Conformal Nets and KK-Theory

Abstract

‎Given a completely rational conformal net $\mathcal{A}$ on $S^1$‎, ‎its fusion‎ ‎ring acts faithfully on the K-group $K_0(\mathfrak{K}_{\mathcal{A}})$ of a certain‎ ‎universal $C^*$-algebra $\mathfrak{K}_{\mathcal{A}}$ associated to $\mathcal{A}$‎, ‎as shown in a‎ ‎previous paper‎. ‎We prove here that this action can actually be‎ ‎identified with a Kasparov product‎, ‎thus paving the way for a‎ ‎fruitful interplay between conformal field theory and KK-theory‎.