Taming Supersymmetric Defects in 3d-3d Correspondence

Abstract: We study knots in 3d Chern-Simons theory with complex gauge group $SL(N,\mathbb{C})$, in the context of its relation with 3d $\mathcal{N}=2$ theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d $(2,0)$ theory, which is compactified on a 3-manifold $\hat{M}$. We identify such defects in various corners of the 3d-3d correspondence, namely in 3d $SL(N,\mathbb{C})$ Chern-Simons theory, in 3d $\mathcal{N}=2$ theory, in 5d $\mathcal{N}=2$ super Yang-Mills theory, and in the M-theory holographic dual. We can make quantitative checks of the 3d-3d correspondence by computing partition functions at each of these theories. This Letter is a companion to a longer paper, which contains more details and more results.