Aspects of defects in 3d-3d correspondence

In this paper we study supersymmetric co-dimension 2 and 4 defects in the compactification of the 6d (2,0) theory of type A_N−1 on a 3-manifold M. The so-called 3d-3d correspondence is a relation between complexified Chern-Simons theory (with gauge group SL(N,ℂ)) on M and a 3d N=2 theory T_N[M]. We study this correspondence in the presence of supersymmetric defects, which are knots/links inside the 3-manifold. Our study employs a number of different methods: state-integral models for complex Chern-Simons theory, cluster algebra techniques, domain wall theory T[SU(N)], 5d N=2 SYM, and also supergravity analysis through holography. These methods are complementary and we find agreement between them. In some cases the results lead to highly non-trivial predictions on the partition function. Our discussion includes a general expression for the cluster partition function, which can be used to compute in the presence of maximal and certain class of non-maximal punctures when N>2. We also highlight the non-Abelian description of the 3d N=2T_N[M] theory with defect included, when such a description is available. This paper is a companion to our shorter paper http://dx.doi.org/10.1088/1751-8113/49/30/30LT02, which summarizes our main results.