Holonomy spin foam models: Definition and coarse graining

We propose a new holonomy formulation for spin foams, which naturally extends the theory space of lattice gauge theories. This allows current spin foam models to be defined on arbitrary 2-complexes as well as to generalize current spin foam models to arbitrary, in particular, finite groups. The similarity with standard lattice gauge theories allows us to apply standard coarse graining methods, which for finite groups can now be easily considered numerically. We will summarize other holonomy and spin network formulations of spin foams and group field theories and explain how the different representations arise through variable transformations in the partition function. A companion paper will provide a description of boundary Hilbert spaces as well as a canonical dynamic encoded in transfer operators.

Figure 1

Holonomy variables associated to half edges ($g_{ve}$) and wedges ($g_{ef}$) of a face.

Figure 2

Holonomy variables in the dual description of operator spin foam models.

Figure 3

Graphical notation of the relation between the OSFM version of the edge propagator $P_e$ and the one used in (6), which is also used in the BC and EPRL models.

Figure 4

Variables in the group field theory formulation.

Figure 5

Holonomy variables in the vertex amplitude formulation.

Figure 6

Wedge holonomies in the alternative vertex amplitude description.

Figure 7

Removing a trivial subdivision of an edge.

Figure 8

Removing a subdivision of a face.

Figure 9

The structure of the $C$ function for an edge shared by three faces.

Figure 10

This figure shows the face weights ${\tilde{\omega }}_{j^{+},j^{-}}$ for the $SU(2)\times SU(2)$ Barrett-Crane model as a function of $2j^{+}$, $2j^{-}$, which label the $x$ and $y$ axes respectively. The initial configuration ${\tilde{\omega }}_{j^{+},j^{-}}={\delta }_{j^{+},j^{-}}$ of face weights is the diagonal line of dots at the top (blue). The dots starting at around 0.2 (red) and spreading out flatly give the configuration after one iteration, where we show only a dot for values of ${\tilde{\omega }}_{j^{+},j^{-}}>{10}^{-4}$ from the picture the initial diagonal configuration is spread to nondiagonal arguments, but the values for the face weights are decreasing fast. After two iterations we arive at the dots on the floor of the graph (yellow-green). A large portion of the ${\tilde{\omega }}_{j^{+},j^{-}}$ is already smaller than ${10}^{-4}$ and the configuration is almost at the high temperature fixed point.

Figure 11

The 1–4 Pachner move can be iterated to generate an actual flow in the space of models.

Figure 12

Truncated RG flow of the $S_3$ model under a 1–4 move. The diagram includes four fixed points of the flow.