## – Europe/Lisbon — Online

Bruce Bartlett, Stellenbosch University

Asymptotics of the classical and quantum $6j$ symbols

Topological Quantum Field Theory Seminar

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.

Dmitry Melnikov, International Institute of Physics*To be announced*

Ingmar Saberi, University of Heidelberg*To be announced*

Theo Johnson-Freyd, Dalhousie University and Perimeter Institute

Higher S-matrices

Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter.

Fabian Haiden, Mathematical Institute, University of Oxford

Categorical Kähler Geometry

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.