Signatures in TQFT : Asymptotics and Modularity

TL;DR

This paper investigates the asymptotic behavior and modular properties of the signature of SU(2) TQFT vector spaces associated with surfaces, revealing convergence to a special function and proposing a conjectural transformation law.

Contribution

It establishes the convergence of scaled signatures to a specific function and links this to modular forms, proposing a new reciprocity-like transformation law.

Findings

Scaled signatures converge to a function involving sine sums.

The limit function is related to an Eichler integral of a modular form.

A conjecture for a reciprocity law for genus 2 signatures is proposed.

Abstract

We study the signature $\sigma_g(\frac q p)$ of $\mathrm{SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $\zeta=e^{i\pi q/p}$. We prove that $\frac{1}{p^2}\sigma_2(\frac{q}{p})$ converges to $\Lambda(\theta)=\frac{16}{\pi^3}\sum\limits_{n\ge 1, \textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}$ when $\frac{q}{p}$ goes to an irrational number $\theta\in [0,1]$ under certain conditions. We also observe that the function $\Lambda(\theta)$ 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.