Holonomy spin foam models: Asymptotic geometry of the partition function

TL;DR

This paper analyzes the asymptotic geometry of spin foam partition functions for various models, revealing suppression of geometric boundary data unless certain curvature constraints are met, and introduces wave front sets as a new analytical tool.

Contribution

It provides a criterion for the existence of the partition function, identifies curvature constraints affecting geometric data, and introduces wave front sets for spin foam analysis.

Findings

Most Regge manifolds are suppressed asymptotically.

Accidental curvature constraints arise from face amplitude twisting.

Wave front sets are effective tools for spin foam asymptotic analysis.

Abstract

We study the asymptotic geometry of the spin foam partition function for a large class of models, including the models of Barrett and Crane, Engle, Pereira, Rovelli and Livine, and, Freidel and Krasnov. The asymptotics is taken with respect to the boundary spins only, no assumption of large spins is made in the interior. We give a sufficient criterion for the existence of the partition function. We find that geometric boundary data is suppressed unless its interior continuation satisfies certain accidental curvature constraints. This means in particular that most Regge manifolds are suppressed in the asymptotic regime. We discuss this explicitly for the case of the configurations arising in the 3-3 Pachner move. We identify the origin of these accidental curvature constraints as an incorrect twisting of the face amplitude upon introduction of the Immirzi parameter and propose a way to resolve this problem, albeit at the price of losing the connection to the SU(2) boundary Hilbert space. The key methodological innovation that enables these results is the introduction of the notion of wave front sets, and the adaptation of tools for their study from micro local analysis to the case of spin foam partition functions.