A TQFT of Turaev-Viro type on shaped triangulations

TL;DR

This paper introduces a quantum invariant for shaped triangulations of 3-manifolds, extending Turaev-Viro theory with hyperbolic functions, invariance properties, and conjectured relations to Teichmüller TQFT and supersymmetric theories.

Contribution

It develops a new TQFT based on shaped triangulations with hyperbolic gamma functions, invariance under Pachner moves, and conjectured connections to existing quantum invariants.

Findings

Defined a convergent state integral invariant for shaped triangulations.

Proved invariance under shaped 3-2 Pachner moves and shape gauge transformations.

Conjectured a relationship between the new invariant and Teichmüller TQFT.

Abstract

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.