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Citation: Bruillard, Paul, et al. "Rank-finiteness for modular categories." Journal of the American Mathematical Society 29.3 (2016): 857-881.
This paper proves that there are finitely many modular tensor categories of every rank. This paper is a beautiful example of how finiteness results in MTC theory lead to powerful applications of number theory. In particular, the construction of MTCs corresponds to inductively solving a large number of algebraic equations. Each step in this process can be bounded, and so we get an overall finite bound on the count of MTCs. As is highlighted in this paper, the bounds are effective but spectacularly far from being tight.
The first major finiteness result used is Ocneanu rigidity, which says that given fixed fusion rules there are finitely many fusion categories which satisfy those rules:
> Gainutdinov, Azat M., Jonas Haferkamp, and Christoph Schweigert. "Davydov-Yetter cohomology, comonads and Ocneanu rigidity." Advances in Mathematics 414 (2023): 108853.
After this, one uses explicit algebraicity results a-la
> Etingof, Pavel. "On Vafa's theorem for tensor categories." Mathematical Research Letters 9.5-6 (2002): 651-657.
to deduce that the formula D^2=\sum_i d_i only has finitely many solutions, giving the desired final result. As such, one of the key improvements of this paper is a new algebraicity result for MTCs.