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## "Fusion rules and modular transformations in 2D conformal field theory", Erik Verlinde, 1988

*Reviewed December 29, 2023*

*Citation:* Verlinde, Erik. "Fusion rules and modular transformations in 2D conformal field theory." Nuclear Physics B 300 (1988): 360-376.

*Web:* https://www.sciencedirect.com/science/article/pii/0550321388906037

*Tags:* Physical, Foundational, Conformal-field-theory

This paper proves the so-called "Verlinde formula" for conformal field theory. This formula
is an important foundational result, which has since been generalized and extended to a large number of related areas.
In particular, it applies to modular tensor categories.

The Verline formula for modular tensor categories goes back to the definition of
modular tensor categories. Immediately after first defining them in

> Moore, Gregory, and Nathan Seiberg. "Lectures on RCFT." Physics, geometry and topology. Boston, MA: Springer US, 1990. 263-361.

Moore and Seiberg assert that, quote, "the above axioms are sufficient for establishing
the relation Sa=bS". This formula is proved
earlier in the book (as an exercise to the reader) in the language of fusion algebras, and they
make no care to repeat the proof for MTCs. Every in-depth treatment of MTCs since has included a proof of the formula.