Rinat Kashaev

A TQFT of Turaev-Viro type on shaped triangulations


A shaped triangulation is a finite triangulation of an oriented pseudo three manifold where each tetrahedron carries dihedral angles of an ideal hyberbolic tetrahedron. I will explain a construction which associates to each shaped triangulation a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3-2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann-Zagier Poisson bracket. Similarly to Turaev-Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. At least for shaped triangulations of closed 3-manifolds, this partition function seems to be twice the absolute value squared of the partition function of Techmüller TQFT. This is similar to the known relationship between the Turaev-Viro and the Witten-Reshetikhin-Turaev invariants of three manifolds. The talk will be based on the joint work with Feng Luo and Grigory Vartanov arXiv:1210.8393.