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An axiomatic formulation of rational two-dimensional conformal field theory essentially leads to what we now call a modular tensor category:
G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989) 177–254.
Cain Edie-Michell, Auto-equivalences of the modular tensor categories of type A, B, C and G, 2020. arXiv:2002.03220