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**Title. ** A unified mathematical theory of gapped and gapless boundaries of 2d topological orders

**Abstract.** A 2d topological order can be described by a unitary modular tensor category (UMTC) of anyons and a chiral central charge. The mathematical theory for gapped boundaries of 2d topological orders has been established a few years ago. A gapped boundary is described by a unitary fusion category whose Drinfeld center gives precisely the UMTC associated to the bulk. The mathematical theory for gapless boundaries is not known, but it was long believed that a chiral gapless boundary should be described by a ``chiral conformal field theory'', the precise meaning of which was not known. In this talk, I will establish this missing theory and show that observables on the 1+1D world sheet of such a boundary form an enriched monoidal category, whose Drinfeld center is again precisely the UMTC associated to the bulk. This is joint work with Hao Zheng.