We discuss elliptic genera of supersymmetric two-dimensional conformal field theories with continuous spectra. We argue that they consist of a mock modular term, and a non-holomorphic modular completion. We illustrate the general arguments with a large class of examples, computed using conformal field theory, string theory and localization techniques. We discuss applications, amongst others to black hole physics.

We will argue that the cigar conformal field theory SL(2,R)/U(1) is a prime example of a non-rational conformal field theory and we exhibit some of its characteristic features that might generalize to other non-rational conformal field theories. We discuss how this particular non-rational conformal field theory (with or without boundary) arises in string theory backgrounds with NS5-branes, and how the technical advances reached in the conformal field theory can illuminate aspects of holography in these backgrounds.