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"Lectures on RCFT", Gregory Moore, Nathan Seiberg, 1990

Reviewed December 28, 2023

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

Web: https://www.physics.rutgers.edu/~gmoore/LecturesRCFT.pdf

Tags: Foundational, Physical, Modular-tensor-categories, Conformal-field-theory

This is the paper which introduced modular tensor categories.

The goal of this paper was to conjecture a classification of rational conformal field theories. So far, it was known that quantum groups gave rise to rational conformal field theories. In fact, all known rational conformal field theories came from quantum groups. The conjecture was that this was really it - every rational conformal field theories comes from a quantum groups. The authors describe the situation beautifully: "A cynical version of this conjecture would state that nothing new has been found since 1986, so we must be done. The purpose of these lectures is to make a case that the conjecture is not cynical but based on the insight that RCFT is closely related to group theory, and in fact must be defined by axioms closely related to groups".

I found the above quote to be confusing, so allow me to clarify. What Moore and Seiberg are saying is that every RCFT is described by a modular functor, and modular functors are in bijection with modular tensor categories. The axioms of a modular tensor category are very analogous to the axioms of a Tannakian category. Just like Tannakian categories are in bijection with groups, Moore and Seiberg see that modular tensor categories must be in bijection with something sort of like groups. These sort-of-like-group things should be quantum groups.

Nowadays, the Moore-Seiberg conjecture is known to be wrong. Not all modular tensor categories come from quantum groups, group algebras, or any of the related Hopf-algebra theoretic constructions. That is: exotic MTCs exist. For instance, some of these exotic MTCs come from vertex operator algebras associated to subfactors. Two exotic subfactor MTCs known to exist are the Haagerup MTC and the E6 MTC:

> Hong, Seung-Moon, Eric Rowell, and Zhenghan Wang. "On exotic modular tensor categories." Communications in Contemporary Mathematics 10.supp01 (2008): 1049-1074.

Another important piece of historical information in this paper: "The name modular tensor category was suggested by Igor Frenkel and we will adopt it. We thank him for discussions on this subject and for urging us to express the definition of S in terms of simple objects." It is good to know that Igor Frenkel coined the term, and had key inputs on its original definition.

Another fantastic feature of this paper is the following quote: "As we have mentioned, the above axioms are sufficient for establishing the relation Sa=bS. Thus we may summarize the main result of [our earlier works on rational conformal field theory] in the statement that a modular tensor category is equivalent to a modular functor".

The fact that MTCs correspond to TQFTs/modular functors takes a lot of effort to prove. It is an amusing quirk of physics research that this deep theorem, which was only proved 25 years later in the mathematical literature, was assumed to be already proven before TQFTs were even defined.