Bounding bubbles: The vertex representation of 3d group field theory and the suppression of pseudomanifolds

Abstract

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the group field theory (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 pseudomanifold configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.

Spin Foam Models with Finite Groups

“…Moreover, many techniques may be applied to these field theories that are idiosyncratic to quantum field theories (as opposed to quantum mechanics). Perhaps the most striking is the study of their renormalisation group properties [208][209][210][211][212][213][214][215][216][217][218], but also work has commenced on the study of mean field theory properties [219], matter coupling [220][221][222], various symmetry analyses [223,224], and instantonic field theory solutions [225][226][227][228] (including cosmological applications [229]). …”

Singular topologies in the Boulatov model

“…It is therefore natural to look for a formulation of the model in which the fields are defined in such variables. Thanks to the gauge invariance (2), this transformation can actually be performed, and was extensively studied in [5]. One obtains a theory with four complex fields ψ (G u , G v , G w ), and the same structure of colored action (6), but with two notable differences: a) the gauge condition (2) is traded for a closure condition, encoded by a distributional factor δ(G u G v G w ) in the propagator; b) the interaction term encodes the tetrahedral geometry through identifications of vertices common to different triangles (as opposed to edges in the usual formulation), which is reflected in a stranded structure with 3-valent interactions.…”

“…3 The present article is a summary of the motivations and results of the recent work [5], where more technical details can be found. center of the edge i.…”

“…4 ht is also the auxiliary variable appearing in the gauge condition (2). 5 The topological structure used is the obvious one, induced by the identifications of the boundaries of the tetrahedra, themselves having the topology of a ball in R 3 . …”

Group Field Theory and Loop Quantum Gravity

“…These considerations affect work on GFT renormalisation only to a limited extent, as one can proceed guided only by the mathematical behaviour of GFT amplitudes. Their scaling with N is the first thing that has been studied, with focus on (coloured) topological models, in particular showing the suppression of singular topologies [53]. Still in the context of topological GFTs, remarkable calculations of radiative corrections were performed [54], and one interesting implication was that, in order to achieve renormalizability, these models need to be augmented by a kinetic term given by the Laplace-Beltrami operator on the group manifold…”