Generalized volume conjecture and the [formula omitted]-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function

Abstract

We introduce and study the partition function Z γ ( 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 Z γ ( M u ) 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.