Quantum modularity of signatures in TQFT and generalized Dedekind sums

TL;DR

This paper proves the quantum modularity of the SU(2)-TQFT signature for genus 2 surfaces, linking it to generalized Dedekind sums and Eichler integrals, thus confirming a conjecture from 2025.

Contribution

It establishes the quantum modularity of the SU(2)-TQFT signature and connects it to generalized Dedekind sums and Eichler integrals, providing a new perspective on TQFT invariants.

Findings

Proved quantum modularity of SU(2)-TQFT signature for genus 2 surfaces.

Connected generalized Dedekind sums with TQFT signatures via trigonometric sums.

Expressed TQFT and Dedekind sums as radial limits of Eichler integrals.

Abstract

We prove the quantum modularity of the signature of $ \mathrm{SU}(2) $-TQFT for a genus 2 surface, which was conjectured by March\'{e}--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 $ \Gamma(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.