  We propose an equivalence of the partition functions of two different 3d
gauge theories. On one side of the correspondence we consider the partition
function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a
punctured Riemann surface times an interval. On the other side we have a
partition function of a 3d N=2 superconformal field theory on S^3, which is
realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the
proof of this conjecture using connections with quantum Liouville theory and
quantum Teichmuller theory, and study in detail the example of the
once-punctured torus. Motivated by these results we advocate a direct
Chern-Simons interpretation of the ingredients of (a generalization of) the
Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as
well as on possible generalizations of our proposals.
