litterature. The issue can be summarized in the following way. There exist two constructions:

{semisimple categories} ⇆ {sets}

One construction sends a category to its set of isomorphism classes of simple objects. The

other construction takes a set and promotes its elements to be the objects of a category.

Those two constructions are each other's inverses (up to non-canonical isomorphism). But

the corresponding claim becomes false in the presence of involutions. The two constructions

{semisimple categories with involution} ⇆ {sets with involution}

are

letter below for the definition of a category with involution), construct the associated set with

involution, and then turn that into a category with involution, then the category with involution

that you get at the end of the day is

Any definition of modular functor based on the concept of a set with involution is therefore

likely to be problematic. This is the case for example with Andersen and Ueno's definition,

and I have written a little note to explain the adverse consequences of their unfortunate choice

of definition: the WZW modular functor that they construct is not uniquely defined, and the

main theorem of their Inventiones paper is therefore ill-formulated.

That being said, I should also say that Andersen and Ueno's papers are full of great ideas, very

nicely written, and that the mistake which I am pointing out is likely to be fixable.

Letter on “Geometric construction of modular functors from conformal field theory” and

“Construction of the Reshetikhin-Turaev TQFT from conformal field theory” by Jørgen Andersen and Kenji Ueno

Here is the response of Andersen and Ueno:

Response to André Henriques' statements about our papers [AU1, AU2, AU3, AU4],

Andersen has posted a fix to some of the problems which I raised. The left hand side of the

main isomorphism in [AU15, Thm 1.1] (the modular functor associated to a quantum

group modular tensor category) is now well-defined. The fix relies on the fact that a quantum

group modular tensor category admits a canonical "fundamental symplectic character" –

a fact which appears to be specific to those modular tensor categories. The right hand side

of [AU15, Thm 1.1] can also be made well-defined using the same ideas, even though this is

not discussed in the linked paper.

Note that if one adopted a different definition of a modular functor, such as the one in Bakalov-

Kirillov, then these problems would have been avoided.

As explained in my letter, Andersen-Ueno modular functors, Walker modular functors, and

Bakalov-Kirillov modular functors are three

point, their exact relationship is not well understood.