Genus two partition functions and Renyi entropies of large c CFTs

I will explain how to compute the genus two partition function of holographic two dimensional CFTs. I will focus on the Riemann surfaces relevant for computing the third Renyi entropy of two intervals. Unlike the torus partition function that computes the second Renyi entropy, I will show that the result is no longer universal at large central charge. It depends on details of the theory and in particular wether there are light operators in the spectrum. I will show that if the theory contains an operator below a certain bound, the third Renyi entropy has a new phase transition.