Generalized volume conjecture and the A-polynomials: The Neumann-Zagier potential function as a classical limit of the partition function

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46 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1895-1940
Number of pages46
JournalJournal of Geometry and Physics
Volume57
Issue number9
DOIs
Publication statusPublished - Aug 1 2007
Externally publishedYes

Fingerprint

Neumann function
Classical Limit
A-polynomial
Potential Function
Partition Function
partitions
polynomials
Dilogarithm
Hyperbolic Knot
Hyperbolic Structure
Hyperbolic 3-manifold
Triangular pyramid
Quantum Groups
Triangulation
Bundle
Torus
Circle
Complement
triangulation
tetrahedrons

All Science Journal Classification (ASJC) codes

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

Cite this

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abstract = "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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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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