Quantum modularity of signatures in TQFT and generalized Dedekind sums

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 $\Gamma (N)$, 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.

arXiv comment26 pages, 6 figures