Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function
By Kazuhiro Hikami. 2006
98 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/aaea8f98106b3a4c5eb0d64583a14cad65f9e9b3
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
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In Mathematical Physics
Citations in the corpus
- Exact results for perturbative Chern-Simons theory with complex gauge group. (2009, methodology): “16) is invariant under 2–3 Pachner moves on the underlying triangulations and produces a true hyperbolic invariant [25, 26, 29].”