Spin-foams as semi-classical vertices: gluing constraints and a hybrid algorithm

Numerical methods in spin-foam models have significantly advanced in the last few years, yet challenges remain in efficiently extracting results for amplitudes with many quantum degrees of freedom. In this paper we sketch a proposal for a ``hybrid algorithm'' that would use both the full quantum amplitude and its asymptotic approximation in the relevant regimes. As a first step towards the algorithm, we derive a new representation of the partition function where each spin-foam vertex possesses its own coherent data, such that it can be individually asymptotically approximated. We do this through the implementation of gluing constraints between vertices, which we study numerically. We further derive an asymptotic expression for the constraints for arbitrary boundary data, including data for which there are no critical points. From this new representation we conjecture an intermediate quasi-geometric spin-foam regime describing a superposition of semi-classical vertices glued in a non-matching way via the gluing constraints.