Citation: Kong, Liang, and Xiao-Gang Wen. "Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions." arXiv preprint arXiv:1405.5858 (2014).
Web: https://arxiv.org/abs/1405.5858
Tags: Higher-dimensional, Modular-tensor-categories
In this lovely paper, the authors paint a vision for what the algebraic framework for topological order should be in higher dimensions. This includes introducing the term "BF_n categories", which are conjectural algebraic structures which exist in all dimensions, admit certain constructions, and satisfy a gambit of properties. As one of my friends put it: "[This Kong-Wen paper] paints a beautiful picture, and it is very inspirational, and it is so vague in places that you can decide that it is as correct or misguided as you like".
The definitions in 2D (modular tensor categories) are well-established by now. The definition in 3D (fusion 2-categories) is now established, and the central theorem about fusion 2-categories has been proved:
> Décoppet, Thibault D., et al. "The Classification of Fusion 2-Categories." arXiv preprint arXiv:2411.05907 (2024).
The 4D definition (fusion 3-categories) has only recently been given in the literate:
> Bhardwaj, Lakshya, et al. "Fusion 3-Categories for Duality Defects." arXiv preprint arXiv:2408.13302 (2024).
Making their picture work even for 3D seems like hard work - it might be a long time before category theorists can make the picture painted by this article rigorous.