We propose an equivalence of the two quantities defined from a punctured Riemann surface together with an element of the mapping class group. One is the partition function of the 3d N=2 theory on duality domain wall inside a 4d N=2 theory. Another is a hyperbolic volume of the mapping torus. We have proven that the classical limit of the former reproduces the latter in the case of the once-punctured torus. We will also explain this equality by a chain of connections involving Liouville/Toda theory and quantum Teichmuller theory. This suggests a ``categorification'' of the Alday-Gaiotto-Tachikawa relation.