By M. Aizenman (Chief Editor)

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**Example text**

The map τ −→ zτ = wz τ wz−1 + τz0 is an affine transformation of AW that permutes the vertices: zτi ≡ τzi . Let bz,z ∈ t be such that −1 wz wz wzz = cz,z = e2πi bz,z . 5) We shall take bz,1 = 0 = b1,z . The Cartan subgroup valued chain c = (cz,z ) is a 2-cocycle on group Z : ,−1 −1 (δc)z,z z = (wz cz ,z wz−1 ) czz ,z cz,z z cz,z = 1 , see Appendix A of [41] for a brief summary on finite group cohomology. The 3-form H descends to a 3-form H on G . For (integer) k for which H has periods in 2πZ there exists a gerbe G = (Y , B , L , µ ) over G with curvature H .

1 Different realizations of spaces Mλλ0 λ are more convenient in different contexts. We shall need still another realization that is derived from the one based on the fusion rule intertwiners of the Lie algebra action. We shall need a more concrete description of the spaces HomFR g (Vλ1 , Vλ0 ⊗ Vλ ). Consider the linear mapping Hom g (Vλ1 , Vλ0 ⊗ Vλ ) ψ −→ |ψ ∈ Vλ such that for all |v ∈ Vλ and the highest weight vectors |λs ∈ Vλs annihilated by the step generators eα for positive roots α of g, v | ψ = λ0 ⊗ v | ψ |λ1 .

When restricted to Cτ , the 3-form H becomes exact. In particular, H |Cτ = dQτ , where Qτ = = k 4π tr (h−1 dh) e2π iτ (h−1 dh) e−2πiτ is a (smooth) 2-form on Cτ . Let G = (Y, B, L, µ) be a gerbe over G with curvature H as described above. Recall that a G-brane D supported by D is a pair (D, E) , where E is a G|D -module. Such module determines, in turn, a vector bundle E with connection over YD = π −1 (D) for π denoting the projection from Y to G. With D = Cτ , the additional restriction, imposed by the conservation of the diagonal current algebra, fixes the curvature of bundle E to be the scalar 2-form F = π ∗ Qτ − B|YD .

### Communications in Mathematical Physics - Volume 258 by M. Aizenman (Chief Editor)

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