## – Europe/Lisbon — Online

Bruce Bartlett, Stellenbosch University

Asymptotics of the classical and quantum $6j$ symbols

The classical (resp. quantum) $6j$ symbols are real numbers which encode the associator information for the tensor category of representations of $SU(2)$ (resp. the quantum group of $SU(2)$ at level $k$). They form the building blocks for the Turaev-Viro $3$-dimensional TQFT. I will review the intriguing asymptotic formula for these symbols in terms of the geometry of a Euclidean tetrahedron (in the classical case) or a spherical tetrahedron (in the quantum case), due to Ponzano-Regge and Taylor-Woodward respectively. There is a wonderful integral formula for the square of the classical $6j$ symbols as a group integral over $SU(2)$, and I will report on investigations into a similar conjectural integral formula for the quantum $6j$ symbols. In the course of these investigations, we observed and proved a certain reciprocity formula for the Wigner derivative for spherical tetrahedra. Joint with Hosana Ranaivomanana.