15:45
Fix a loop group LG, a level k∈ℕ, and let Repᵏ(LG) be corresponding category of positive energy representations.
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 widely 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 is not the case: there is an anomaly.
More precisely, there is an ℝ₊-torsor canonically associated to Σ. It is only after trivialising of this ℝ₊-torsor that the fusion product acquires an LG-invariant inner product. (The same statement applies when Σ is an arbitrary Riemann surface with boundary.)
Joint work with James Tener.