### 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 |
---|---|

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 1 2007 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

**Generalized volume conjecture and the A-polynomials : The Neumann-Zagier potential function as a classical limit of the partition function.** / Hikami, Kazuhiro.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Generalized volume conjecture and the A-polynomials

T2 - The Neumann-Zagier potential function as a classical limit of the partition function

AU - Hikami, Kazuhiro

PY - 2007/8/1

Y1 - 2007/8/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=34248578237&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2007.03.008

DO - 10.1016/j.geomphys.2007.03.008

M3 - Article

AN - SCOPUS:34248578237

VL - 57

SP - 1895

EP - 1940

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 9

ER -