We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the 't Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

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We study correlation functions involving extended defect operators in the fourdimensional N = 4 super-Yang-Mills (SYM). The main tool is supersymmetric localization with respect to the supercharge Q introduced in [1] which computes observables in the Qcohomology. We classify general defects of different codimensions in the N = 4 SYM that belong to the Q-cohomology, which form 1 16 -BPS defect networks. By performing the Qlocalization of the N = 4 SYM on the four-dimensional hemisphere, we discover a novel defect-Yang-Mills (dYM) theory on a submanifold given by the two-dimensional hemisphere and described by (constrained) two-dimensional Yang-Mills coupled to topological quantum mechanics on the boundary circle. This also generalizes to interface defects in N = 4 SYM by the folding trick. We provide explicit dictionary between defect observables in the SYM and those in the dYM, which enables extraction of general 1 16 -BPS defect network observables of the SYM from two-dimensional gauge theory and matrix model techniques. Applied to the D5 brane interface in the SU(N ) SYM, we explicitly determine a set of defect correlation functions in the large N limit and obtain precise matching with strong coupling results from IIB supergravity on AdS 5 × S 5 .