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Global symmetries in two dimensions are implemented by topological defect lines (TDLs). For continuous global symmetries, the TDLs are nothing but the Noether charges. However there can be TDLs that are not associated to any global symmetry, and can be thought of as a generalized notion of symmetry. 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-symmetry TDLs are preserved along a RG flow, then the vacuum cannot be a nondegenerate 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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