Asymptotics of the classical and quantum $6j$ symbols

Abstract: 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.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

( video )

Topological Quantum Field Theory Club (IST, Lisbon)

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