Modular invariance and holographic CFTs

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Modular invariance plays an important role in AdS3/CFT2 holography. I discuss the structure of non-holomorphic CFT partition functions, namely in what sense the light spectrum determines the heavy spectrum and how to construct example partition functions using Poincare series. This yields necessary conditions on the spectrum of holographic CFTs. Finally I will discuss permutation orbifolds as examples of such theories.