A Systematic Approach to K\"ahler Moduli Stabilisation

Achieving full moduli stabilisation in type IIB string compactifications for generic Calabi-Yau threefolds with hundreds of K\"ahler moduli is notoriously hard. This is due not just to the very fast increase of the computational complexity with the number of moduli, but also to the fact that the scalar potential depends in general on the supergravity variables only implicitly. In fact, the supergravity chiral coordinates are 4-cycle volume moduli but the K\"ahler potential is an explicit function of the 2-cycle moduli and inverting between these two variables is in general impossible. In this paper we propose a general method to fix all type IIB K\"ahler moduli in a systematic way by working directly in terms of 2-cycle moduli: on one side we present a `master formula' for the scalar potential which can depend on an arbitrary number of K\"ahler moduli, while on the other we perform a computer-based search for critical points, introducing a hybrid Genetic/Clustering/Amoeba algorithm and other computational techniques. This allows us to reproduce several known minima, but also to discover new examples of both KKLT and LVS models, together with novel classes of LVS minima without diagonal del Pezzo divisors and hybrid vacua which share some features with KKLT and other with LVS solutions.