On modular functors

I herby would like to point out a subtlety in some definitions of modular functor used in the
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 not each other's inverses. If you take a semisimple category with involution (see my
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 not equivalent to the original one.

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],

Update (Feb 2016)
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 genuinely distinct mathematical notions. At this
point, their exact relationship is not well understood.