On a class of fusion 2-category symmetry: condensation completion of braided fusion category

Abstract

<p>Recently, many studies have 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 the fusion 2-category symmetry ΣB where B is a braided fusion category. In particular, we elaborate on the monoidal structure of ΣB, which not only determines the fusion rules but also controls the dynamics of topological operators/defects. We then take ΣsVec as an example to demonstrate how weUbKnZ9XrM8EV0KFIXCLgRteBJwhcalculate fusion rule, quantum dimension and 10j-symbol of the fusion 2-category. With our algorithm, all these data can be effciently encoded and computed in the computer program. The complete program has been uploaded to github¹ . 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 is in analogy to the Clebsch-Gordon coeffcients, and the 10j-symbol are in analogy to the 6j-symbol.</p>