Papers › On a class of fusion 2-category symmetry: condensation completion of braided fusion category
On a class of fusion 2-category symmetry: condensation completion of braided fusion category
Wenjie Xi, Tian Lan, Longye Wang, Chenjie Wang, Wei-Qiang Chen
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Recently, many studies are focused on generalized global symmetry, a mixture of both invertible and non-invertible symmetries in various space-time dimensions. The complete structure of generalized global symmetry is described by higher fusion category theory. In this paper, We first review the construction of fusion 2-category symmetry ΣB where B is a a braided fusion category. In particular, we elaborate on the monoidal structure of ΣB which determines fusion rules and controls the dynamics of topological operators/defects. We then take ΣsVec as an example to demonstrate how we calculate fusion rule, quantum dimension and 10j-symbol of the fusion 2-category. With our algorithm, all these data can be efficiently encoded and computed in computer program. The complete program will be uploaded to github soon. Our work 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 are in analogy to the Clebsch-Gordon coefficients and the 10j-symbol are in analogy to the 6j-symbol.
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