Abstract
Symmetry serves as a guiding principle in modern physics. Recently, many stud ies are focused on generalized global symmetry, a mixture of both invertible and non invertible symmetries in various space-time dimensions. The complete structure of gen eralized global symmetry is described by higher-fusion category theory which is widely applied in the study of many physical systems. Fusion 1-category has been applied in the study of 1+1D conformal field theory(CFT) and 2+1D topological order. The data of fusion 2-category can be used to study 3+1D topological order. In this thesis, we first review the construction of a fusion 2-category symmetry ΣB where B is a braided fusion category. In particular, we elaborate on the monoidal structure of ΣB which determines the fusion rules and controls the dynamics of topological operators/defects. We take ΣsVec as an example to demonstrate how we calculate all categorical data including the 10j-symbol of the fusion 2-category. With our algorithm, all these data can be efficiently encoded and computed with a computer program. Our paper can be thought as explicitly computing the representation theory of B, in analogy to, for example, the representation theory of SU(2). The choice of basis bimodule maps is in analogy to the Clebsch-Gordon coefficients, and the 10j-symbol are in analogy to the 6j-symbol. We then construct a 3D membrane-net model and the corresponding Topological Quantum Field Theory (TQFT) to realize a 3+1D topological order on lattice whose canonical boundary is described by the fusion 2-category ΣsVec. Finally, we compute the ground state degeneracy of this model, which is the simplest topological invariant.
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