We introduce and study the partition function Zγ (M) for the cusped hyperbolic 3-manifold M. We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and we define the partition function Zγ (Mu) correspondingly. This function is shown to give the Neumann-Zagier potential function in the classical limit γ → 0, and the A-polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.
All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology