We provide quantitative evidence for our previous conjecture which states an equivalence of the partition function of a 3d $\mathcal{N}=2$ gauge theory on a duality wall and that of the $SL(2,\mathbb{R})$ Chern-Simons theory on a mapping torus, for a class of examples associated with once-punctured torus. In particular, we demonstrate that a limit of the 3d $\mathcal{N}=2$ partition function reproduces the hyperbolic volume and the Chern-Simons invariant of the mapping torus. This is shown by analyzing the classical limit of the trace of an element of the mapping class group in the Hilbert space of the quantum Teichmüller theory. We also show that the subleading correction to the partition function reproduces the Reidemeister torsion.

• Received 27 April 2013

DOI:https://doi.org/10.1103/PhysRevD.88.026011