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## "Classical and quantum conformal field theory", Gregory Moore, Nathan Seiberg, 1989

*Reviewed December 27, 2023*

*Citation:* Moore, Gregory, and Nathan Seiberg. "Classical and quantum conformal field theory." Communications in Mathematical Physics 123 (1989): 177-254.

*Web:* https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-123/issue-2/Classical-and-quantum-conformal-field-theory/cmp/1104178762.full

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

This is the paper that "discovered" modular tensor categories, along with the sister-paper

> Moore, Gregory, and Nathan Seiberg. "Polynomial equations for rational conformal field theories." Physics Letters B 212.4 (1988): 451-460.

The goal of these papers was to deeply think through the basics of rational conformal field theory,
figure out what the fundamental algebraic identities present are, and then re-build the theory
using these fundamental algebraic identities. The algebraic structures underlying
rational conformal field theory are modular tensor categories. In a very real sense,
these are the first two papers on modular tensor categories, and they each give
their own set of axioms. These axioms have since been refined, but the core idea is very present.

Another interesting feature of this work is that they discuss quantum groups.
In Moore and Seiberg's words: "Section 9 is more speculative. It suggests a possible connection
between conformal field theory and quantum groups". It is fantastic to see how correct they were.