Group field theory renormalization in the 3D case: Power counting of divergences

We take the first steps in a systematic study of group field theory (GFT) renormalization, focusing on the Boulatov model for 3D quantum gravity. We define an algorithm for constructing the 2D triangulations that characterize the boundary of the 3D bubbles, where divergences are located, of an arbitrary 3D GFT Feynman diagram. We then identify a special class of graphs for which a complete contraction procedure is possible, and prove, for these, a complete power counting. These results represent important progress towards understanding the origin of the continuum and manifoldlike appearance of quantum spacetime at low energies, and of its topology, in a GFT framework.

Figure 1

The GFT vertex.

Figure 2

The GFT propagator.

Figure 3

Fully labeled truncated tetrahedron.

Figure 4

Fully labeled 3D interaction (dual vertex).

Figure 5

2D dual vertices descendent from a 3D dual vertex.

Figure 6

2D dual lines descending from a 3D dual line.

Figure 7

The sunshine graph ${\mathcal{G}}_1$.

Figure 8

The 2D projection ${\overline{\mathcal{G}}}_1$.

Figure 9

A typical set of faces of $b$ having common 3D ancestors with the faces of $b^{\prime }$.

Figure 10

Two neighboring faces.

Figure 11

The joining of two faces separated by one line, $F_1{\bigsqcup }_L{F}_2$.

Figure 12

Two faces, $F_1$ and $F$, that share two lines in a planar graph.

Figure 13

The graph ${\overline{\mathcal{G}}}_1^{\prime }$.

Figure 14

The graph ${\overline{\mathcal{G}}}_1^{\prime \prime }$.

Figure 15

The graph ${\overline{\mathcal{G}}}_1^{\prime \prime \prime }$.

Figure 16

The twisted sunshine graph ${\mathcal{G}}_2$.

Figure 17

Bubbles of the graph ${\mathcal{G}}_2$ (the graph ${\overline{\mathcal{G}}}_2$).

Figure 18

The permuted sunshine graph ${\mathcal{G}}_3$.

Figure 19

Connected components associated to the bubbles of the graph ${\mathcal{G}}_3$ (the ${\overline{\mathcal{G}}}_3$ graph).

Figure 20

The three 2D descendents of the same 3D ancestor.

Figure 21

The first case.

Figure 22

The second case.

Figure 23

The third case.

Figure 24

${\overline{\mathcal{G}}}^{b_1}\cap b_j$ after the contraction of $b_1$.

Figure 25

The fourth case.