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By M. Aizenman (Chief Editor)

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This is well-defined. Moreover if g ∈ P U (H) then π(yg) = ˆ ˆ ˆ π(y) so that [φ(y), ρ(y)]P U (H) = [φ(yg), ρ(yg)]P U (H) = [φ(y)g, ρ(yg)]P U (H) = −1 ˆ [φ(yg), gρ(yg)]P U (H) , and hence ρ(yg) = g ρ(y) proving equivariance. 1. Notice that if we worked with Fred instead of BU K then it has connected components Fredn consisting of operators of index n. We can then consider sections of Y ×P U (H) Fredn for every n, not just zero. Such a section will pull back a K class and if we take the determinant of this K class we will obtain a line bundle on Y on which the gerbe Ln acts.

2. Product of one- and two-sphere. The case X = S 1 × S 2 is interesting since an explicit realization of the principle P U (H)-bundles over X, for [H ] ∈ H 3 (S 1 ×S 2 , Z) = Z, is known [6]. We can compute K • (S 1 × S 2 , [H ]) using the Mayer-Vietoris sequence as in Sect. 1, but the analogue of the procedure in Sect. 3 is more convenient. Take a point x0 ∈ S 1 . We have 0 −−−−→ C0 (R × S 2 ) −−−−→ C(S 1 × S 2 ) −−−−→ C({x0 } × S 2 ) −−−−→ 0. 12), K 1 (S 2 ) −−−−→ K 0 (S 1 × S 2 , [H ]) −−−−→ K 0 (S 2 )    .

Publ. Math. 34, 129–151 (1968) 36. : Equivariant contractibility of the general linear group of Hilbert space. Bull. London Math. Soc. 1, 329–331 (1969) 37. : The Geometry of Bundle Gerbes. DG/0004117 38. : Fusion of Positive Energy Representations of LSpin2n . PhD Dissertation, University of Cambridge, 1997 39. : K-theory and C ∗ -algebras. Oxford: Oxford University Press, 1993 40. : D-Branes and K-theory. J. High Energy Phys. 12, 019 (1998), hep-th/9810188 41. : Overview of K-theory applied to strings.

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