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1/r,m+rδ i 1/r,m Recall that we have the natural isomorphism between Mg,n and Mg,n . We do not distinguish them. Let α(t) be the family of the weights given as follows: αj (t) = 0 (j = i), αi (t) = t. 32 T. Mochizuki ∨i i Let E (m + r · δ i , α(t)) and E (m + r · δ i , α(t)) denote the vector bundles for the tuple m + r · δ i with the weight α(t). For each t, we obtain the cohomology classes c(t) ∈ H ∗ M by the construction in Subsubsect. 4. In the case t = 0, the coho1/r,m+r·δ i mology class is the virtual class cg,n .

The twisted multiplication of kernels was used in [9] to study the hyperbolic quantum Hall effect, but in that two-dimensional situation the multiplication is automatically associative. Once one gets to three dimensions the analogous algebra is nonassociative. 1. 6) and θx θy = ad(σ (x, y))θxy , where ad(σ (x, y))[K](z, w) = φ(x, y, z)φ(x, y, w)−1 K(z, w) comes from the multiplier σ (x, y)(v) = φ(x, y, v). Nonassociative Tori and Applications to T-Duality 49 Proof. Using the tricharacter property of φ, we see that (θx [K1 ]∗θx [K2 ])(z, w) = φ(z, v, w)φ(x, z, v)φ(x, v, w)K1 (zx, vx)K2 (vx, wx)dv G = φ(zx, vx, wx)φ(x, z, w)K1 (zx, vx)K2 (vx, wx) dv G = φ(x, z, w)(K1 ∗ K2 )(zx, wx), = θx [K1 ∗ K2 ](z, w).

Such (Leptin-)Busby-Smith twistings have long been known but, for non-trivial H0 , v(x, y) is not a cocycle and that means that associativity fails in the twisted crossed product. In Sect. 2 we give the relationship between the differential forms and the multicharacters on G which will be used in our later constructions. Section 3 reviews the generalised Busby-Smith twisted crossed products. An example of a twisting is given in Sect. 4, together with a proof that it is the only type up to stability.

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Communications in Mathematical Physics - Volume 264 by M. Aizenman (Chief Editor)

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