$ {\text{SL}}\left( {2,\mathbb{R}} \right) $ Chern-Simons, Liouville, and gauge theory on duality walls

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.