Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the GFT fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudo-manifolds configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.Comment: 28 pages, 17 figures. Few typos fixed. Minor corrections in figure 6 and theorem

Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice

A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy $\Delta E$ with system size $L$, $\sigma(\Delta E)\sim L^y$, is obtained as $y_3=0.25546(3)$ by reducing the equivalent of lattices up to $L=2^{100}$ in $d=3$, and as $y_4=0.76382(4)$ for up to $L=2^{35}$ in $d=4$. The reduction rules allow the exact determination of the ground state energy, entropy, and also provide an approximation to the overlap distribution. With these methods, some well-know and some new features of diluted hierarchical lattices are calculated.Comment: 7 pages, RevTex, 6 figures (postscript), added results for d=4, some corrections; final version, as to appear in EPJ

Dynamics and the Emergence of Geometry in an Information Mesh

The idea of a graph theoretical approach to modeling the emergence of a quantized geometry and consequently spacetime, has been proposed previously, but not well studied. In most approaches the focus has been upon how to generate a spacetime that possesses properties that would be desirable at the continuum limit, and the question of how to model matter and its dynamics has not been directly addressed. Recent advances in network science have yielded new approaches to the mechanism by which spacetime can emerge as the ground state of a simple Hamiltonian, based upon a multi-dimensional Ising model with one dimensionless coupling constant. Extensions to this model have been proposed that improve the ground state geometry, but they require additional coupling constants. In this paper we conduct an extensive exploration of the graph properties of the ground states of these models, and a simplification requiring only one coupling constant. We demonstrate that the simplification is effective at producing an acceptable ground state. Moreover we propose a scheme for the inclusion of matter and dynamics as excitations above the ground state of the simplified Hamiltonian. Intriguingly, enforcing locality has the consequence of reproducing the free non-relativistic dynamics of a quantum particle

Group field theories

Group field theories are particular quantum field theories defined on D copies of a group which reproduce spin foam amplitudes on a space-time of dimension D. In these lecture notes, we present the general construction of group field theories, merging ideas from tensor models and loop quantum gravity. This lecture is organized as follows. In the first section, we present basic aspects of quantum field theory and matrix models. The second section is devoted to general aspects of tensor models and group field theory and in the last section we examine properties of the group field formulation of BF theory and the EPRL model. We conclude with a few possible research topics, like the construction of a continuum limit based on the double scaling limit or the relation to loop quantum gravity through Schwinger-Dyson equationsComment: Lectures given at the "3rd Quantum Gravity and Quantum Geometry School", march 2011, Zakopan

A geometric approach to free variable loop equations in discretized theories of 2D gravity

We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity theories which correspond to matrix models, our method is a generalization of the technique of Schwinger-Dyson equations and is closely related to recent work describing the master field in terms of noncommuting variables; the important differences are that we derive a single equation for the generating function using purely graphical arguments, and that the approach is applicable to a broader class of theories than those described by matrix models. Several example applications are given here, including theories of gravity coupled to a single Ising spin ($c = 1/2$), multiple Ising spins ($c = k/2$), a general class of two-matrix models which includes the Ising theory and its dual, the three-state Potts model, and a dually weighted graph model which does not admit a simple description in terms of matrix models.Comment: 40 pages, 8 figures, LaTeX; final publication versio

The Ising Model on a Quenched Ensemble of c = -5 Gravity Graphs

We study with Monte Carlo methods an ensemble of c=-5 gravity graphs, generated by coupling a conformal field theory with central charge c=-5 to two-dimensional quantum gravity. We measure the fractal properties of the ensemble, such as the string susceptibility exponent gamma_s and the intrinsic fractal dimensions d_H. We find gamma_s = -1.5(1) and d_H = 3.36(4), in reasonable agreement with theoretical predictions. In addition, we study the critical behavior of an Ising model on a quenched ensemble of the c=-5 graphs and show that it agrees, within numerical accuracy, with theoretical predictions for the critical behavior of an Ising model coupled dynamically to two-dimensional quantum gravity, provided the total central charge of the matter sector is c=-5. From this we conjecture that the critical behavior of the Ising model is determined solely by the average fractal properties of the graphs, the coupling to the geometry not playing an important role.Comment: 23 pages, Latex, 7 figure

From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity Ns of the inherent structures generically has a lognormal distribution. In addition, the large volume limit of ln/ differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.Comment: 10 pages, 9 figs, slightly improved presentation, more references, accepted for publication in Phys Rev