A new anomaly in chiral CFT

In the late 80-ies, Segal and Kontsevich formulated a mathematical definition of two-dimensional
conformal field theory (CFT). Unfortunately, within the framework of their definition, the main
expected examples of CFTs (associated to loop groups, and known as the "WZW models") could
never be constructed. It has been long predicted by Graeme Segal that a construction of these CFTs
will feature in an prominent way the fact that they are unitary CFTs. The axioms of unitary CFTs
have never been written down, but they were expected to be a straightforward adaptation of the
axioms of CFT. We show that this is not the case: there's a new anomaly which only becomes
visible when there's inner products around.

Let us explain, at a more technical level, the nature of this anomaly. Fix a loop group LG, a level
k∈ℕ, and let Repᵏ(LG) be category of positive energy representations at level k. For any pair of
pants Σ (with complex structure in the interior and parametrized boundary), there is an associated
functor Repᵏ(LG) × Repᵏ(LG) → Repᵏ(LG): (H,K) ↦ H⊠K, called the fusion product. It had been
expected (but never proven) that this operation should be unitary. Namely, that the choice of
LG-invariant inner products on H and on K should induce an LG-invariant inner product on H⊠K.
We show that this expectation was not quite correct, and that the inner product on H⊠K is only
well defined up to a positive scalar.