SU(2) graph invariants, Regge actions and polytopes

TL;DR

This paper analyzes the large spin asymptotics of SU(2) graph invariants, extending previous results to more general configurations, and confirms the asymptotic formulas numerically at relatively low spins.

Contribution

It generalizes the large spin asymptotic analysis of 15j symbols to broader SU(2) graph invariants and explores the geometric interpretations of saddle points.

Findings

Asymptotic formulas match numerical data within a few percent at spins around 10.

Identifies new geometric configurations called angle-matched twisted geometries.

Extends the understanding of Regge actions to more general polytope-based structures.

Abstract

We revisit the the large spin asymptotics of 15j symbols in terms of cosines of the 4d Euclidean Regge action, as derived by Barrett and collaborators using a saddle point approximation. We bring it closer to the perspective of area-angle Regge calculus and twisted geometries, and compute explicitly the Hessian and phase offsets. We then extend it to more general SU(2) graph invariants, showing that saddle points still exist and have a similar structure. For graphs dual to 4d polytopes we find again two distinct saddle points leading to a cosine asymptotic formula, however a conformal shape-mismatch is allowed by these configurations, and the asymptotic action is thus a generalisation of the Regge action. The allowed mismatch correspond to angle-matched twisted geometries, 3d polyhedral tessellations with adjacent faces matching areas and 2d angles, but not their diagonals. We study these geometries, identify the relevant subsets corresponding to 3d Regge data and flat polytope data, and discuss the corresponding Regge actions emerging in the asymptotics. Finally, we also provide the first numerical confirmation of the large spin asymptotics of the 15j symbol. We show that the agreement is accurate to the per cent level already at spins of order 10, and the next-to-leading order oscillates with the same frequency and same global phase.