Superconformal index and 3d-3d correspondence for mapping cylinder/torus

We probe the 3d-3d correspondence for mapping cylinder/torus using the superconformal index. We focus on the case when the fiber is a once-punctured torus (Sigma(1,1)). The corresponding 3d field theories can be realized using duality domain wall theories in 4d N = 2* theory. We show that the superconformal indices of the 3d theories are the SL(2,C) Chern-Simons partition function on the mapping cylinder/torus. For the mapping torus, we also consider another realization of the corresponding 3d theory associated with ideal triangulation. The equality between the indices from the two descriptions for the mapping torus theory is reduced to a simple basis change of the Hilbert space for the SL(2, C) Chern-Simons theory on R x Sigma(1,1).