[Paper Review] A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifolds

Tian Yang|arXiv (Cornell University)|Sep 9, 2020

Geometric and Algebraic Topology30 references4 citations

TL;DR

This paper introduces a relative version of the Turaev-Viro invariants for ideally triangulated 3-manifolds with boundary, incorporating edge colorings to encode cone angles in a hyperbolic polyhedral metric. It proves the Volume Conjecture for these invariants in the regime of sufficiently small cone angles, showing that the logarithmic asymptotics of the invariants recover the hyperbolic volume of the manifold under the corresponding metric.

ABSTRACT

We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also propose the Volume Conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the Volume Conjecture for the Turaev-Viro invariants proposed by Chen-Yang [8] for hyperbolic 3-manifolds with totally geodesic boundary.

Motivation & Objective

• To define a relative version of the Turaev-Viro invariants for ideally triangulated 3-manifolds with non-empty boundary and edge colorings.
• To extend the Volume Conjecture for Turaev-Viro invariants to include hyperbolic polyhedral metrics with singular locus along edges and cone angles determined by colorings.
• To establish the asymptotic volume conjecture in the regime of sufficiently small cone angles, where the hyperbolic polyhedral metric is uniquely realizable.
• To connect quantum invariants to geometric structures via the hyperbolic volume of polyhedral metrics with prescribed cone angles.

Proposed method

• Define the relative Turaev-Viro invariant using quantum 6j-symbols and edge-dependent H-functions, with colorings assigned to edges satisfying r-admissibility conditions.
• Construct the invariant as a sum over r-admissible colorings, incorporating quantum parameters q = e^{2πi/r} and q = e^{πi/r} for different normalization regimes.
• Establish a duality between the relative Turaev-Viro invariants and relative Reshetikhin-Turaev invariants of the double of the manifold.
• Use stationary phase and complex analysis techniques to estimate the asymptotic behavior of the quantum invariants in the large r limit.
• Apply the method of steepest descent to the associated quantum partition function, identifying the dominant contribution from the critical point corresponding to the hyperbolic polyhedral metric.
• Prove that for small cone angles, the logarithmic growth rate of the invariant converges to the hyperbolic volume of the manifold under the polyhedral metric.

Experimental results

Research questions

• RQ1 Can a relative version of the Turaev-Viro invariant be defined for 3-manifolds with boundary and edge colorings, generalizing the standard invariant?
• RQ2 Does the asymptotic behavior of the relative Turaev-Viro invariant recover the hyperbolic volume of the manifold when the cone angles are determined by the coloring?
• RQ3 Is the Volume Conjecture for Turaev-Viro invariants valid in the case of hyperbolic polyhedral metrics with small cone angles?
• RQ4 How does the relative Turaev-Viro invariant relate to the geometry of the double of the 3-manifold and its hyperbolic structure?
• RQ5 Can the stationary phase method be applied to prove the asymptotic volume conjecture in the small cone angle regime?

Key findings

• The relative Turaev-Viro invariant is defined for ideally triangulated 3-manifolds with non-empty boundary and edge colorings, generalizing the standard Turaev-Viro invariant.
• The invariant satisfies a duality with the relative Reshetikhin-Turaev invariant of the double of the manifold, with explicit normalization factors depending on r and the topology of M.
• For sufficiently small cone angles, the Volume Conjecture holds: lim_{r→∞} (2π/r) log TV_r(M,E,b^{(r)}) = Vol(M_{E_θ}), where θ is determined by the coloring.
• The proof relies on complex analysis and the method of steepest descent, showing that the dominant contribution to the invariant comes from the critical point corresponding to the hyperbolic polyhedral metric.
• The error term in the asymptotic expansion is bounded by O(e^{r/(2π)(Vol - ε')}), confirming the convergence to the volume in the large r limit.
• The non-vanishing of the leading coefficient in the asymptotic expansion is established via a non-degeneracy condition on the Hessian of the action functional.