SU(2) graph invariants, Regge actions and polytopes
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 areaangle 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 shapemismatch is allowed by these configurations, and the asymptotic action is thus a generalisation of the Regge action. The allowed mismatch correspond to anglematched 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 nexttoleading order oscillates with the same frequency and same global phase.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.01727
 arXiv:
 arXiv:1708.01727
 Bibcode:
 2017arXiv170801727D
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 J.2
 EPrint:
 v2: Section added on the implications of our results for spin foam models of quantum gravity, few amendments, references updated. 36 pages, many figures and many footnotes v3: Comparison with more recent work added at the end of the conclusions