Lecture - Rinat Kashaev (University of Geneva)

I will describe a relation between quantum dilogarithms and 3-dimensional hyperbolic geometry obtained by quantising the dihedral angles of an ideal hyperbolic tetrahedron with respect to the Neumann-Zagier symplectic structure. In this way, one constructs a
(metaplectic) quantum operator $Q$ realising the 3-3 Pachner move for 4-dimensional triangulations. This realisation admits a natural generalisation to any self-dual locally compact abelian group, together with a fixed gaussian exponential. The 5-term operator identity, satisfied by a quantum dilogarithm over such a group, is equivalent to an integral identity involving the operator kernel of $Q$.