometrodynamics is provided by the Ponzano–Regge state-sum model, which defines 3D quantum gravity as a discrete topological quantum field theory (TQFT). This formulation provides an explicit and detailed definition of the quantum boundary states, which allows a rich correspondence between quantum boundary conditions and boundary theories, thereby leading to holographic dualities between 3D quantum gravity and 2D statistical models as used in condensed matter. After presenting the general framework, we focus on the concrete example of the coherent twisted torus boundary, which allows for a direct comparison with other approaches to 3D/2D holography at asymptotic infinity. We conclude with the most interesting questions to pursue in this framework.

Semiclassical geometry of integrable systems**Dedicated to the memory of P P Kulish.

The main result of this paper is a formula for the scalar product of semiclassical eigenvectors of two integrable systems on the same symplectic manifold. An important application of this formula is the Ponzano–Regge type of asymptotic of Racah–Wigner coefficients.

Projective Ponzano–Regge spin networks and their symmetries

We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingredients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn–Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quantum gravity models carried out successfully over the last few decades. The role of Regge symmetry–an intriguing discrete symmetry of the 6j which goes beyond the standard tetrahedral symmetry of this symbol–will be also discussed in brief to highlight its role in providing a natural regularization of projective spin networks that somehow mimics the standard regularization through a q-deformation of SU(2).

FROM DIMENSIONAL REDUCTION OF 4d SPIN FOAM MODEL TO ADDING NON-GRAVITATIONAL FIELDS TO 3d SPIN FOAM MODEL

A Kaluza–Klein-like approach for a 4d spin foam model is considered. By applying this approach to a model based on group field theory in 4d (TOCY model), and using the Peter–Weyl expansion of the gravitational field, reconstruction of new non-gravitational fields and interactions in the action are found. The perturbative expansion of the partition function produces graphs colored with SU(2) algebraic data, from which one can reconstruct a 3d simplicial complex representing space-time and its geometry (like in the Ponzano–Regge formulation of pure 3d quantum gravity), as well as the Feynman graph for typical matter fields. Thus a mechanism for generation of matter and construction of new dimensions are found from pure gravity.

Holographic hierarchy in the Gaussian matrix model via the fuzzy sphere

The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano–Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of a holographic hierarchy linking, via the large N expansion, the zero-dimensional QFT of the Matrix model to 2D strings and 3D membranes.

Divergences and orientation in spinfoams

We suggest that at least some of the large radiative corrections appearing in the spinfoam framework might be tied to the implicit sum over orientations. Specifically, we show that in a suitably simplified context the characteristic ‘spike’ (large-spin) divergence of the Ponzano–Regge model disappears when restricting the theory to just one of the two orientations appearing in the asymptotic limit of the vertex amplitude.

Path integral measure and triangulation independence in discrete gravity

A path integral measure for gravity should also preserve the fundamental symmetry of general relativity, which is diffeomorphism symmetry. In previous work, we argued that a successful implementation of this symmetry into discrete quantum gravity models would imply discretization independence. We therefore consider the requirement of triangulation independence for the measure in (linearized) Regge calculus, which is a discrete model for quantum gravity, appearing in the semi--classical limit of spin foam models. To this end we develop a technique to evaluate the linearized Regge action associated to Pachner moves in 3D and 4D and show that it has a simple, factorized structure. We succeed in finding a local measure for 3D (linearized) Regge calculus that leads to triangulation independence. This measure factor coincides with the asymptotics of the Ponzano Regge Model, a 3D spin foam model for gravity. We furthermore discuss to which extent one can find a triangulation independent measure for 4D Regge calculus and how such a measure would be related to a quantum model for 4D flat space. To this end, we also determine the dependence of classical Regge calculus on the choice of triangulation in 3D and 4D.

Holonomy observables in Ponzano–Regge-type state sum models

Observables on group elements in the Ponzano–Regge model are studied. These observables are shown to have a natural interpretation in terms of Feynman diagrams on a sphere and are contrasted to the well-studied observables on the spin labels. The physical interpretation is elucidated by showing how they arise from the no-gravity limit of the Turaev–Viro model and Chern–Simons theory.

Asymptotics of Wigner 3nj-symbols with small and large angular momenta: an elementary method

Yu and Littlejohn recently studied in (2011 Phys. Rev. A 83 052114 (arXiv:1104.1499)) some asymptotics of Wigner symbols with some small and large angular momenta. They found that in this regime the essential information is captured by the geometry of a tetrahedron, and gave new formulae for 9j-, 12j- and 15j-symbols. We present here an alternative derivation which leads to a simpler formula, based on the use of the Ponzano–Regge formula for the relevant tetrahedron. The approach is generalized to Wigner 3nj-symbols with some large and small angular momenta, where more than one tetrahedron are needed, leading to new asymptotics for Wigner 3nj-symbols. As an illustration, we present 15j-symbols with one, two and four small angular momenta, and give an alternative formula to Yu’s recent 15j-symbol with three small spins.

Semiclassical mechanics of the Wigner 6j-symbol

The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems to explore the geometrical issues surrounding the Ponzano–Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano–Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson) and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.

The Hamiltonian constraint in 3d Riemannian loop quantum gravity

We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the extrinsic curvature from dihedral angles. The Wheeler–DeWitt equation takes the form of difference equations, which are actually recursion relations satisfied by Wigner symbols. On the boundary of a tetrahedron, the Hamiltonian generates the exact recursion relation on the 6j-symbol which comes from the Biedenharn–Elliott (pentagon) identity. This fills the gap between the canonical quantization and the symmetries of the Ponzano–Regge state-sum model for 3d gravity.

A deformed Poincaré invariance for group field theories

In the context of quantum gravity, group field theories are field theories that generate spinfoam amplitudes as Feynman diagrams. They can be understood as generalizations of the matrix models used for 2D quantum gravity. In particular, Boulatov's theory reproduces the amplitudes of the Ponzano–Regge spinfoam model for 3D quantum gravity. Motivated by recent works on field theories on non-commutative flat spaces, we show that Boulatov's theory (and its colored version) is actually invariant under a global deformed Poincaré symmetry. This allows us to define a notion of flat or excited geometry states when considering scalar perturbations around classical solutions of the group field equations of motion. As a side result, our analysis seems to point out that the notion of braiding of group field theories should be a key feature to study further in this context.

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called `bubble divergences'. A common expectation is that the degree of these divergences is given by the number of `bubbles' of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian -- in both cases, the divergence degree is given by the second Betti number of the 2-complex.

Asymptotics of the Wigner 9j-symbol

We present the asymptotic formula for the Wigner 9j-symbol, valid when all quantum numbers are large, in the classically allowed region. As in the Ponzano–Regge formula for the 6j-symbol, the action is expressed in terms of lengths of edges and dihedral angles of a geometrical figure, but the angles require care in definition. Rules are presented for converting spin networks into the associated geometrical figures. The amplitude is expressed as the determinant of a 2 × 2 matrix of Poisson brackets. The 9j-symbol possesses caustics associated with the fold and elliptic and hyperbolic umbilic catastrophes. The asymptotic formula obeys the exact symmetries of the 9j-symbol.