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Quantum modularity of signatures in TQFT and generalized Dedekind sums

Yuya Murakami

TL;DR

This work proves the quantum modularity of the SU(2) TQFT signature for genus 2 by tying it to generalized Dedekind sums attached to modular forms and their Eichler integrals. The authors develop a comprehensive framework: (i) reciprocity for generalized Dedekind sums via completed twisted L-functions and regularized period polynomials, (ii) multiple proofs using integral representations, Eichler integral asymptotics, and Mellin summation, and (iii) specialization to Eisenstein series and level-2 sums to obtain explicit quantum modularity statements. They then express the genus-2 signature $\sigma_2(x)$ in terms of higher-level Dedekind sums and prove a concrete modular relation $\sigma_2(x/(2x+1)) - \sigma_2(x) = 2r^2 + 2rp + p^2 - 1$, illustrating the deep connection between TQFT invariants and modular-form arithmetic. The results provide explicit radial limits and a robust framework for quantum modularity linking topology and number theory, with potential extensions to higher genus and other modular objects.

Abstract

We prove the quantum modularity of the signature of $ \mathrm{SU}(2) $-TQFT for a genus 2 surface, which was conjectured by Marché--Masbaum in 2025. Our approach is based on a quantum modularity of generalized Dedekind sums associated with general modular forms. In the case of Eisenstein series for $ Γ(N) $, these generalized Dedekind sums admit trigonometric sum expressions, which coincide with the formula for the $ \mathrm{SU}(2) $-TQFT signature. Furthermore, we express both the $ \mathrm{SU}(2) $-TQFT and generalized Dedekind sums as radial limits of Eichler integrals.

Quantum modularity of signatures in TQFT and generalized Dedekind sums

TL;DR

This work proves the quantum modularity of the SU(2) TQFT signature for genus 2 by tying it to generalized Dedekind sums attached to modular forms and their Eichler integrals. The authors develop a comprehensive framework: (i) reciprocity for generalized Dedekind sums via completed twisted L-functions and regularized period polynomials, (ii) multiple proofs using integral representations, Eichler integral asymptotics, and Mellin summation, and (iii) specialization to Eisenstein series and level-2 sums to obtain explicit quantum modularity statements. They then express the genus-2 signature in terms of higher-level Dedekind sums and prove a concrete modular relation , illustrating the deep connection between TQFT invariants and modular-form arithmetic. The results provide explicit radial limits and a robust framework for quantum modularity linking topology and number theory, with potential extensions to higher genus and other modular objects.

Abstract

We prove the quantum modularity of the signature of -TQFT for a genus 2 surface, which was conjectured by Marché--Masbaum in 2025. Our approach is based on a quantum modularity of generalized Dedekind sums associated with general modular forms. In the case of Eisenstein series for , these generalized Dedekind sums admit trigonometric sum expressions, which coincide with the formula for the -TQFT signature. Furthermore, we express both the -TQFT and generalized Dedekind sums as radial limits of Eichler integrals.
Paper Structure (17 sections, 27 theorems, 137 equations, 2 figures)

This paper contains 17 sections, 27 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reciprocity for generalized Dedekind sums associated to modular forms
  3. Definition and main result
  4. Proof
  5. Eichler Integrals and Generalized Dedekind Sums
  6. Asymptotic formulas
  7. Asymptotic expansions and Mellin transform
  8. Asymptotic expansions for sums in Fourier coefficients of modular forms
  9. Generalized Dedekind sums for Eisenstein series
  10. Preliminaries
  11. Discrete Fourier transform
  12. Periodic Bernoulli polynomials
  13. Hurwitz zeta function
  14. Basic facts for Eisenstein series of $\Gamma(N)$
  15. Generalized Dedekind sums for Eisenstein series of $\Gamma(N)$ with periodic maps
  16. ...and 2 more sections

Key Result

Theorem 1.2

conj:Marche-Masbaum is true.

Figures (2)

  • Figure 1: Quantum modularity for $S_0^{\mathrm{odd}} (x)$.
  • Figure 2: Quantum modularity for $S_2^{\mathrm{odd}} (x)$.

Theorems & Definitions (65)

  • Conjecture 1.1: Marché--Masbaum Marche-Masbaum
  • Theorem 1.2
  • Theorem 1.3: prop:sign_TQFT_simple_expression_g=2,cor:sign_TQFT_Eichler_int
  • Theorem 1.4: thm:reciprocity_gen'd_Ded_sum
  • Corollary 1.5: cor:reciprocity_gen'd_Ded_sum_Gamma(N)
  • Definition 2.1
  • Theorem 2.2: Reciprocity for generalized Dedekind sums
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 55 more