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 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.