Generating series and asymptotics of classical spin networks

TL;DR

This paper analyzes classical SU(2) spin networks by deriving their generating series with holonomies and studying their asymptotic behavior through stationary phase approximation, extending previous formulas and providing detailed asymptotics.

Contribution

It introduces a generalized formula for the generating series of spin networks with holonomies and establishes their asymptotic behavior under label rescaling using stationary phase methods.

Findings

Derived a generalized generating series formula for spin networks with holonomies.

Provided asymptotic formulas for spin networks as labels grow large.

Extended Westbury's formula to a broader class of networks.

Abstract

We study classical spin networks with group SU(2). In the first part, using gaussian integrals, we compute their generating series in the case where the networks are equipped with holonomies; this generalizes Westbury's formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.