Linearized group field theory and power-counting theorems

Abstract

We introduce a linearized version of group field theory. It can be viewed either as a group field theory over the additive group of a vector space or as an asymptotic expansion of any group field theory around the unit group element. We prove exact power counting theorems for any graph of such models. For linearized colored models the power counting of any amplitude is further computed in term of the homology of the graph. An exact power counting theorem is also established for a particular class of graphs of the nonlinearized models, which satisfy a planarity condition. Examples and connections with previous results are discussed.Pacs numbers: 04.60.-m, 04.60.Pp

Group field theory in dimension4−ϵ

“…A general power-counting theorem for TGFTs with gauge invariance condition was first derived in [24,25], with the help of older results [47]. A classification of potentially renormalizable models could thus be deduced, in terms of the rank d of the fields, the dimension D of the group and the maximal valency of renormalizable interactions v max .…”

Tensor models and embedded Riemann surfaces

“…The importance of the fact that the coupling constant scales as λ/ √ N 3 has been stressed in [11] and subsequently in [8], where the relevant details are expressed. To explain briefly, consider two graphs Γ 1 and Γ 2 in the expansion, such that Γ 2 is just Γ 1 supplemented with a single insertion of Fig.…”

“…Moreover, the appropriate scaling of the coupling constant in this case is λ/ √ N [11]. The graph amplitude now takes the form:…”

“…These ribbon graphs are algebraic objects that capture the topological properties of the Feynman graphs. They contain two classes of subgraphs, known as bubbles [9] and jackets [11], which are of particular significance. While bubbles are easily identified as Riemann surfaces embedded in the dual triangulation, the topological properties of the jackets have remained more obscure.…”

“…A stumbling block seemed to be that they generated a plethora of unwanted structures; not only simplicial manifolds, but simplicial pseudomanifolds [4]. Recently, after much work [5,11], a promising step has been made in that direction with the construction of a 1/N -topological expansion [6][7][8] for the so-called coloured tensor models [9,10]. In that context, it was shown that for arbitrary dimension d, only graphs corresponding to d-spheres arise at leading order in the 1/N -expansion.…”