AN ENRICHED FUSION CATEGORICAL DESCRIPTION OF A 1+1D ℤ𝑁 ANOMALOUS TOPOLOGICAL PHASE TRANSITION

In this thesis, we explicitly compute an anyon condensation in the 2+1D double parafermion topological order, the modular fusion category associated to which is denoted by Z_1(PF_𝑁). Here, PF_𝑁 denotes the ℤ_𝑁 parafermion modular fusion category and Z_1(−) is its Drinfeld center. When 𝑁 is odd, we prove rigorously that the condensed 2+1D phase is precisely the finite gauge theory with the gauge group ℤ_𝑁. We compute explicitly the fusion category Z_1(PF𝑁)𝐵 associated to the 1+1D gapped domain wall between the condensed and uncondensed phases where 𝐵 is a condensable algebra in Z_1(PF_𝑁). We also compute a Z_1(PF_𝑁)-enriched fusion category obtained from the Z_1(PF_𝑁)-action on Z_1(PF_𝑁)_𝐵 via the so-called canonical construction. This thesis is the mathematical part of the work arXiv:2208.01572. Its physical part is to show how the ingredients of above enriched fusion category emerge naturally from the lattice model realization of a purely boundary topological phase transition between two gapped boundaries of the 2+1D finite gauge theory. Together, they provide non-trivial new examples for the unified mathematical theory of gapped and gapless boundaries of 2+1D topological orders only recently established by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760).

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