home | literature reviews

"On Vafa's theorem for tensor categories", Pavel Etingof, 2002

Reviewed October 7, 2023

Citation: Etingof, Pavel. "On Vafa's theorem for tensor categories." Mathematical Research Letters 9.5-6 (2002): 651-657.

Web: https://arxiv.org/abs/math/0207007

Tags: Mathematical, Monoidal-categories, Modular-tensor-categories, Conformal-field-theory

This paper proves a generalization of Vafa's theorem for tensor categories. One very interesting aspect of tensor category theory is the number of finiteness results present: relevant numbers keep being algebraic, relevant phases keep being roots of unity, and relevant vector spaces keep being finite dimensional. Vafa's theorem is the quintessential example of this ("phases in an MTC are roots of unity"), as well as the other results discussed in this paper. It is important to note that Vafa's original statement is both less relevant and less intelligible to mathematicians than more modern formulations:

> Vafa, Cumrun. "Toward classification of conformal theories." Physics Letters B 206.3 (1988): 421-426.

Etingof's paper is now the go-to reference for Vafa's theorem among mathematicians, outside of textbook accounts such as the one found in

>Bakalov, Bojko, and Alexander A. Kirillov. Lectures on tensor categories and modular functors. Vol. 21. American Mathematical Soc., 2001.