## 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.

Original language | English |
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Pages (from-to) | 1895-1940 |

Number of pages | 46 |

Journal | Journal of Geometry and Physics |

Volume | 57 |

Issue number | 9 |

DOIs | |

Publication status | Published - Aug 2007 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology