  A shaped triangulation is a finite triangulation of an oriented pseudo three
manifold where each tetrahedron carries dihedral angles of an ideal hyberbolic
tetrahedron. To each shaped triangulation, we associate 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. We conjecture that for shaped triangulations of closed 3-manifolds,
our partition function is twice the absolute value squared of the partition
function of Techm\"uller TQFT defined by Andersen and Kashaev. This is similar
to the known relationship between the Turaev-Viro and the
Witten-Reshetikhin-Turaev invarints of three manifolds. We also discuss
interpretations of our construction in terms of three-dimensional
supersymmetric field theories related to triangulated three-dimensional
manifolds.
