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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.

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    $\begingroup$ 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. $\endgroup$
    – Prahar
    Oct 16, 2021 at 10:49

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Of course there are CFTs that exist on the sphere but not on higher genus Riemann surfaces.

To build an example, you can start with a modular invariant CFT, and consider a subset of fields that closes under OPEs. In the Ising model, this could be the identity, or the identity plus the energy operator. In the compactified free boson, you could take fields with zero winding (or even winding, etc).

Conversely, you can break modular invariance by adding extra fields to a modular invariant CFT. In the Ising model, you can add disorder operators.

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