Genus two partition functions of extremal conformal field theories

Recently Witten conjectured the existence of a family of "extremal" conformal field theories (ECFTs) of central charge c = 24k, which are supposed to be dual to three-dimensional pure quantum gravity in AdS3. Assuming their existence, we determine explicitly the genus two partition functions of k = 2 and k = 3 ECFTs, using modular invariance and the behavior of the partition function in degenerating limits of the Riemann surface. The result passes highly nontrivial tests and in particular provides a piece of evidence for the existence of the k = 3 ECFT. We also argue that the genus two partition function of ECFTs with k ≤ 10 are uniquely fixed (if they exist). © SISSA 2007.