Revisiting random tensor models at large N via the Schwinger-Dyson equations

The Schwinger-Dyson Equations (SDEs) of matrix models are known to form (half) a Virasoro algebra and have become a standard tool to solve matrix models. The algebra generated by SDEs in tensor models (for random tensors in a suitable ensemble) is a specific generalization of the Virasoro algebra and it is important to show that these new symmetries determine the physical solutions. We prove this result for random tensors at large N. Compared to matrix models, tensor models have more than a single invariant at each order in the tensor entries and the SDEs make them proliferate. However, the specific combinatorics of the dominant observables allows to restrict to linear SDEs and we show that they determine a unique physical perturbative solution. This gives a new proof that tensor models are Gaussian at large N, with the covariance being the full 2-point function.Comment: 17 pages, many figure

A New Recursion Relation for the 6j-Symbol

The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano-Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes.Comment: 10 pages, v2: title and introduction changed, paper re-structured; Annales Henri Poincare (2011

Constraint algebra in LQG reloaded : Toy model of a U(1)^{3} Gauge Theory I

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework.Comment: 60 pages, 6 figure

Generating Functions for Coherent Intertwiners

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.Comment: 31 page

Coupling of hard dimers to dynamical lattices via random tensors

We study hard dimers on dynamical lattices in arbitrary dimensions using a random tensor model. The set of lattices corresponds to triangulations of the d-sphere and is selected by the large N limit. For small enough dimer activities, the critical behavior of the continuum limit is the one of pure random lattices. We find a negative critical activity where the universality class is changed as dimers become critical, in a very similar way hard dimers exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents are calculated exactly. An alternative description as a system of `color-sensitive hard-core dimers' on random branched polymers is provided.Comment: 12 page

Holonomy observables in Ponzano-Regge type state sum models

We study observables on group elements in the Ponzano-Regge model. We show that these observables have a natural interpretation in terms of Feynman diagrams on a sphere and contrast them to the well studied observables on the spin labels. We elucidate this interpretation by showing how they arise from the no-gravity limit of the Turaev-Viro model and Chern-Simons theory.Comment: 15 pages, 2 figure

Bubbles and jackets: new scaling bounds in topological group field theories

We use a reformulation of topological group field theories in 3 and 4 dimensions in terms of variables associated to vertices, in 3d, and edges, in 4d, to obtain new scaling bounds for their Feynman amplitudes. In both 3 and 4 dimensions, we obtain a bubble bound proving the suppression of singular topologies with respect to the first terms in the perturbative expansion (in the cut-off). We also prove a new, stronger jacket bound than the one currently available in the literature. We expect these results to be relevant for other tensorial field theories of this type, as well as for group field theory models for 4d quantum gravity.Comment: v2: Minor modifications to match published versio

Holomorphic Simplicity Constraints for 4d Spinfoam Models

Within the framework of spinfoam models, we revisit the simplicity constraints reducing topological BF theory to 4d Riemannian gravity. We use the reformulation of SU(2) intertwiners and spin networks in term of spinors, which has come out from both the recently developed U(N) framework for SU(2) intertwiners and the twisted geometry approach to spin networks and spinfoam boundary states. Using these tools, we are able to perform a holomorphic/anti-holomorphic splitting of the simplicity constraints and define a new set of holomorphic simplicity constraints, which are equivalent to the standard ones at the classical level and which can be imposed strongly on intertwiners at the quantum level. We then show how to solve these new holomorphic simplicity constraints using coherent intertwiner states. We further define the corresponding coherent spin network functionals and introduce a new spinfoam model for 4d Riemannian gravity based on these holomorphic simplicity constraints and whose amplitudes are defined from the evaluation of the new coherent spin networks.Comment: 27 page

Codimension-2 surfaces and their Hilbert spaces: low-energy clues for holography from general covariance

We argue that the holographic principle may be hinted at already from low-energy considerations, assuming diffeomorphism invariance, quantum mechanics and Minkowski-like causality. We consider the states of finite spacelike hypersurfaces in a diffeomorphism-invariant QFT. A low-energy regularization is assumed. We note a natural dependence of the Hilbert space on a codimension-2 boundary surface. The Hilbert product is defined dynamically, in terms of transition amplitudes which are described by a path integral. We show that a canonical basis is incompatible with these assumptions, which opens the possibility for a smaller Hilbert-space dimension than canonically expected. We argue further that this dimension may decrease with surface area at constant volume, hinting at holographic area-proportionality. We draw comparisons with other approaches and setups, and propose an interpretation for the non-holographic space of graviton states at asymptotically-Minkowski null infinity.Comment: 13 pages, 9 eps figures. Added Section VI, improved presentation. Expanded and split the Introduction into two sections. Added Section VII. Added reference

Classical Setting and Effective Dynamics for Spinfoam Cosmology

We explore how to extract effective dynamics from loop quantum gravity and spinfoams truncated to a finite fixed graph, with the hope of modeling symmetry-reduced gravitational systems. We particularize our study to the 2-vertex graph with N links. We describe the canonical data using the recent formulation of the phase space in terms of spinors, and implement a symmetry-reduction to the homogeneous and isotropic sector. From the canonical point of view, we construct a consistent Hamiltonian for the model and discuss its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the dynamics from the spinfoam approach. We compute exactly the transition amplitude between initial and final coherent spin networks states with support on the 2-vertex graph, for the choice of the simplest two-complex (with a single space-time vertex). The transition amplitude verifies an exact differential equation that agrees with the Hamiltonian constructed previously. Thus, in our simple setting we clarify the link between the canonical and the covariant formalisms.Comment: 38 pages, v2: Link with discretized loop quantum gravity made explicit and emphasize