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Signatures in TQFT : Asymptotics and Modularity

Julien Marché, Gregor Masbaum

Abstract

We study the signature $σ_g(\frac q p)$ of $\mathrm{SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $ζ=e^{iπq/p}$. We prove that $\frac{1}{p^2}σ_2(\frac{q}{p})$ converges to $Λ(θ)=\frac{16}{π^3}\sum\limits_{n\ge 1, \textrm{ odd}}\frac{1}{n^3\sin(nπθ)}$ when $\frac{q}{p}$ goes to an irrational number $θ\in [0,1]$ 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.

Signatures in TQFT : Asymptotics and Modularity

Abstract

We study the signature of -TQFT vector spaces associated to surfaces of genus , as a function of the defining root of unity . We prove that converges to when goes to an irrational number under certain conditions. We also observe that the function is the boundary value of an Eichler integral of a level modular form of weight , 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.

Paper Structure

This paper contains 12 sections, 16 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Signatures in TQFT and signed Verlinde algebras
  3. A trigonometric formula for the signature in genus 2
  4. Asymptotic behavior of the signature in genus $2$
  5. The higher genus case
  6. Modularity properties and transformation laws for the signature
  7. A genus 0 example related to Dedekind sums
  8. Relation to Dedekind sums
  9. A modular interpretation for $S(\frac{q}{p})$
  10. Transformation law for $S(\frac{q}{p})$
  11. A modular form related to the genus 2 signature
  12. Appendix

Key Result

Theorem 1

For almost all irrational $\theta\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\theta$, then one has

Figures (3)

  • Figure 1: Graph of the signature function $\sigma_2$
  • Figure 2: The graph of the map $\frac{q}{p}\mapsto \sigma_3(\frac{q}{p})/{p^4}$ for $p\le 31$.
  • Figure 3: Normalized signature of the punctured torus with colors $2k=2,4,6$.

Theorems & Definitions (41)

  • Theorem
  • Conjecture
  • Theorem 2.1: DM
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • Corollary 3.2: YMu
  • ...and 31 more