Linearized Group Field Theory and Power Counting Theorems

TL;DR

The paper introduces linearized (Abelianized) group field theories and proves exact power counting for all graphs, with a complementary formulation for colored GFTs in terms of graph homology. It shows how the ultraviolet scaling can be captured by a Pfaffian/Symanzik-polynomial framework and connects colored graph power counting to Betti numbers and bubble counts. For a class of nonlinearized graphs obeying a planarity condition, an exact, group-independent power counting is established, aided by a jacket-based momentum-routing argument. These results sharpen the understanding of ultraviolet behavior in GFTs and point to future work on richer polynomial structures and planarity-driven simplifications.

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.

Figures (4)

• Figure 1: Propagator and vertex $\phi^{D+1}$ in $D$ dimensional GFT.
• Figure 2: A graph, its dual and a spanning tree on the dual.
• Figure 3: A chain with power counting $2+n/2$
• Figure 4: An other chain with power counting $2+n/2$

Theorems & Definitions (12)

• Definition 2.1
• Proposition 2.1
• Theorem 2.1
• Definition 3.1
• Theorem 3.1
• Corollary 3.1
• Lemma 3.1
• Theorem 3.2
• Definition 4.1
• Theorem 4.1