CFTs that are not modular invariant

Are there any 2d CFTs that do not have modular invariant partition functions? All the examples that I know of, like the free boson, WZW models, etc. have modular invariant partition functions.

• You could take the compact free boson and NOT include the twisted sectors. That's a totally well defined CFT on the sphere but it cannot be defined on the torus. Oct 16, 2021 at 10:49