Signatures in TQFT : Asymptotics and Modularity

2025/12/15 by Julien Marché, Marché, Julien, Gregor Masbaum +1
Mathematics · #11F11 #11K50 #57K16 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2512.13450

openalex publication_date 2025/12/15 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

We study the signature σ_g(\frac q p) of SU₂-TQFT vector spaces associated to surfaces of genus g, as a function of the defining root of unity ζ=e^iπq/p. We prove that (1)/(p²)σ₂((q)/(p)) converges to Λ(θ)=(16)/(π³)∑_n≥ 1, \textrm odd(1)/(n³sin(nπθ)) when (q)/(p) goes to an irrational number θ∈ [0,1] under certain conditions. We also observe that the function Λ(θ) is the boundary value of an Eichler integral of a level 2 modular form of weight 4, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums.

Citations

• Reidemeister torsion of two-bridge knots and signatures of TQFT
• Signatures of TQFTs and trace fields of two-bridge knots
• Toledo invariants of Topological Quantum Field Theories
• D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods

Related

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• Reidemeister torsion of two-bridge knots and signatures of TQFT
• Asymptotic Fermat for signatures (r,r,p) using the modular approach
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• Signature for flat unitary bundles over surfaces with boundary