Abstract
Abstract 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.
Highlights
Still, the models constructed on this simple basis are extremely rich, and a wealth of interesting results have been obtained recently about them [3, 22]
Many power counting results have been proven recently, with a special focus on topological group field theories [19,20,21, 23, 47, 61,62,63, 71, 72]. These include: very general power counting theorems based on the lattice gauge theory formulation of the GFT Feynman amplitudes [71, 72], showing how these amplitudes depend on both topological properties of the corresponding cellular complex and on the details of its combinatorial structure; the scaling bounds that are at the root of the results on the large-N limit of such models [19,20,21], in which we know that only a restricted class of complexes corresponding to spherical manifolds dominate; analyses targeting restricted, more focused topological issues like the relative weight of pseudomanifolds over manifolds in the GFT perturbative expansion [23]
The new formulations, on the other hand, seem more natural from a field theory point of view, in light of the fact that the symmetry group of the theory acts on such variables, and, as we have shown in this paper, allows a more direct and powerful analysis of the scaling of the amplitudes when the cut-off is removed
Summary
The amplitudes as written are divergent and need to be regularized to be given rigorous meaning. We will use variables involved in flatness conditions around edges of colours {345}, {145} and {135}, a choice that corresponds to a tree of the first kind Since each strand is connected to two interactions, the integrals to compute are not simple convolutions, difficult To circumvent this problem we bound all the heat-kernels implementing flatness constraints of colours {345}, {145} and {135} by their value at the identity, and integrate the propagators. The last line is exactly a product of bubble 3d amplitudes: b∈B2 Abt. As for the integral in the first line, it is associated with the graph made of all the strands of colour (125) which, as in the previous case, we bound by:.
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