I will explain how the characters of various rational and non-rational VOAs of type A are obtained from sl_r invariants of torus links. Specifically, we will consider principal W algebras, (1,p) singlet and (1,p) triplet VOAs.

In this talk, I will discuss some of the main results in https://arxiv.org/abs/2208.14605. For a pair of C*-algebras (A,B), representing an (A, B) C*-correspondence on a pair of Hilbert spaces (H_1, H_2) roughly consists in naturally realizing the correspondence as a closed subspace of L(H_1, H_2). This concept is a generalization of R. Exel theory for Hilbert A-B bimodules, originally introduced in 1993. Exel's methods were used as a tool to prove, in its full generality, that any two Morita equivalent C*-algebras have isomorphic K-theory. Extending this theory to C*-correspondences yields necessary and sufficient conditions for an (A,B) C*-correspondence to be a Hilbert A-B bimodule. Another consequence is that, if a right Hilbert A-module X is represented on (H_1, H_2), we then get faithful representations of L_A(X) and K_A(X), the algebras of adjointable and compact-adjointable maps, on the Hilbert space H_1. This will play a crucial role in my talk next week, where I will talk about the objects we get when the Hilbert spaces are replaced by general L^p spaces for p in (1, infinity).