Determine whether the holonomy-only closure insertion preserves amplitude factorization

Determine whether the holonomy-only implementation of geometric closure, which couples all wedge holonomies meeting at an edge, can be formulated so that the EPRL-FK amplitude continues to factorize into vertex contributions.

Background

The paper proposes imposing geometric closure by inserting a delta function enforcing the compatibility between spins and normals extracted from wedge holonomies. When rewritten solely in terms of the holonomies, the resulting expression depends on all wedges meeting at an edge and therefore on data associated with both vertices adjacent to that edge. This makes the proposed modification nonlocal with respect to the usual vertex decomposition of the spinfoam amplitude.

The authors explicitly identify preservation of the standard factorization into vertex contributions as unresolved. Resolving this issue is necessary for determining whether the proposed closure condition can be incorporated into a conventional spinfoam amplitude without fundamentally changing its factorized structure.

References

The important caveat is that, while “local” at the edge, this expression still depends on the wedge holonomies associated with all the wedges meeting at the edge, and therefore on data from both vertices surrounding the edge. This is a nonlocal modification of the amplitude, and it is not clear whether it can be implemented in a way that preserves the factorization of the amplitude into vertex contributions.

— Closing the Loop: from EPRL-FK spinfoams to Regge dynamics  (2609.37179 - Bruno et al., 29 Sep 2026) in Section 5, “A candidate implementation of geometric closure in the EPRL-FK amplitude”; Discussion