Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant

Abstract

We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and correspondingly we define quantum invariant Z(M_u). This quantum invariant is shown to give the Neumann--Zagier potential function in the classical limit, 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 punctured torus bundle over the circle.

References in corpus (22)

• Quantum Dilogarithm
• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Dilogarithm identities
• Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
• A Link Invariant from Quantum Dilogarithm
• The colored Jones function is q-holonomic
• Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
• Classical 6j-symbols and the tetrahedron
• Quantum Dilogarithm as a 6j-Symbol
• R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
• On the spectrum of Dehn twists in quantum Teichmuller theory
• Modular Double of Quantum Group
• Difference equation of the colored Jones polynomial for torus knot
• Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds
• Representations of the quantum Teichmuller space, and invariants of surface diffeomorphisms
• Hyperbolic Structure Arising from a Knot Invariant
• A dilogarithmic formula for the Cheeger-Chern-Simons class
• Hyperbolicity of Partition Function and Quantum Gravity
• A version of the volume conjecture
• Asymptotics of the Colored Jones Polynomial and the A-Polynomial
• Some limits of the colored Jones polynomials of the figure-eight knot
• The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces

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• Rotating Black Hole Entropy from M5-branes
• Semiclassical Analysis of the 3d/3d Relation
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• Peacock patterns and new integer invariants in topological string theory
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• Chern-Simons Theory and S-duality
• The Volume Conjecture and Topological Strings
• A 3d-3d appetizer
• The resurgent structure of quantum knot invariants
• S-duality resurgence in SL(2) Chern-Simons theory
• Expanding 3d $\mathcal{N}=2$ Theories around the Round Sphere
• Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
• Challenges of beta-deformation
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• 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories
• Gluing I: Integrals and Symmetries
• A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs
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• Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group
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• Complex Chern-Simons theory at level k via the 3d-3d correspondence
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• Lens partition function, pentagon identity and star-triangle relation
• A Spectral Perspective on Neumann-Zagier
• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
• Optimistic limits of colored Jones polynomials and complex volumes of hyperbolic links
• Pentagon identities arising in supersymmetric gauge theory computations
• SU(2)/SL(2) knot invariants and KS monodromies
• Reconstructing GKZ via topological recursion
• Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Resurgence in complex Chern-Simons theory at generic levels
• Asymptotic aspects of the Teichmüller TQFT
• Braiding Operator via Quantum Cluster Algebra
• Quantum Trilogy: Discrete Toda, Y-System and Chaos
• Cluster Partition Function and Invariants of 3-manifolds
• State integral models and the tetrahedron equation
• Dubrovnik Skein Theory and Power Sum Elements