AdS₃ Partition Functions Reconstructed

For pure gravity in AdS_3, Witten has given a recipe for the construction of holomorphically factorizable partition functions of pure gravity theories with central charge c=24k. The partition function was found to be a polynomial in the modular invariant j-function. We show that the partition function can be obtained instead as a modular sum which has a more physical interpretation as a sum over geometries. We express both the j-function and its derivative in terms of such a sum.