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Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $W$, whose entries are the fusion-space dimensions $W_{ia}$, to label different types of gapped domain walls. Mathematically, we propose a classification of bimodule categories between modular tensor categories. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

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In this paper we obtain a stability theorem of generalized K¨ahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of K¨ahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact K¨ahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized K¨ahler structures on Fano surfaces and toric K¨ahler manifolds. In particular, we show that every nonzero holomorphic Poisson structure on a K¨ahler manifold induces deformations of non-trivial generalized K¨ahler structures.

The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in$S$^{$n$+1}under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in$S$^{4}.