A \(K\)-theoretic note on geometric quantization (Q1819131)

The possibility that one could find a purely topological proof that quantization commutes with reduction was conjectured by Guillemin, based on earlier observations by Bott on the index of Dirac operators. From the abstract: ``We give a purely K-theoretic proof of a case of the ``quantization commutes with reduction'' result, conjectured by Guillemin and Sternberg and proven by Meinrenken and Vergne. We show that the quantization is simply a pushforward in K-theory, and use Lerman's symplectic cutting and the localization theorem in equivariant K-theory to prove that quantization commutes with reduction. The case where \(G= S^1\) and the action is free on the zero level of the moment map is addressed''.