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.

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